# Triangular Numbers Multiple of Triangular Numbers and Solutions of Pell
Equations

Vladimir Pletser  European Space Agency (ret<EMAIL_ADDRESS>
###### Abstract.

For all positive non-square integer multiplier $k$, there is an infinity of
multiples of triangular numbers which are also triangular numbers. With a
simple change of variables, these triangular numbers can be found using
solutions of Pell equations. With some conditions on parities of fundamental
solutions of the simple and generalized Pell equations, only odd solutions of
the generalized Pell equation are retained to provide many infinitely
solutions found on branches corresponding to each of the generalized
fundamental solutions. General algebraic expressions of fundamental solutions
of the Pell equations are found for some values of the multiplier $k$ in
function of the closest natural square. Further, among the expressions of Pell
equation solutions, a set of recurrent relations is identical to those found
previously without the Pell equation solving method. It is found also that two
constants of the problem of multiples of triangular numbers are directly
related to the fundamental solutions of the simple Pell equation, which is an
unexpected result as it means that simple Pell equation fundamental solutions
in all generality, are related to constants in recurrent relations of the
problem of finding triangular numbers multiple of other triangular numbers.

###### Key words and phrases:

Triangular Numbers, Multiple of Triangular Numbers, Recurrent Relations, Pell
Equations, Fundamental Solutions

##  1\. Introduction

Triangular numbers $T_{t}=\frac{t\left(t+1\right)}{2}$ are figurate numbers
with several interesting properties and formula (see, e.g., [1, 2]). In this
paper, we investigate triangular numbers $T_{\xi}$ that are multiples of other
triangular numbers $T_{t}$

$T_{\xi}=kT_{t}$ (1)

Several authors have investigated this Diophantine equation; see, e.g., [3, 4,
5, 6, 7, 8, 9]. Further historical accounts can be found in [6]. Recently,
Pletser showed [10] that, for non-square integer values of $k$, the four
variables $t,\xi,T_{t}$ and $T_{\xi}$ can be represented by recurrent
relations involving a rank $r$ and parameters $\kappa$ and $\gamma$ which are
respectively the sum and the product of the $\left(r-1\right)^{\text{th}}$ and
the $r^{\text{th}}$ values of $t$. The rank is being defined as the number of
successive values of $t$ solutions of (1) such that their successive ratios
are slowly decreasing without jumps.

We only consider solutions of (1) for $k>1$ as, for $k=0$ and $k=1$, solutions
are trivial, respectively, $\xi=0$ and $\xi=t$ for any positive integer $t$.

In this paper, we investigate how to find all solutions to (1) using the
method of resolution of the simple and generalized Pell equations associated
to (1). We show that the rank $r$ and parameters $\kappa$ and $\gamma$ of
recurrent relations can be deduced from fundamental solutions of Pell
equations. Section 2 introduces the rank $r$ and recurrent relations. Section
3 give a short reminder on how to find solutions of Pell equations. In Section
4, Pell equation methods are applied to find all multiples of triangular
numbers that are triangular numbers. In certain cases, general expressions of
fundamental solutions of the Pell equations associated to (1) are given for
values of the multiplier $k$ in function of the closest natural square values
$s^{2}$.

##  2\. Rank and recurrent relations

The Online Encyclopedia of Integer Sequences (OEIS) [11] lists sequences of
solutions of (1) for $k=2,3,5,6,7,8$. Let us note first that, among all
solutions, $\left(t_{0},\xi_{0}\right)=\left(0,0\right)$ is always a first
solution of (1) for all non-square integer value of $k$.

Let’s consider the two cases of $k=3$ and $k=6$ yielding the successive
solution pairs as shown in Table 1. We indicate also the ratios
$t_{n}/t_{n-1}$ for both cases and $t_{n}/t_{n-2}$ for $k=6$. It is seen that
for $k=3$, the ratio $t_{n}/t_{n-1}$ varies between close values, from 5 down
to 3.737, while for $k=6$, the ratio $t_{n}/t_{n-1}$ alternates between values
3 … 2.385 and 4.667 … 4.206, while the ratio $t_{n}/t_{n-2}$ decreases more
regularly from 14 to 10.029 (corresponding approximately to the product of the
alternating values of the ratio $t_{n}/t_{n-1}$). We call rank $r$ the integer
value such that $t_{n}/t_{n-r}$ is approximately constant or, better,
decreases regularly without jumps (a more precise definition is given
further). So, here, the case $k=3$ has rank $r=1$ and the case $k=6$ has rank
$r=2$.

Table 1. Solutions of (1) for $k=3,6$ |  | $k=3$ |  |  |  | $k=6$ |
---|---|---|---|---|---|---|---
$n$ | $t_{n}$ | $\xi_{n}$ | $t_{n}/t_{n-1}$ | $t_{n}$ | $\xi_{n}$ | $t_{n}/t_{n-1}$ | $t_{n}/t_{n-2}$
0 | 0 | 0 |  | 0 | 0 |  |
1 | 1 | 2 | – | 1 | 3 | – | –
2 | 5 | 9 | 5 | 3 | 8 | 3 | –
3 | 20 | 35 | 4 | 14 | 35 | 4.66667 | 14
4 | 76 | 132 | 3.8 | 34 | 84 | 2.42857 | 11.33333
5 | 285 | 494 | 3.75 | 143 | 351 | 4.20588 | 10.21429
6 | 1065 | 1845 | 3.73684 | 341 | 836 | 2.38461 | 10.02941

Pletser showed [10] that the rank $r$ is the index of $t_{r}$ and $\xi_{r}$
solutions of (1) such that

$\kappa=t_{r}+t_{r-1}=\xi_{r}-\xi_{r-1}-1$ (2)

The rank $r$ is also such that the ratio $t_{2r}/t_{r}$, corrected by the
ratio $t_{r-1}/t_{r}$, is equal to a constant $2\kappa+3$

$\frac{t_{2r}-t_{r-1}}{t_{r}}=2\kappa+3$ (3)

For example, for $k=6$ and $r=2$, $\kappa=t_{2}+t_{1}=3+1=4$, and
$\kappa=\xi_{2}-\xi_{1}-1=8-3-1=4$, yielding $2\kappa+3=11$.

Pletser found [10] four recurrent equations for $t_{n},\xi_{n},T_{t_{n}}$ and
$T_{\xi_{n}}$for each non-square integer value of $k$

$\displaystyle t_{n}$
$\displaystyle=2\left(\kappa+1\right)t_{n-r}-t_{n-2r}+\kappa$ (4)
$\displaystyle\xi_{n}$
$\displaystyle=2\left(\kappa+1\right)\xi_{n-r}-\xi_{n-2r}+\kappa$ (5)
$\displaystyle T_{t_{n}}$
$\displaystyle=\left(4\left(\kappa+1\right)^{2}-2\right)T_{t_{n-r}}-T_{t_{n-2r}}+\left(T_{\kappa}-\gamma\right)$
(6) $\displaystyle T_{\xi_{n}}$
$\displaystyle=\left(4\left(\kappa+1\right)^{2}-2\right)T_{\xi_{n-r}}-T_{\xi_{n-2r}}+k\left(T_{\kappa}-\gamma\right)$
(7)

where coefficients are functions of two constants $\kappa$ and $\gamma$,
respectively the sum (2) and the product $\gamma=t_{r-1}t_{r}$. Note that
these four relations are independent from the value of $k$.

##  3\. Pell equations: A Reminder

The Diophantine bivariate quadratic equation

$X^{2}-DY^{2}=N,$ (8)

with integers $X,Y,D,N$ and square free $D$, is called the Pell equation.
Several mathematicians have investigated this equation (see historical
accounts in [12, 13, 14, 15, 16]), Treatments and solutions are described in
several classical text books (see e.g. [17, 19, 18, 20] and references
therein). We remind here some general formulas and how to calculate solutions.
Details can be found in references.

For $N=1$, (8) is called the simple Pell equation

$x^{2}-Dy^{2}=1$ (9)

This equation admits the obvious trivial solution
$\left(x_{0},y_{0}\right)=\left(1,0\right)$ and infinitely many solutions
given by

$\displaystyle\left(x_{n},y_{n}\right)=$
$\displaystyle\left(\frac{\left(x_{f}+\sqrt{D}y_{f}\right)^{n}+\left(x_{f}-\sqrt{D}y_{f}\right)^{n}}{2},\right.$
$\displaystyle\left.\frac{\left(x_{f}+\sqrt{D}y_{f}\right)^{n}-\left(x_{f}-\sqrt{D}y_{f}\right)^{n}}{2\sqrt{D}}\right)$
(10)

where $n$ are positive integers and $\left(x_{f},y_{f}\right)$ is the least
solution to (9), i.e. the smallest integer solution different from the trivial
solution, $x_{f}>1,y_{f}>0$. We call this least solution the fundamental
solution. Obviously, having found the fundamental solution
$\left(x_{f},y_{f}\right)$ yields directly three other solutions,
$\left(-x_{f},y_{f}\right),\left(x_{f},-y_{f}\right),\left(-x_{f},-y_{f}\right)$.

Lagrange devised a method to find the fundamental solution, based on the
continued fraction expansion of the quadratic irrational $\sqrt{D}$, that can
be summarized as follows. One computes the $j^{\textnormal{th}}$ convergent
$\left(p_{j}/q_{j}\right)$ of the continued fraction
$\left[\alpha_{0};\alpha_{1},\ldots,\alpha_{j},\alpha_{j+1},\ldots\right]$ of
$\sqrt{D}$, with $\alpha_{0}=\left\lfloor\sqrt{D}\right\rfloor$, i.e., the
greatest integer $\leq\sqrt{D}$. This continued fraction becomes periodic
after the following term, $\alpha_{j+1}=2\alpha_{0}$ if $\sqrt{D}$ is a
quadratic irrational. The recurrence relations

$p_{i}=\alpha_{i}p_{i-1}+p_{i-2}\,\,,\,\,\,q_{i}=\alpha_{i}q_{i-1}+q_{i-2}$

yield the terms $p_{i}$ and $q_{i}$ of the convergent, with
$p_{-2}=0,p_{-1}=1,q_{-2}=1,q_{-1}=0$. The fundamental solution is then
$\left(x_{f},y_{f}\right)=\left(p_{j},q_{j}\right)$ if $j$ is odd, or
$\left(x_{f},y_{f}\right)=\left(p_{2j+1},q_{2j+1}\right)$ if $j$ is even.

For $N\neq 1$, (8) is called the generalized Pell equation, which can have
either no solution, or one, or several fundamental solutions
$\left(X_{f_{i}},Y_{f_{i}}\right)$, with positive integers $i$ such that
$1\leq i\leq\rho$, where $\rho$ is the total number of fundamental solutions
admitted by (8). All integer solutions, if they exist, are found on double
infinite branches that can be expressed in terms of the fundamental
solution(s) $\left(X_{f_{i}},Y_{f_{i}}\right)$ and
$\left(-X_{f_{i}},Y_{f_{i}}\right)$. Methods to calculate the fundamental
solution(s) of the generalized Pell equation (see e.g. [17, 19, 20, 21, 22,
23, 24, 25] and references therein) are all based on Lagrange’s method of
continued fractions, sometime adapted (see e.g. [26]). The nearest integer
continued fraction method and the Lagrange-Mollin-Matthews method [25] are
used further to calculate the fundamental solutions of respectively, the
simple and the generalized Pell equations.

Once fundamental solutions are known, the other solutions
$\left(X_{n},Y_{n}\right)$ of (8) are calculated by

$X_{n}+\sqrt{D}Y_{n}=\pm\left(X_{f_{i}}+\sqrt{D}Y_{f_{i}}\right)\left(x_{f}+\sqrt{D}y_{f}\right)^{n}$
(11)

for a proper choice of sign $\pm$ [20], yielding respectively, for $n=0,1,2$
(assuming a $+$ sign),

$\displaystyle\left(X_{0},Y_{0}\right)=$
$\displaystyle\left(X_{f_{i}},Y_{f_{i}}\right)$ (12)
$\displaystyle\left(X_{1},Y_{1}\right)=$
$\displaystyle\left(X_{f_{i}}x_{f}+DY_{f_{i}}y_{f},X_{f_{i}}y_{f}+Y_{f_{i}}x_{f}\right)$
(13) $\displaystyle\left(X_{2},Y_{2}\right)=$
$\displaystyle\left(X_{f_{i}}\left(x_{f}^{2}+Dy_{f}^{2}\right)+2DY_{f_{i}}x_{f}y_{f},\right.$
$\displaystyle\left.Y_{f_{i}}\left(x_{f}^{2}+Dy_{f}^{2}\right)+2X_{f_{i}}x_{f}y_{f}\right)$
(14)

Note that, for each value of $n$, one can have several (up to $\rho$)
solutions depending on the different values of the generalized fundamental
solutions $\left(X_{f_{i}},Y_{f_{i}}\right)$.

The other solutions $\left(X_{n},Y_{n}\right)$ of (8) can also be represented
by recurrence relations

$\left(X_{n},Y_{n}\right)=\left(x_{f}X_{n-1}+Dy_{f}Y_{n-1},x_{f}Y_{n-1}+y_{f}X_{n-1}\right)$
(15)

that can also be written as

$\left(X_{n},Y_{n}\right)=\left(2x_{f}X_{n-1}-X_{n-2},2x_{f}Y_{n-1}-Y_{n-2}\right)$
(16)

or by Chebyshev polynomials of the first kind $T_{n-1}\left(x_{f}\right)$ and
of the second kind $U_{n-2}\left(x_{f}\right)$, evaluated at $x_{f}$ (see
[27]),

$\displaystyle\left(X_{n},Y_{n}\right)=$
$\displaystyle\left(X_{f_{i}}T_{n-1}\left(x_{f}\right)+DY_{f_{i}}y_{f}U_{n-2}\left(x_{f}\right),\right.$
$\displaystyle\left.X_{f_{i}}y_{f}U_{n-2}\left(x_{f}\right)+Y_{f_{i}}T_{n-1}\left(x_{f}\right)\right)$
(17)

One notices that the second recurrent relations (16) is similar in form to
recurrent relations (4) and (5) found in [10].

##  4\. Pell Equations and Multiples of Triangular Numbers

###  4.1. Solutions of Pell equations

For non-square integers $k$ and with the change of variables

$\left(X,Y\right)=\left(2\xi+1,2t+1\right)$ (18)

(1) becomes a generalized Pell equation [7, 8], with $D=k$ and $N=1-k$
negative, as $k>1$,

$X^{2}-kY^{2}=1-k$ (19)

and the associated simple Pell equation reads

$x^{2}-ky^{2}=1$ (20)

Odd solutions $\left(X,Y\right)$ of (19) provide then pairs
$\left(\xi,t\right)$, solutions of (1). Following the procedure of Section 2,
the fundamental solutions of the simple and generalized Pell equations are
calculated and shown in Tables 2 to (4) for non-square $k$ between 2 and 102.
The second and third columns give the rank $r$ found in [10] and the total
number $\rho$ of fundamental solutions of the generalized Pell equation.The
fourth column shows the single fundamental solution of the simple Pell
equations; the fifth and sixth columns give the fundamental solutions of the
generalized Pell equations, the fifth column for those solutions with both
$X_{f_{i}}$ and $Y_{f_{i}}$ odd or having different parities, while the sixth
column give those solutions with both $X_{f_{i}}$ and $Y_{f_{i}}$ even (except
for $k=56$, see discussion further).

Table 2. Fundamental solutions of simple (19) and generalized (20) Pell equations $k$ | $r$ | $\rho$ | $\left(x_{f},y_{f}\right)$ | $\left(X_{f_{i}},Y_{f_{i}}\right)$ |
---|---|---|---|---|---
2 | 1 | 1 | $\left(3,2\right)$ | $\left(1,1\right)$ |
3 | 1 | 1 | $\left(2,1\right)$ | $\left(1,1\right)$ |
5 | 2 | 3 | $\left(9,4\right)$ | $\left(\pm 1,1\right)$ | $\left(4,2\right)$
6 | 2 | 2 | $\left(5,2\right)$ | $\left(\pm 1,1\right)$ |
7 | 2 | 2 | $\left(8,3\right)$ | $\left(\pm 1,1\right)$ |
8 | 2 | 2 | $\left(3,1\right)$ | $\left(\pm 1,1\right)$ |
10 | 3 | 3 | $\left(19,6\right)$ | $\left(\pm 1,1\right),\left(9,3\right)$ |
11 | 2 | 2 | $\left(10,3\right)$ | $\left(\pm 1,1\right)$ |
12 | 2 | 2 | $\left(7,2\right)$ | $\left(\pm 1,1\right)$ |
13 | 4 | 6 | $\left(649,180\right)$ | $\left(\pm 1,1\right),\left(\pm 25,7\right)$ | $\left(\pm 14,4\right)$
14 | 2 | 2 | $\left(15,4\right)$ | $\left(\pm 1,1\right)$ |
15 | 2 | 2 | $\left(4,1\right)$ | $\left(\pm 1,1\right)$ |
17 | 2 | 3 | $\left(33,8\right)$ | $\left(\pm 1,1\right)$ | $\left(16,4\right)$
18 | 2 | 2 | $\left(17,4\right)$ | $\left(\pm 1,1\right)$ |
19 | 3 | 3 | $\left(170,39\right)$ | $\left(\pm 1,1\right),\left(39,9\right)$ |
20 | 2 | 2 | $\left(9,2\right)$ | $\left(\pm 1,1\right)$ |
21 | 4 | 6 | $\left(55,12\right)$ | $\left(\pm 1,1\right),\left(\pm 13,3\right)$ | $\left(\pm 8,2\right)$
22 | 4 | 4 | $\left(197,42\right)$ | $\left(\pm 1,1\right),\left(\pm 23,5\right)$ |
23 | 2 | 2 | $\left(24,5\right)$ | $\left(\pm 1,1\right)$ |
24 | 2 | 2 | $\left(5,1\right)$ | $\left(\pm 1,1\right)$ |
26 | 3 | 3 | $\left(51,10\right)$ | $\left(\pm 1,1\right),\left(25,5\right)$ |
27 | 2 | 2 | $\left(26,5\right)$ | $\left(\pm 1,1\right)$ |
28 | 4 | 4 | $\left(127,24\right)$ | $\left(\pm 1,1\right),\left(\pm 15,3\right)$ |
29 | 4 | 6 | $\left(9801,1820\right)$ | $\left(\pm 1,1\right),\left(\pm 59,11\right)$ | $\left(\pm 86,16\right)$
30 | 2 | 2 | $\left(11,2\right)$ | $\left(\pm 1,1\right)$ |
31 | 4 | 4 | $\left(1520,273\right)$ | $\left(\pm 1,1\right),\left(\pm 61,11\right)$ |
32 | 2 | 2 | $\left(17,3\right)$ | $\left(\pm 1,1\right)$ |
33 | 2 | 4 | $\left(23,4\right)$ | $\left(\pm 1,1\right)$ | $\left(\pm 10,2\right)$
34 | 2 | 2 | $\left(35,6\right)$ | $\left(\pm 1,1\right)$ |
35 | 2 | 2 | $\left(6,1\right)$ | $\left(\pm 1,1\right)$ |
37 | 2 | 3 | $\left(73,12\right)$ | $\left(\pm 1,1\right)$ | $\left(36,6\right)$
38 | 2 | 2 | $\left(37,6\right)$ | $\left(\pm 1,1\right)$ |
39 | 2 | 2 | $\left(25,4\right)$ | $\left(\pm 1,1\right)$ |
40 | 4 | 4 | $\left(19,3\right)$ | $\left(\pm 1,1\right),\left(\pm 11,2\right)$ |
41 | 4 | 4 | $\left(2049,320\right)$ | $\left(\pm 1,1\right),\left(\pm 83,13\right)$ |
42 | 2 | 2 | $\left(13,2\right)$ | $\left(\pm 1,1\right)$ |
43 | 4 | 4 | $\left(3482,531\right)$ | $\left(\pm 1,1\right),\left(\pm 85,13\right)$ |
44 | 2 | 2 | $\left(199,30\right)$ | $\left(\pm 1,1\right)$ |
45 | 4 | 6 | $\left(161,24\right)$ | $\left(\pm 1,1\right),\left(\pm 19,3\right)$ | $\left(\pm 26,4\right)$
Table 3. Fundamental solutions of simple (19) and generalized (20) Pell equations $k$ | $r$ | $\rho$ | $\left(x_{f},y_{f}\right)$ | $\left(X_{f_{i}},Y_{f_{i}}\right)$ |
---|---|---|---|---|---
46 | 6 | 6 | $\left(24335,3588\right)$ | $\left(\pm 1,1\right),\left(\pm 47,7\right),\left(\pm 183,27\right)$ |
47 | 2 | 2 | $\left(48,7\right)$ | $\left(\pm 1,1\right)$ |
48 | 2 | 2 | $\left(7,1\right)$ | $\left(\pm 1,1\right)$ |
50 | 3 | 3 | $\left(99,14\right)$ | $\left(\pm 1,1\right),\left(49,7\right)$ |
51 | 3 | 3 | $\left(50,7\right)$ | $\left(\pm 1,1\right),\left(35,5\right)$ |
52 | 4 | 4 | $\left(649,90\right)$ | $\left(\pm 1,1\right),\left(\pm 79,11\right)$ |
53 | 4 | 6 | $\left(66249,9100\right)$ | $\left(\pm 1,1\right),\left(\pm 211,29\right)$ | $\left(\pm 160,22\right)$
54 | 2 | 2 | $\left(485,66\right)$ | $\left(\pm 1,1\right)$ |
55 | 4 | 4 | $\left(89,12\right)$ | $\left(\pm 1,1\right),\left(\pm 21,3\right)$ |
56 | 2 | 4 | $\left(15,2\right)$ | $\left(\pm 1,1\right)$ | $\left(\pm 13,2\right)$
57 | 4 | 4 | $\left(151,20\right)$ | $\left(\pm 1,1\right),\left(\pm 37,5\right)$ |
58 | 4 | 4 | $\left(19603,2574\right)$ | $\left(\pm 1,1\right),\left(\pm 175,23\right)$ |
59 | 2 | 2 | $\left(530,69\right)$ | $\left(\pm 1,1\right)$ |
60 | 2 | 2 | $\left(31,4\right)$ | $\left(\pm 1,1\right)$ |
61 | 8 | 12 | $\left(1766319049,\right.$ | $\left(\pm 1,1\right),\left(\pm 367,47\right),$ | $\left(\pm 62,8\right),$
|  |  | $\left.226153980\right)$ | $\left(\pm 6709,859\right),\left(\pm 94793,12137\right)$ | $\left(\pm 5186,664\right)$
62 | 2 | 2 | $\left(63,8\right)$ | $\left(\pm 1,1\right)$ |
63 | 2 | 2 | $\left(8,1\right)$ | $\left(\pm 1,1\right)$ |
65 | 2 | 5 | $\left(129,16\right)$ | $\left(\pm 1,1\right)$ | $\left(\pm 14,2\right),\left(64,8\right)$
66 | 4 | 4 | $\left(65,8\right)$ | $\left(\pm 1,1\right),\left(\pm 23,3\right)$ |
67 | 4 | 4 | $\left(48842,5967\right)$ | $\left(\pm 1,1\right),\left(\pm 401,49\right)$ |
68 | 2 | 2 | $\left(33,4\right)$ | $\left(\pm 1,1\right)$ |
69 | 4 | 6 | $\left(7775,936\right)$ | $\left(\pm 1,1\right),\left(\pm 91,11\right)$ | $\left(\pm 116,14\right)$
70 | 4 | 4 | $\left(251,30\right)$ | $\left(\pm 1,1\right),\left(\pm 41,5\right)$ |
71 | 4 | 4 | $\left(3480,413\right)$ | $\left(\pm 1,1\right),\left(\pm 143,17\right)$ |
72 | 2 | 2 | $\left(17,2\right)$ | $\left(\pm 1,1\right)$ |
73 | 6 | 6 | $\left(2281249,267000\right)$ | $\left(\pm 1,1\right),\left(\pm 145,17\right),\left(\pm 1461,171\right)$ |
74 | 2 | 2 | $\left(3699,430\right)$ | $\left(\pm 1,1\right)$ |
75 | 2 | 2 | $\left(26,3\right)$ | $\left(\pm 1,1\right)$ |
76 | 6 | 6 | $\left(57799,6630\right)$ | $\left(\pm 1,1\right),\left(\pm 113,13\right),\left(\pm 305,35\right)$ |
77 | 4 | 6 | $\left(351,40\right)$ | $\left(\pm 1,1\right),\left(\pm 43,5\right)$ | $\left(\pm 34,4\right)$
78 | 4 | 4 | $\left(53,6\right)$ | $\left(\pm 1,1\right),\left(\pm 25,3\right)$ |
79 | 2 | 2 | $\left(80,9\right)$ | $\left(\pm 1,1\right)$ |
80 | 2 | 2 | $\left(9,1\right)$ | $\left(\pm 1,1\right)$ |
82 | 3 | 3 | $\left(163,18\right)$ | $\left(\pm 1,1\right),\left(81,9\right)$ |
83 | 2 | 2 | $\left(82,9\right)$ | $\left(\pm 1,1\right)$ |
84 | 2 | 2 | $\left(55,6\right)$ | $\left(\pm 1,1\right)$ |
85 | 8 | 12 | $\left(285769,\right.$ | $\left(\pm 1,1\right),\left(\pm 101,11\right),$ | $\left(\pm 16,2\right),$
|  |  | $\left.30996\right)$ | $\left(\pm 341,37\right),\left(\pm 1429,155\right)$ | $\left(\pm 424,46\right)$
Table 4. Fundamental solutions of simple (19) and generalized (20) Pell equations $k$ | $r$ | $\rho$ | $\left(x_{f},y_{f}\right)$ | $\left(X_{f_{i}},Y_{f_{i}}\right)$ |
---|---|---|---|---|---
86 | 4 | 4 | $\left(10405,1122\right)$ | $\left(\pm 1,1\right),\left(\pm 343,37\right)$ |
87 | 2 | 2 | $\left(28,3\right)$ | $\left(\pm 1,1\right)$ |
88 | 4 | 4 | $\left(197,21\right)$ | $\left(\pm 1,1\right),\left(\pm 65,7\right)$ |
89 | 4 | 4 | $\left(500001,53000\right)$ | $\left(\pm 1,1\right),\left(\pm 179,19\right)$ |
90 | 2 | 2 | $\left(19,2\right)$ | $\left(\pm 1,1\right)$ |
91 | 6 | 6 | $\left(1574,165\right)$ | $\left(\pm 1,1\right),\left(\pm 27,3\right),\left(\pm 181,19\right)$ |
92 | 4 | 4 | $\left(1151,120\right)$ | $\left(\pm 1,1\right),\left(\pm 47,5\right)$ |
93 | 4 | 6 | $\left(12151,1260\right)$ | $\left(\pm 1,1\right),\left(\pm 125,13\right)$ | $\left(\pm 154,16\right)$
94 | 4 | 4 | $\left(2143295,\right.$ | $\left(\pm 1,1\right),\left(\pm 281,29\right)$ |
|  |  | $\left.221064\right)$ |  |
95 | 2 | 2 | $\left(39,4\right)$ | $\left(\pm 1,1\right)$ |
96 | 4 | 4 | $\left(49,5\right)$ | $\left(\pm 1,1\right),\left(\pm 17,2\right)$ |
97 | 4 | 8 | $\left(62809633,\right.$ | $\left(\pm 1,1\right),\left(\pm 581,59\right)$ | $\left(\pm 98,10\right),$
|  |  | $\left.6377352\right)$ |  | $\left(\pm 12902,1310\right)$
98 | 2 | 2 | $\left(99,10\right)$ | $\left(\pm 1,1\right)$ |
99 | 2 | 2 | $\left(10,1\right)$ | $\left(\pm 1,1\right)$ |
101 | 2 | 3 | $\left(201,20\right)$ | $\left(\pm 1,1\right)$ | $\left(100,10\right)$
102 | 2 | 2 | $\left(101,10\right)$ | $\left(\pm 1,1\right)$ |

From these Tables, we deduce the following.

First, the rank of solutions of (1) is equal to, or less than, the total
number of fundamental solutions of the generalized Pell equations,
$r\leq\rho$, as was expected.

Second, for all the single fundamental solutions $\left(x_{f},y_{f}\right)$ of
the simple Pell equation, both $x_{f}$ and $y_{f}$ are of different parities,
i.e., one is odd, the other even (except for some cases of $k\equiv
0\left(\text{mod}8\right)$, where both $x_{f}$ and $y_{f}$ are odd; see
further). It is easy to see why: for (20) to hold, the following three
conditions must hold:

(C1) $x_{f}$ and $y_{f}$ can not be simultaneously even, whatever the value of
$k$ is;

(C2) if $k$ is even, $x_{f}$ must necessarily be odd and $y_{f}$ can be either
even or odd;

(C3) if $k$ is odd, $x_{f}$ and $y_{f}$ must have different parities, one odd
and the other even.

Third, the sets of fundamental solutions of the generalized Pell equation
always include the two fundamental solutions
$\left(X_{f_{1}},Y_{f_{1}}\right)=\left(1,1\right)$ and
$\left(X_{f_{2}},Y_{f_{2}}\right)=\left(-1,1\right)$, which is quite obvious
from (19). The only two exceptions are for the cases $k=2$ and $3$. Although
$\left(-1,1\right)$ is also a solution to (19) for these two cases, it does
not bring a new branch of solutions calculated by (11) to (13) different from
the one obtained with $\left(1,1\right)$. Therefore, there is only one
fundamental solution, i.e., $\rho=1$ for these two cases. Furthermore, the two
pairs $\left(1,-1\right)$ and $\left(-1,-1\right)$ are also solutions of (19),
but they do not yield new branches of solutions different from those obtained
with $\left(-1,1\right)$ and $\left(1,1\right)$.

Fourth, all generalized fundamental solutions
$\left(X_{f_{i}},Y_{f_{i}}\right)$ with $i>2$, i.e., other than $\left(\pm
1,1\right)$, have both $X_{f_{i}}$ and $Y_{f_{i}}$ odd, except for
$k=40,96,208,\ldots$ where $Y_{f_{i}}$ is even.

Fifth, the generalized fundamental solutions with both $X_{f_{i}}$ and
$Y_{f_{i}}$ even are shown separately as they do not bring any solutions to
(1), and there are $\rho-r$ such solutions.

With the two generalized fundamental solutions
$\left(X_{f_{1,2}},Y_{f_{1,2}}\right)=\left(\pm 1,1\right)$, one has from (12)
$\left(X_{0_{1,2}},Y_{0_{1,2}}\right)=\left(\pm 1,1\right)$ and it yields the
two trivial solutions $\left(\xi_{0_{1,2}},t_{0_{1,2}}\right)=\left(\frac{\pm
1-1}{2},\frac{1-1}{2}\right)=\left(0,0\right)$ and $\left(-1,0\right)$ of (1).
The next generalized solution (13) reads

$\left(X_{1_{1,2}},Y_{1_{1,2}}\right)=\left(\left(\pm
x_{f}+ky_{f}\right),\left(\pm y_{f}+x_{f}\right)\right)$

yielding, from (18),

$\left(\xi_{1,2},t_{1,2}\right)=\left(\frac{\pm x_{f}+ky_{f}-1}{2},\frac{\pm
y_{f}+x_{f}-1}{2}\right)$ (21)

with both terms integers under the three conditions C1 to C3 above.

For other generalized fundamental solutions $\left(X_{f_{i}},Y_{f_{i}}\right)$
(with $i>2$) different from $\left(\pm 1,1\right)$, one has from (12)
$\left(X_{0_{i}},Y_{0_{i}}\right)=\left(X_{f_{i}},Y_{f_{i}}\right)$, yielding

$\left(\xi_{0_{1,2}},t_{0_{1,2}}\right)=\left(\frac{X_{f_{i}}-1}{2},\frac{Y_{f_{i}}-1}{2}\right)$
(22)

integer solutions of (1) if $X_{f_{i}}$ and $Y_{f_{i}}$ are both odd. The next
generalized solution (13) reads
$\left(X_{1_{i}},Y_{1_{i}}\right)=\left(X_{f_{i}}x_{f}+kY_{f_{i}}y_{f},X_{f_{i}}y_{f}+Y_{f_{i}}x_{f}\right)$,
yielding

$\left(\xi_{1_{i}},t_{1_{i}}\right)=\left(\frac{X_{f_{i}}x_{f}+kY_{f_{i}}y_{f}-1}{2},\frac{X_{f_{i}}y_{f}+Y_{f_{i}}x_{f}-1}{2}\right)$
(23)

One sees clearly that $X_{f_{i}}$ and $Y_{f_{i}}$ can not be simultaneously
even for $\xi_{1_{i}}$ and $t_{1_{i}}$ to be integers. For $X_{f_{i}}$ and
$Y_{f_{i}}$ both odd, the three conditions C1 to C3 above ensure that
$\xi_{1_{i}}$ and $t_{1_{i}}$ are integers.

For the cases of $X_{f_{i}}$ odd and $Y_{f_{i}}$ even, like for $k=40$ and
$96$ in Tables 2 to 4, one has that $x_{f}$ and $y_{f}$ must be simultaneously
odd and, by condition C2 above, $k$ must be even for (23) to provide integer
solutions.

Finally, for all single fundamental solutions $\left(x_{f},y_{f}\right)$ of
the simple Pell equation with both $x_{f}$ and $y_{f}$ odd,they appear for
most of the values of $k$ such that $k\equiv 0\left(\text{mod}8\right)$. The
exceptions to this are for $k=56$, 72, 112, 184, 240, 248, 264, 272, 376, …,
i.e., for some values of $k$ such that $k\equiv\pm 8,\pm
16\left(\text{mod}64\right)$ (but not all), where $y_{f}$ is even. In these
cases, one has that $k$ and $y_{f}$ are both even, then $Y_{f_{i}}$ can not be
even for (23) to provide integer solutions. If this is not the case, i.e., if
$Y_{f_{i}}$ is even, then the generalized fundamental solutions
$\left(X_{f_{i}},Y_{f_{i}}\right)$ must be discarded as it does not provide
integer solutions for $t$ in (23).

For the general case of $k\equiv 0\left(\text{mod}8\right)$, the fact that
$y_{f}$ is not odd can be explained as follows. As $k\equiv
0\left(\text{mod}8\right)$ is not square free, the simple Pell equation (20)
can be simplified posing $k=c^{2}k^{\prime}$, with $k^{\prime}$ square free,
yielding

$x^{2}-k^{\prime}y^{\prime}{}^{2}=1$ (24)

with $y^{\prime}=cy$. The fundamental solution
$\left(x_{f},y_{f}^{\prime}\right)$ of (24) yields then the fundamental
solution
$\left(x_{f},y_{f}\right)=\left(x_{f},\frac{y_{f}^{\prime}}{c}\right)$ of
(20). For example, for $k=8$, let $k^{\prime}=2$ and $c=2,$(24) yields
$\left(x_{f},y_{f}^{\prime}\right)=\left(3,2\right)$ and
$\left(x_{f},\frac{y_{f}^{\prime}}{c}\right)=\left(x_{f},y_{f}\right)=\left(3,1\right)$.
For most of the cases of $k$ such that $k\equiv 0\left(\text{mod}8\right)$,
$y_{f}^{\prime}$ is divisible by c such that $\frac{y_{f}^{\prime}}{c}$ is odd
yielding then $y_{f}$ odd.

For the exceptions of some values of $k$ such that $k\equiv\pm 8,\pm
16\left(\text{mod}64\right)$, this procedure does not lead to an odd value of
$\frac{y_{f}^{\prime}}{c}$. For example, for $k=56$, let $k^{\prime}=14$ and
$c=2,$yielding $\left(x_{f},y_{f}^{\prime}\right)=\left(15,4\right)$ and
$y_{f}=\frac{y_{f}^{\prime}}{c}=2$. For $k=72$, let $k^{\prime}=2$ and
$c=6,$yielding $\left(x_{f},y_{f}^{\prime}\right)=\left(3,2\right)$. However,
$y_{f}^{\prime}$ is not divisible by $c=6$ and one must consider not the first
fundamental solution of the simple Pell equation for $k^{\prime}=2$, but the
second solution given by (10) for $n=2$, yielding
$\left(x_{2},y_{2}^{\prime}\right)=\left(17,12\right)$ that gives
$y_{f}=\frac{y_{2}^{\prime}}{c}=2$ and finally
$\left(x_{f},y_{f}\right)=\left(17,2\right)$.

Furthermore, for some expressions of $k$ in function of the closest natural
square $s^{2}$, one can find general expressions of $\left(x_{f},y_{f}\right)$
and $\left(X_{f_{i}},Y_{f_{i}}\right)$ in addition to $\left(\pm 1,1\right)$
(i.e., for $i>2$) as shown in Table 5. All these expressions can easily be
demonstrated by replacing the appropriate variables in (19) and (20).

Table 5. Expressions of
$k,s,r,\left(x_{f},y_{f}\right),\left(X_{f_{i}},Y_{f_{i}}\right)$ for $i>2$

$k$ | $s$ | $r$ | $\left(x_{f},y_{f}\right)\mid\left(X_{f_{i}},Y_{f_{i}}\right)$
---|---|---|---
$s^{2}+1$ | even | 2 | $\left(\pm\left(2s^{2}+1\right),2s\right)\mid\left(-,-\right)$
| odd | 3 | $\left(\pm\left(2s^{2}+1\right),2s\right)\mid\left(s^{2},s\right)$
$s^{2}+2$ | any | $2^{(a)}$ | $\left(\pm\left(s^{2}+1\right),s\right)\mid\left(-,-\right)$
$s^{2}+4$ | even | $2^{(b)}$ | $\left(\pm\left(\frac{s^{2}}{2}+1\right),\frac{s}{2}\right)\mid\left(-,-\right)$
| $1\left(\text{mod}4\right)$ | $4^{(c)}$ | $\left(\pm\left(\frac{s^{2}\left(s^{2}+3\right)^{2}}{2}+1\right),\frac{s\left(s^{2}+1\right)\left(s^{2}+3\right)}{2}\right)\mid$
|  |  | $\left(\pm\left(s\left(\frac{s^{2}-s+4}{2}\right)-1\right),\frac{s\left(s-1\right)}{2}+1\right)$
| $3\left(\text{mod}4\right)$ | 4 | $\left(\pm\left(\frac{\left(s^{2}+1\right)^{2}\left(s^{2}+4\right)}{2}-1\right),\frac{s\left(s^{2}+1\right)\left(s^{2}+3\right)}{2}\right)\mid$
|  |  | $\left(\pm\left(s\left(\frac{s^{2}+s+4}{2}\right)+1\right),\frac{s\left(s+1\right)}{2}+1\right)$
$s^{2}+8$ | $0\left(\text{mod}4\right)$ | 2 | $\left(\pm\left(\frac{s^{2}}{4}+1\right),\frac{s}{4}\right)\mid\left(-,-\right)$
| $2\left(\text{mod}4\right)$ | 2 | $\left(\pm\left(\frac{s^{2}\left(s^{2}+8\right)}{8}+1\right),s\left(\frac{s^{2}+4}{8}\right)\right)\mid\left(-,-\right)$
$s^{2}+s$ | any | 2 | $\left(\pm\left(2s+1\right),2\right)\mid\left(-,-\right)$
$s^{2}\widehat{\pm}\frac{2s}{\sigma}$ |  $0\left(\text{mod}\sigma\right),\forall\sigma$ odd | $\geq 2$ | $\left(\pm\left(\sigma s\widehat{\pm}1\right),\sigma\right)\mid\left(*,*\right)$
|  $0\left(\text{mod}\frac{\sigma}{2}\right),\forall\sigma$ even | $\geq 2$ | $\left(\pm\left(\sigma s\widehat{\pm}1\right),\sigma\right)^{(d)}\mid\left(*,*\right)$
$s^{2}+s-1$ | any | $\geq 2$ | $\left(*,*\right)\mid\left(\pm\left(2s^{2}+2s-1\right),2s+1\right)^{(e)}$
$s^{2}+s-2$ | $0\left(\text{mod}3\right)$ | >4 | $\left(*,*\right)\mid\left(\pm\left(\frac{2s^{2}+4s}{3}-1\right),\frac{2s}{3}+1\right)^{(f)}$
| $1\left(\text{mod}3\right)$ | 2 | $\left(*,*\right)\mid\left(-,-\right)$
| $2\left(\text{mod}3\right)$ | 4 | $\left(*,*\right)\mid\left(\pm\frac{2s^{2}-5}{3},\frac{2\left(s-2\right)}{3}+1\right)$
$s^{2}+s+1$ | $1\left(\text{mod}3\right)$ | 4 | $\left(\pm\left(\frac{2\left(2s+1\right)^{2}}{3}+1\right),4\left(\frac{2\left(s-1\right)}{3}+1\right)\right)\mid$
|  |  | $\left(\pm\frac{2s^{2}+2s-1}{3},\frac{2s+1}{3}\right)$
| $0,2\left(\text{mod}3\right)$ | $\geq 4$ | $\left(*,*\right)\mid\left(\pm\left(2s^{2}+2s+1\right),2s+1\right)$
$s^{2}+2s$ | any | 2 | $\left(\pm\left(s+1\right),1\right)\mid\left(-,-\right)$
$s^{2}+2s-1$ | any | 2 | $\left(\pm\left(s^{2}+2s\right),s+1\right)\mid\left(-,-\right)$
$s^{2}+2s-2$ | $2\left(\text{mod}3\right)$ | $2^{(g)}$ | $\left(\pm\frac{2s^{2}+4s-1}{3},\frac{2\left(s+1\right)}{3}\right)\mid\left(-,-\right)$
$s^{2}+2s-3$ | $0\left(\text{mod}4\right)$ | 4 | $\left(\pm\frac{\left(s+1\right)\left(s^{2}+2s-2\right)}{2},\frac{s\left(s+2\right)}{2}\right)\mid\left(\pm\frac{s^{2}+3s-2}{2},\frac{s+2}{2}\right)$
| $2\left(\text{mod}4\right)$ | $4^{(h)}$ | $\left(\pm\frac{\left(s+1\right)\left(s^{2}+2s-2\right)}{2},\frac{s\left(s+2\right)}{2}\right)\mid\left(\pm\frac{s^{2}+s-4}{2},\frac{s}{2}\right)$
| odd | $2^{(i)}$ | $\left(\pm\frac{s^{2}+2s-1}{2},\frac{s+1}{2}\right)\mid\left(-,-\right)$
$s^{2}+2s-7$ | $3\left(\text{mod}4\right)$ | $2^{(j)}$ | $\left(\pm\frac{s^{2}+2s-3}{4},\frac{s+1}{4}\right)\mid\left(-,-\right)$
| $1\left(\text{mod}4\right)$ | 4 | $\left(\pm\frac{s^{2}\left(s^{2}-2\right)+4s\left(s^{2}-3\right)+1}{8},\frac{\left(s^{2}-1\right)\left(s+3\right)}{8}\right)\mid$
|  |  | $\left(\pm\frac{2s^{2}+3s-5}{4},\frac{s+1}{2}\right)$
$s^{2}+\frac{\left(3s+1\right)}{2}$ | odd | 2 | $\left(\pm\left(4s+3\right),4\right)\mid\left(-,-\right)$

$\left(-,-\right)$: no solutions exist as $r=2$;$\left(*,*\right)$: no
apparent pattern; $\widehat{\pm}$: plus/minus sign independent from other
$\pm$ sign; (a) except for $k=51,66$ ($r=3,4$); (b) except for $k=40$ ($r=4$);
(c) except for $k=85$ ($r=8$); (d) except for $k=\sigma^{2}-1$, with $\sigma$
even ; (e) except for $k=5,11,55,\ldots$; (f) except for $k=40$; (g) except
for $k=78$ ($r=4$); (h) except for $k=5$ ($r=2$); (i) except for $k=96$
($r=4$); (j) except for $k=136$ ($r=4$)

Note that these general expressions for the fundamental solutions
$\left(x_{f},y_{f}\right)$ are valid in all generality for the simple Pell
equation (20).

###  4.2. First $r$ solutions of (1) for multiple of triangular numbers

Before calculating all solutions of (1) yielding triangular numbers that are
multiple of other triangular numbers, we have to find the first $r$ solutions
$\left(\xi_{i},t_{i}\right)$ (with $0\leq i\leq r$) of (1), arranged in
increasing value order, i.e.,
$\xi_{0}=0<\xi_{1}<\ldots<\xi_{i}<\ldots<\xi_{r}$ and ,
$t_{0}=0<t_{1}<\ldots<t_{i}<\ldots<t_{r}$, and that correspond to the $r$
fundamental solutions $\left(X_{f_{i}},Y_{f_{i}}\right)$ of the generalized
Pell equation (19), with both $X_{f_{i}}$ and $Y_{f_{i}}$ odd or of different
parities.

The generalized fundamental solutions
$\left(X_{f_{1}},Y_{f_{1}}\right)=\left(1,1\right)$ and
$\left(X_{f_{2}},Y_{f_{2}}\right)=\left(-1,1\right)$ provide respectively, the
solutions $\left(\xi_{r},t_{r}\right)$ and $\left(\xi_{r-1},t_{r-1}\right)$ of
(1) from (13), yielding successively

$\displaystyle\left(X_{1_{1}},Y_{1_{1}}\right)$
$\displaystyle=\left(X_{f_{1}}x_{f}+kY_{f_{1}}y_{f},X_{f_{1}}y_{f}+Y_{f_{1}}x_{f}\right)$
$\displaystyle=\left(x_{f}+ky_{f},y_{f}+x_{f}\right)$
$\displaystyle\left(X_{1_{2}},Y_{1_{2}}\right)$
$\displaystyle=\left(X_{f_{2}}x_{f}+kY_{f_{2}}y_{f},X_{f_{2}}y_{f}+Y_{f_{2}}x_{f}\right)$
$\displaystyle=\left(-x_{f}+ky_{f},-y_{f}+x_{f}\right)$

and

$\displaystyle\left(\xi_{r},t_{r}\right)=$
$\displaystyle\left(\frac{X_{1_{1}}-1}{2},\frac{Y_{1_{1}}-1}{2}\right)$
$\displaystyle=$
$\displaystyle\left(\frac{x_{f}+ky_{f}-1}{2},\frac{y_{f}+x_{f}-1}{2}\right)$
(25) $\displaystyle\left(\xi_{r-1},t_{r-1}\right)=$
$\displaystyle\left(\frac{X_{1_{2}}-1}{2},\frac{Y_{1_{2}}-1}{2}\right)$
$\displaystyle=$
$\displaystyle\left(\frac{-x_{f}+ky_{f}-1}{2},\frac{-y_{f}+x_{f}-1}{2}\right)$
(26)

Then for $r>2$, the next two generalized fundamental solutions
$\left(X_{f_{3}},Y_{f_{3}}\right)$ and
$\left(X_{f_{4}},Y_{f_{4}}\right)=\left(-X_{f_{3}},Y_{f_{3}}\right)$ yield
respectively $\left(\xi_{1},t_{1}\right)$ and $\left(\xi_{2},t_{2}\right)$. If
both $X_{f_{3}}$ and $Y_{f_{3}}$ are odd, then (12) ($n=0$) can be used for
$\left(\xi_{1},t_{1}\right)$, yielding

$\left(\xi_{1},t_{1}\right)=\left(\frac{X_{f_{3}}-1}{2},\frac{Y_{f_{3}}-1}{2}\right)$
(27)

Equation (12) could also be used for $\left(\xi_{2},t_{2}\right)$ with
$\left(-X_{f_{3}},Y_{f_{3}}\right)$, but it would provide a negative value for
$\xi_{2}$. Instead, we use (13) ($n=1$), giving

$\left(\xi_{2},t_{2}\right)=\left(\frac{-X_{f_{3}}x_{f}+kY_{f_{3}}y_{f}-1}{2},\frac{-X_{f_{3}}y_{f}+Y_{f_{3}}x_{f}-1}{2}\right)$
(28)

The next two generalized fundamental solutions
$\left(X_{f_{5}},Y_{f_{5}}\right)$ and $\left(X_{f_{6}},Y_{f_{6}}\right)$
$=\left(-X_{f_{5}},Y_{f_{5}}\right)$ yield similarly the next two solutions
$\left(\xi_{i},t_{i}\right)$ that are put in the right increasing order.

For example, for $k=13$, $r=4$,
$\left(x_{f},y_{f}\right)=\left(649,180\right)$,
$\left(X_{f_{i}},Y_{f_{i}}\right)=\left(\pm 1,1\right),\left(\pm 25,7\right)$,
(25) and (26) yield respectively,
$\left(\xi_{r},t_{r}\right)=\left(\xi_{4},t_{4}\right)=\left(1494,414\right)$
and
$\left(\xi_{r-1},t_{r-1}\right)=\left(\xi_{3},t_{3}\right)=\left(845,234\right)$;
(27) and (28) yield respectively,
$\left(\xi_{1},t_{1}\right)=\left(12,3\right)$ and
$\left(\xi_{2},t_{2}\right)=\left(77,21\right)$.

Another example, for $k=46$, $r=6$, $\left(x_{f},y_{f}\right)=(24335,3588)$,
$\left(X_{f_{i}},Y_{f_{i}}\right)=\left(\pm 1,1\right),\left(\pm
47,7\right),\left(\pm 183,27\right)$. With
$\left(X_{f_{1,2}},Y_{f_{1,2}}\right)=\left(\pm 1,1\right)$, (25) and (26)
yield respectively, $\left(\xi_{6},t_{6}\right)=\left(94691,13961\right)$,
$\left(\xi_{5},t_{5}\right)=\left(70356,10373\right)$. With
$\left(X_{f_{3,4}},Y_{f_{3,4}}\right)=\left(\pm 47,7\right)$, (12) yields
$\left(\xi_{1},t_{1}\right)=\left(\frac{X_{f_{3}}-1}{2},\frac{Y_{f_{3}}-1}{2}\right)=\left(23,3\right)$
and (13) yields

$\left(\xi_{4},t_{4}\right)=\left(\frac{-X_{f_{3}}x_{f}+kY_{f_{3}}y_{f}-1}{2},\frac{-X_{f_{3}}y_{f}+Y_{f_{3}}x_{f}-1}{2}\right)=\left(5795,854\right)$

Finally, with $\left(X_{f_{5,6}},Y_{f_{5,6}}\right)=\left(\pm 183,27\right)$,
(12) yields
$\left(\xi_{2},t_{2}\right)=\left(\frac{X_{f_{5}}-1}{2},\frac{Y_{f_{5}}-1}{2}\right)=\left(91,13\right)$
and (13) yields

$\left(\xi_{3},t_{3}\right)=\left(\frac{-X_{f_{5}}x_{f}+kY_{f_{5}}y_{f}-1}{2},\frac{-X_{f_{5}}y_{f}+Y_{f_{5}}x_{f}-1}{2}\right)=\left(1495,220\right)$.

For the case where $Y_{f_{i}}$ is even, i.e., $k=40,96,120,\ldots$, (25), (26)
and (27) cannot be used with $\left(X_{f_{1,2}},Y_{f_{1,2}}\right)=\left(\pm
1,1\right)$ as both $k$ and $Y_{f_{i}}$ are even, yielding non-integer
solutions for $\xi$ and $t$. Instead, the other generalized fundamental
solution have to be used with (13) ($n=1$) and (14) ($n=2$). For example, for
$k=40$, $r=4$, $\left(x_{f},y_{f}\right)=(19,3)$,
$\left(X_{f_{i}},Y_{f_{i}}\right)=\left(\pm 1,1\right),\left(\pm 11,2\right)$,
(13) yields, first, with $\left(X_{f_{3}},Y_{f_{3}}\right)=\left(11,2\right)$,
$\left(X_{1_{3}},Y_{1_{3}}\right)=\left(X_{f_{3}}x_{f}+kY_{f_{3}}y_{f},X_{f_{3}}y_{f}+Y_{f_{3}}x_{f}\right)=\left(449,71\right)$,
yielding $\left(\xi_{2},t_{2}\right)=\left(224,35\right)$, and second, with
$\left(X_{f_{4}},Y_{f_{4}}\right)=\left(-11,2\right)$,

$\left(X_{1_{4}},Y_{1_{4}}\right)=\left(X_{f_{4}}x_{f}+kY_{f_{4}}y_{f},X_{f_{4}}y_{f}+Y_{f_{4}}x_{f}\right)=\left(31,5\right)$,
giving $\left(\xi_{1},t_{1}\right)=\left(15,2\right)$. Next, (14) yields,
first, with $\left(X_{f_{1}},Y_{f_{1}}\right)=\left(1,1\right)$,

$\left(X_{2_{1}},Y_{2_{1}}\right)=\left(x_{f}^{2}+ky_{f}^{2}+2kx_{f}y_{f},x_{f}^{2}+ky_{f}^{2}+2x_{f}y_{f}\right)=\left(5281,835\right)$,
yielding $\left(\xi_{4},t_{4}\right)=\left(2640,417\right)$, and second, with
$\left(X_{f_{2}},Y_{f_{2}}\right)=\left(-1,1\right)$,
$\left(X_{2_{2}},Y_{2_{2}}\right)=\left(-\left(x_{f}^{2}+ky_{f}^{2}\right)+2kx_{f}y_{f},x_{f}^{2}+ky_{f}^{2}-2x_{f}y_{f}\right)=\left(3839,697\right)$,
yielding $\left(\xi_{3},t_{3}\right)=\left(1919,303\right)$.

###  4.3. All solutions of (1) for multiple of triangular numbers

Once that the first $r$ values of $\left(\xi_{i},t_{i}\right)$ have been
found, each corresponding to one of the $r$ generalized fundamental solutions
$\left(X_{f_{i}},Y_{f_{i}}\right)$, the $r$ branches of infinitely many other
solutions can be found using either :

1) the $r$ general solutions (11) (assuming a $+$ sign), yielding

$\xi_{n}+\sqrt{k}t_{n}=\left(\xi_{i}+\sqrt{k}t_{i}+\left(\frac{1+\sqrt{k}}{2}\right)\right)\left(x_{f}+\sqrt{k}y_{f}\right)^{n}-\left(\frac{1+\sqrt{k}}{2}\right)$
(29)

where $\left(\xi_{i},t_{i}\right)$ must be replaced successively by the $r$
values of $\left(\xi_{i},t_{i}\right)$; or

2) the first recurrence relation (15), yielding

$\displaystyle\left(\xi_{n},t_{n}\right)=$
$\displaystyle\left(\left(x_{f}\xi_{n-r}+ky_{f}t_{n-r}\right)+\left(\frac{x_{f}+ky_{f}-1}{2}\right),\right.$
$\displaystyle\left.\left(x_{f}t_{n-r}+y_{f}\xi_{n-r}\right)+\left(\frac{x_{f}+y_{f}-1}{2}\right)\right)$
(30)

where indices of $\xi_{n-r}$ and $t_{n-r}$ (instead of $\xi_{n-1}$ and
$t_{n-1}$) in the right part of (30) refer to the preceding values of $\xi$
and $t$ in the same branch of solutions; or,

3) the second recurrence relation (16), yielding

$\left(\xi_{n},t_{n}\right)=\left(2x_{f}\xi_{n-r}-\xi_{n-2r}+\left(x_{f}-1\right),2x_{f}t_{n-r}-t_{n-2r}+\left(x_{f}-1\right)\right)$
(31)

where indices of $\xi_{n-r},\xi_{n-2r}$ and $t_{n-r},t_{n-2r}$ (instead of
$\xi_{n-1},\xi_{n-2}$ and $t_{n-1},t_{n-2}$) in the right part of (31) refer
to the preceding and the one before values of $\xi$ and $t$ in the same branch
of solutions; or,

4) the Chebyshev polynomial solution (17), yielding

$\displaystyle\left(\xi_{n},t_{n}\right)=$
$\displaystyle\left(\left(\xi_{i}+\frac{1}{2}\right)T_{n-1}\left(x_{f}\right)+k\left(t_{i}+\frac{1}{2}\right)y_{f}U_{n-2}\left(x_{f}\right)-\frac{1}{2},\right.$
$\displaystyle\left.\left(\xi_{i}+\frac{1}{2}\right)y_{f}U_{n-2}\left(x_{f}\right)+\left(t_{i}+\frac{1}{2}\right)T_{n-1}\left(x_{f}\right)-\frac{1}{2}\right)$
(32)

where $\left(\xi_{i},t_{i}\right)$ must be replaced successively by the $r$
values of $\left(\xi_{i},t_{i}\right)$.

###  4.4. Relation between Pell equation solutions and recurrent relations

We can give now a new definition of the rank $r$ introduced in Section 2. The
rank $r$ is the number of fundamental solutions
$\left(X_{f_{i}},Y_{f_{i}}\right)$ of the generalized Pell equation (19), with
$X_{f_{i}}$ odd and $Y_{f_{i}}$ odd or even (if $y_{f}$ is not even) , with
$r\leq\rho$, the total number of generalized solution of (19).

Furthermore, we see that the second recurrent relations (31) for both
$\xi_{n}$ and $t_{n}$ have $x_{f}$ as the only parameter, and that the two
relations are independent from the value of $k$ and $y_{f}$. This fundamental
solution $x_{f}$ of the simple Pell equation (20) acts like a constant of the
problem for each value of $k$. Note further that summing the expressions of
$t_{r}$ and $t_{r-1}$ in (25) and (26) yields $t_{r}+t_{r-1}=x_{f}-1$. As this
sum $t_{r}+t_{r-1}$ was already defined in (2), the constant $\kappa$ is
related to $x_{f}$

$\kappa=x_{f}-1$ (33)

Furthermore, (33) yields also that $y_{f}$ is related to the difference
$\delta=t_{r}-t_{r-1}$ through the simple Pell equation 20
$\left(\kappa+1\right)^{2}-ky_{f}^{2}=1$, which is verified if
$y_{f}^{2}=\kappa^{2}-4t_{r}t_{r-1}=\left(t_{r}-t_{r-1}\right)^{2}=\delta^{2}$,
giving

$\delta=y_{f}$ (34)

for all non-square values of $k$ (except for some values of $k$ such that
$k\equiv 0\left(\text{mod}8\right)$, see further). Replacing in the simple
Pell equation $\left(\kappa+1\right)^{2}-k\delta^{2}=1$ yields the condition
between the sum and the difference of $t_{r}$ and $t_{r-1}$

$\delta=\sqrt{\frac{\kappa^{2}+\kappa}{k}}$ (35)

With the exception of $k=56$, 72, 112, 184, 240, 248, 264, 272, 376, …, i.e.,
for some values of $k$ such that $k\equiv\pm 8,\pm
16\left(\text{mod}64\right)$ (for which (34) is valid), the relation (34) is
not valid for the other cases of $k\equiv 0\left(\text{mod}8\right)$. In these
cases, $\delta>y_{f}$ and one must find the next pair of solutions to the
simple Pell equation by (10) for $n=2$, i.e.,
$\left(x_{2},y_{2}\right)=\left(x_{f}^{2}+ky_{f}^{2},2x_{f}y_{f}\right)$. Then
for these cases,

$\displaystyle\kappa$ $\displaystyle=x_{f}^{2}+ky_{f}^{2}-1$ (36)
$\displaystyle\delta$ $\displaystyle=2x_{f}y_{f}$ (37)

Finally, replacing $x_{f}$ in (31) from 33 yields

$\left(\xi_{n},t_{n}\right)=\left(2\left(\kappa+1\right)\xi_{n-r}-\xi_{n-2r}+\kappa,2\left(\kappa+1\right)t_{n-r}-t_{n-2r}+\kappa\right)$
(38)

which are the same recurrent relations given in (4) and (5).

##  5\. Conclusions

We have shown that the problem of finding all triangular numbers that are
multiples of other triangular numbers with non-square integer multiplier $k$
can be solved using solutions of Pell equations with a simple change of
variables. Only those $r$ fundamental solutions
$\left(X_{f_{i}},Y_{f_{i}}\right)$ of the generalized Pell equation with
$X_{f_{i}}$ odd and $Y_{f_{i}}$ odd or even (if $y_{f}$ is not even) provide
solutions to the problem of finding triangular numbers that are multiple of
other triangular numbers. General expressions of fundamental solutions of the
Pell equations are given for some values of the multiplier $k$ in function of
the closest natural square values $s^{2}$. Many infinitely solutions are then
found on $r$ branches corresponding to each of the $r$ generalized fundamental
solutions $\left(X_{f_{i}},Y_{f_{i}}\right)$ and these solutions can be found
either by a general relation involving $\sqrt{k}$, or by a first set of
recurrent relations, or by a second set of recurrent relations, or by
Chebyshev polynomial solutions. Among these, the second set of recurrent
relations are found to be the same as those found previously without using the
Pell equation solving method.

Furthermore, the number $r$ of generalized fundamental solutions
$\left(X_{f_{i}},Y_{f_{i}}\right)$ with $X_{f_{i}}$ odd and $Y_{f_{i}}$ odd or
even (if $y_{f}$ is not even) corresponds to the rank of these second set
recurrent relations. Finally, the two constants $\kappa=t_{r}+t_{r-1}$ and
$\delta=t_{r}-t_{r-1}$ are also related to respectively the fundamental
solutions $x_{f}$ and $y_{f}$ of the simple Pell equation, as $\kappa=x_{f}-1$
and $\delta=y_{f}$ or $\delta=2x_{f}y_{f}$. These are an unexpected result as
this means that the fundamental solutions of the simple Pell equation, in all
its generality, are related to constants in recurrent relations of the problem
of finding triangular numbers multiple of other triangular numbers.

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c_{2}c_{3}\\\ b_{1}c_{2}c_{3}&|c_{1}c_{2}c_{3}\rightsquigarrow b_{2}\\\
a_{1}b_{2}&|b_{1}b_{2}\\\ \end{split}$ (4.4)

trusting the reader will draw the picture for themselves. To instead put the
$11$-th box on the next-to-left-most position in order to make the partition
$(4,4,2,1)$, we use a blue strap labeled $d_{1}d_{2}|e_{2}$, and remain with
only two endlings $d_{3},d_{4}$. Then our rules produce vestigial lines with
labels

$\begin{split}c_{1}c_{2}d_{3}d_{4}&|d_{1}d_{2}d_{3}d_{4}\rightsquigarrow
c_{3}\\\ b_{1}b_{2}c_{3}&|c_{1}c_{2}c_{3}\\\ \end{split}$ (4.5)

while the last red strap in (4.3) remains untouched, see fig. 20.

Figure 20: Finite web for partition $(4,4,2,1)$.

To, finally222222Partitions $(4,3,2,2)$ and $(4,3,2,1,1)$ follow by
transposition., put the extra box at the center, giving the partition
$(4,3,3,1)$, we use a blue line labeled $d_{2}d_{3}|e_{3}$, remaining with two
teams of endlings labeled $d_{1},d_{4}$. When these meet the first red strap,
they can attach one left and one right to immediately produce the vestigial
line of type $(-+)_{-4}$ with labels

$d_{1}c_{1}c_{2}c_{3}d_{4}|d_{1}d_{2}d_{3}d_{4}\,,$ (4.6)

while the two outer red straps are unaffected. Although (4.6) looks non-
symmetric, it is indeed symmetric under labeling with $x$\- and
$y$-coordinates, which is necessary since we are dealing with a symmetric
partition.

The finite web for the $(4,1,1)$ partition, shown in fig. 21

Figure 21: Finite web for partition $(4,1,1)$.

requires three green and one red straps with labels

$\begin{split}c_{1}&|d_{1}\\\ b_{1}&|c_{1}\\\ b_{2}&|c_{3}\\\
a_{1}&|b_{1}b_{2}\end{split}$ (4.7)

the middle two of which lie on top of each other and have opposite type. At
the intersection with the endling $c_{3}$, our rules stipulate that the
$b_{2}|c_{3}$ strap of type $(++)_{-1}$ turn into a vestigial line with labels
$c_{3}|c_{3}$ of type $(--)_{-1}$, while the endling continues with label
$b_{2}$. As a consequence, in the region above the black endling line, we have
no coinciding $(++)$ and $(--)$ strands, which ensures that there be no non-
canceling holomorphic disks. This example illustrates that endlings do not
have to intersect visibly with straps on the first occasion (in the sense that
the lines pass through each other without changing direction), while the
labels from our complete labeling scheme might still be affected.

The finite web for the $(4,2,1,1)$ partition has somewhat similar features and
moreover contains superimposed vestigial line of mixed type $(++)_{-1}$,
$(--)_{-1}$, see fig. 22.

Figure 22: Finite web for partition $(4,2,1,1)$.

As an example with endlings of three different ages, consider the $(5,3,1)$
partition,

(4.8)

whose web has two red and two green straps labeled

$\begin{split}d_{1}&|e_{1}\\\ c_{1}c_{2}&|d_{1}d_{2}\\\
b_{1}b_{2}&|c_{1}c_{2}c_{3}\\\ a_{1}&|b_{1}b_{2}\\\ \end{split}$ (4.9)

Figure 23: Finite web for partition $(5,3,1)$.

Here, the endling $d_{2}$ interacts with the second green strap for a
vestigial line of type $(--)_{-2}$ labeled $c_{1}d_{2}|d_{1}d_{2}$ emitting
$b_{2}$. $c_{3}$ interacts with the first red one to give
$b_{1}b_{2}c_{3}|c_{1}c_{2}c_{3}$, which then interacts with $c_{2}$ to make
$b_{1}c_{2}c_{3}|c_{1}c_{2}c_{3}$ emitting $b_{2}$. The second red strap
finally absorbs $b_{2}$ to make $a_{1}b_{2}|b_{1}b_{2}$ see fig. 23.

Our final example is the finite web for the partition $(4,2,2)$, shown in fig.
24. It illustrates that while as emphasized above, core lines of boxes in the
same team always lie on top of each other, teams of the same generation (age)
might differ. The core of the box labeled $d_{1}$ is the innermost circle, of
multiplicity $1$, while the core of $d_{3}$ has come to lie on top of the
cores of younger age $3$.

Figure 24: Finite web for partition $(4,2,2)$.

##  5 Beyond fixed points, speculations and future directions

The aim and focus of this paper has been to describe a correspondence between
Young diagrams, parameterizing torus fixed point on ${\rm
Hilb}^{n}({\mathbb{C}}^{2})$, and a certain type of finite webs in critical
exponential networks on the pair of pants. Among the outcomes has been a
clarification of the role of holomorphic disks, and the development of a
somewhat botanical language for linear partitions that is well-adapted to the
finite webs. While we feel that this is evidence enough, we have not claimed
that this correspondence is explicitly one-to-one, how it will extend beyond
the fixed points to a full B-model description of the Hilbert scheme, or even
how exactly wall-crossing witnesses holomorphic disks. The purpose of this
section is to describe our best understanding of the remaining issues.

###  5.1 Navigating through A-brane moduli space

Our matching of partitions with finite webs verifies a rather non-trivial
prediction of (toric) mirror symmetry at the level of the Euler characteristic
of the A-brane moduli, each of our finite web being viewed as the projection
of a special Lagrangian in the mirror geometry (3.1) to the
$\mathbb{C}^{\times}_{x}$-plane. In the following, we give some partial
evidence for projections of special Lagrangians deformed away from the torus
fixed points, but leave it for a future investigation whether one can actually
construct the full family of special Lagrangians corresponding to D4-$n$D0,
and show that the corresponding family of A-branes is indeed parameterized by
${\mathrm{Hilb}}^{n}(\mathbb{C}^{2})$, as it should be [24]. More precisely,
as is explored (for specific examples) in [10, 25] it is expected that the
family of deformations of special Lagrangians $L$ of a fixed homology class
$[L]\in H_{3}(Y,\mathbb{Z})$ forms a foliation of (a region of) $\Sigma$ and a
fibration of (a region of) $Y$. Denote this family by $\mathfrak{M}_{L}$. The
critical leaves of such a foliation are conjectured to correspond to critical
finite webs. Let us describe this family in more detail. At zero string
coupling, each Lagrangian submanifold $L$ of $Y$, can be used to describe a
supersymmetric A-brane, by considering the pair of $L$ together with a unitary
line bundle over it. Then, the family of classical space of deformations of
A-branes $\mathcal{M}_{L}$, specified by $[L]$ takes the form [10, 25]:

$T^{b_{1}(L)}\rightarrow\mathcal{M}_{L}\rightarrow\mathfrak{M}_{L}\,,$ (5.1)

where the fiber $T^{b_{1}(L)}$ parameterizes the flat connections over a
generic point in $\mathfrak{M}_{L}$. However, the fibration (5.1) is not
smooth, since there are points in $\mathfrak{M}_{L}$ where $L$ has some of its
1-cycles pinched, and, on those points, the fiber drops rank. According to the
proposal of [10], there should exist points in $\mathfrak{M}_{L}$ where all
one-cycles in $H_{1}(L,\mathbb{Z})$ pinch and these should be identified with
the finite webs i.e. the critical leaves. Moreover, if
$\mathfrak{D}\subset\mathfrak{M}_{L}$ is the collection of such points, the 5d
BPS count associated with $[L]$ is given by
$\Omega(L)=(-1)^{b_{1}(L)}|\mathfrak{D}|$. However, we should strongly
emphasize that this analysis is based on (5.1) which is the classical space of
deformations of an A-brane of charge $[L]$, in general this should be
corrected by superpotential relations. In the examples considered in [10, 25]
and for instance, for small number of D0-branes, the BPS superpotential
vanishes identically, allowing us to rely on the classical space of
deformations.

Another consequence, suggested by the examples analyzed in [10, 25], is that
$\mathcal{M}_{L}$ has an stratification where each strata is labeled by the
number of pinched cycles at the corresponding points in $\mathfrak{M}_{L}$.
Starting from the zero-dimensional stratum of the critical leaves, one can
parameterize each pinching cycle in terms of some non-contractible cycle in
the corresponding foliation which belongs to a generic point in the moduli
space $\mathfrak{M}_{L}$. Consequently, one can study $\mathfrak{M}_{L}$ near
each fixed point through Lagrangian surgeries that restore one pinched cycle
at a time, and study how it degenerates. Finally, we can recover certain
strata of the moduli space of A-branes $\mathcal{M}_{L}$ via (5.1).

To apply this to the case at hand, we would need to ensure two crucial points:
First of all, the finite webs that we have identified should indeed be the
fixed points under the action of $T^{b_{1}(L)}$ introduced in (5.1) in the
space of bound states of one D4 and $n$ D0-branes. Secondly, there should be a
cell decomposition in the sense of [25] of the A-brane moduli space. This is
also very reasonable to expect. Indeed, the Hilbert scheme of points in
$\mathbb{C}^{2}$ does admit a cell decomposition where the cells retract to
fixed points of the $({\mathbb{C}}^{\times})^{2}$-action on it [45]. Thus,
given our identification of the finite webs precisely with these fixed points,
and assuming that these finite webs indeed arise as $T^{b_{1}(L)}$-fixed
points on the A-brane moduli space, the cell-decomposition of the A-brane
moduli space seems very natural.

Leaving a more thorough investigation of these matters for the future, we will
here be content with exploring some low-dimensional strata of the moduli space
of calibrated finite webs on ${\mathbb{C}}^{\times}_{x}$ that arise (for fixed
$\vartheta=\vartheta_{c}$) by deformation away from the critical exponential
networks for small values of $n$, interpreted as (purported) moduli space of
special Lagrangians in the full three-fold geometry. For finite webs, one
option to do this is via oriented Lagrangian surgery, such as to free up one
of the pinched cycles and then letting it parameterize a part of the moduli
space. We will find substantial evidence that this indeed can be viewed as the
beginning of the above-mentioned cell decomposition.

#### $n=1$

In this case, there is only one finite critical web of the shape in figure 7.
Through surgery one obtains a bubble with a gray line as in figure 25 (see
also fig. 18). In view of
${\mathrm{Hilb}}^{1}(\mathbb{C}^{2})\cong\mathbb{C}^{2}$, the fixed point is
the origin. Performing surgery on the $+$-sheet, the length of the gray line
is the distance of the corresponding point lying on one of the axes of
$\mathbb{R}^{2}$ in the moduli space of foliations, while surgery on the
$-$-sheet corresponds to a point on the other axis. Thus, we conclude
$\mathfrak{M}_{L}\cong(\mathbb{R}_{\geq 0})^{2}$, which reproduces
$\mathcal{M}_{L}\cong\mathbb{C}^{2}$.

Figure 25: Deformation of the finite web for D0-D4 bound state.

#### $n=2$

Factoring our translations and blowing up the symmetric product gives
${\mathrm{Hilb}}^{2}(\mathbb{C}^{2})={\mathbb{C}}^{2}\times{\cal
O}_{{\mathbb{P}}^{1}}(-2)$, which has an exceptional curve with two torus
fixed points. Although we do not yet know how to describe the full moduli
space using foliations, we can identify the stratum containing the two fixed
points by deformation of the critical finite web shown in fig. 12. For
inspiration, we look at the ADHM representation matrices from section 2.1,
which in this stratum take the form [21]

$B_{1}=\begin{pmatrix}\lambda&0\\\ \alpha&\lambda\end{pmatrix},\quad
B_{2}=\begin{pmatrix}\mu&0\\\ \beta&\mu\end{pmatrix},\quad
I=\begin{pmatrix}\sqrt{2\zeta}\\\ 0\end{pmatrix},$ (5.2)

where $[\alpha:\beta]\in\mathbb{P}^{1}$. Inserted in (2.9) one obtains the
equation $|\alpha|^{2}+|\beta|^{2}=\zeta$, and the two torus fixed points
correspond to the endpoints of this interval. Indeed, as we have emphasized
throughout our correspondence between partitions and finite webs, the matrix
elements can be identified with the surgery parameters of the finite webs. In
figure 12 in section 3.4, if the green line is of type $(++)_{-1}$, it
corresponds to the case $\alpha=0$ and if it is of type $(--)_{-1}$,
$\beta=0$.

To recover the rest of the exceptional ${\mathbb{P}}^{1}$, fiber of the
Hilbert-Chow morphism over $0$ parameterized by $[\alpha:\beta]$, we can
follow the same logic with generic values of the surgery parameters, and
indeed find that there is a suitable family of finite webs as shown in fig.
26. Starting from (say) the fixed point with green line of type $(++)_{-1}$,
the blue line (of type $(-+)_{-2}$) starts growing until the green line has
shrunk to zero size, at which point we can grow it back with opposite type
$(--)_{-1}$ to collapse the blue line again and reach the second fixed point.
The exceptional $\mathbb{P}^{1}$ is then the $U(1)$-fibration over the
interval parameterizing this family, realized by $U(1)$ local systems on the
corresponding special Lagrangians.

Figure 26: Interpolating the finite webs for the $(2)$ and $(1,1)$ partitions
along an interval. Figure 27: Midpoint of interpolation in fig. 26.

The finite web representing a bound state with $n=2$, but without any green
line, at the midpoint of the interval, shown in fig. 27, is intriguing in
several ways. From the point of view of moduli, it seems to correspond to a
special locus on the exceptional
${\mathbb{P}}^{1}\subset{\mathrm{Hilb}}^{2}(\mathbb{C}^{2})$ that is otherwise
unknown, or at least we are not aware of any other stratum touching that
point. From the point of view of wall-crossing in exponential networks, one
should emphasize that although this finite web looks tantalizingly similar to
those that we have identified as torus fixed points, one can check that it
does in fact not contain any double ${\cal E}$-walls in the sense of section
3.2, and will hence not contribute to the BPS index of eq. 3.28. How exactly
these observations fit into the broader picture will however, as remarked
before, have to be reserved for a future investigation.

We can also move off the exceptional ${\mathbb{P}}^{1}$, corresponding to
having one D0 at the origin and the other on one of the coordinate axis. We
show a sequence of such deformations in fig. 28, the last of which can be
interpreted as attaching a generic D0 in the moduli space presented in [6] to
the D0-D4 fixed point of fig. 7.

Figure 28: A sequence of deformations of 2D0-D4.

#### $n=3$

Continuing in this vein, a generic point on the Hilbert scheme off the
exceptional divisor looks as in fig. 29, in which each green circle
corresponds to a D0-brane that has been moved away from the origin along one
of the coordinate axes.232323A generic leaf of foliation has some real moduli
which get fixed at the fixed point. A fully generic foliation has the maximal
number of real moduli possible [25].

Figure 29: Fully generic deformation of 3D0-D4.

An interpolation between the $(1,1,1)$ and $(3)$ partitions that generalizes
fig. 26 is shown in figs. 30 and 31.

Figure 30: Interpolating from $(3)$ to $(1,1,1)$, part A. Figure 31:
Interpolating from $(3)$ to $(1,1,1)$, part B.

The ${\mathbb{P}}^{1}$ that is parameterized by the concatenation of these
intervals corresponds, at the level of quiver representations, to the family
of matrices

$B_{1}^{\prime}=\alpha B_{1}+\beta B_{2}\quad,B_{2}^{\prime}=\alpha
B_{2}+\beta B_{1}$ (5.3)

where $B_{1}$ and $B_{2}$ represent the torus fixed points and satisfy
$[B_{1},B_{2}^{\dagger}]=[B_{2},B_{1}^{\dagger}]^{\dagger}=0$ such that each
member of the family satisfies the D-term. The web at the midpoint of the
interval in this case contains a single blue line of type $(-+)_{-3}$ and with
the appropriate changes looks just like fig. 27.

Resolving instead only one of the self-intersections results in the family of
finite webs show in fig. 32, which corresponds to an interpolation between
partitions $(3)$/$(1,1,1)$ and $(2,1)$ along a rational curve in moduli space.

Figure 32: Finite web interpolating $(3)$ and $(2,1)$.

#### $n=4$

Adding another D0-brane by surgery to this last example produces the family of
finite web shown in fig. 33. This corresponds to an interpolation between the
partition $(4)$ and the initially somewhat mysterious $(3,1)$, and sheds some
light on the origin of the “vestigial line” in the latter that we observed in
section 3.4.

Figure 33: Finite web interpolating $(4)$ and $(3,1)$.

Indeed, by a happy numerical coincidence, the two outer green lines coalesce
precisely at the same time as the two black “core” lines running around the
puncture at infinity, at which point we can replace the $2$ $(--)_{-1}$ lines
with the indistinguishable $(--)_{-2}$ to obtain exactly fig. 17.

###  5.2 More on obstructions …

Following observations of section 3.4, finite webs in critical exponential
networks that bound uncanceled holomorphic disks can be interpreted as “stable
non-modules”, by which roughly speaking we mean torus-fixed points in the
quiver configuration space that solve the D-terms242424Algebraically, it would
mean that there is a stability vector generating ${\mathbb{C}}^{n}$ under the
action of $B_{1}$ and $B_{2}$., but violate the F-terms. In the example
specifically, holomorphic disks passing through the self-intersection of the
D0-brane at the branch point can be identified with particular entries in the
ADHM relation $[B_{1},B_{2}]-IJ$ from (1.5), viewed as the derivative of the
superpotential (1.3) with respect to the field $B_{3}$ that originates at that
point.

Figure 34: Finite web corresponding to “non-module” for $n=3$ with $B_{3}\neq
0$.

As further evidence for this interpretation, we show in fig. 34 a critical
finite web in the class of one D4 with $n=3$ D0-branes, two of which have
their “flaps” connected and detached from the branch point. According to the
by now standard identification of surgery parameters with representation data,
if the green line is of type $(--)_{-1}$, this finite web should correspond to
the matrices

$B_{1}=\begin{pmatrix}0&0&0\\\ 1&0&0\\\ 0&0&0\end{pmatrix},\quad B_{2}=0,\quad
B_{3}=\begin{pmatrix}0&0&0\\\ 0&0&0\\\ 0&1&0\end{pmatrix},\quad
I=\begin{pmatrix}1\\\ 0\\\ 0\end{pmatrix}$ (5.4)

viewed as a “torus-fixed stable non-module” of the extended ADHM quiver (1.2),
and indeed it is not hard to locate a holomorphic disk in fig. 34
corresponding to the non-zero entries in $[B_{1},B_{3}]\neq 0$. Exchanging
$B_{1}$ and $B_{2}$ of course corresponds to the green line being of type
$(++)_{-1}$, which then covers all stable obstructed fixed points at
$n=3$.252525The moduli space of obstructed networks for general $n$ is
conjecturally given by
$\left\\{(I,B_{1},B_{2},B_{3})\in{\mathbb{C}}^{n}\oplus\operatorname{End}({\mathbb{C}}^{n})^{3}\,\left|\,\begin{matrix}{\mathbb{C}}\\{B_{1},B_{2},B_{3}\\}\cdot
I={\mathbb{C}}^{n}\\\
B_{3}I=0\end{matrix}\right.\right\\}\mathbin{/\mkern-5.5mu/}{\mathit{G}L}({\mathbb{C}}^{n})$
So again holomorphic disks justify our ignoring fig. 34 for the purposes of
BPS counting, if we did not a priori exclude any finite web involving a
surgery between different D0-branes at the branch point. An interesting aspect
of this identification is that the calibrated surgery is possible at all,
while in general [46] one expects to be able to form bound states between
D0-branes using $B_{3}$ only under framing by a D6-brane, which at the moment
we do not know how to realize at the level of (exponential) networks. That the
requisite entry in $B_{3}$ is in fact tachyonic can be seen for example by
superposing the finite web in fig. 12 with the plain D0 from fig. 5.

###  5.3 … and their possible resolution via ghost webs

The most pressing question in the story however might well be the relation
between the obstruction of finite webs by holomorphic disks and the wall-
crossing formalism for BPS state counting of [8, 9] that we reviewed in
section 3.2. Indeed, while according to our identification obstructed webs are
stable fixed points of the torus action, they do not represent any point in
quiver moduli, which is the main reason for excluding them from the count. Yet
they are included in the double walls ${\cal W}_{c}$ inside the critical
exponential network ${\cal W}(\vartheta_{c})$ at the appropriate angle
$\vartheta_{c}=\vartheta_{(1,n)}$. They would hence appear to contribute in
eq. (3.26), and must either be excluded explicitly by hand, or else cancel
against some other “critical” contribution that we have so far ignored.

To entertain this latter possibility, we recall the observation, recorded
around eq. (3.41), that a general holomorphic disk obstructing one of our
finite webs will end on some closed subcycle of the cycle in
$\widetilde{\Sigma}$ lifting the finite web with vanishing period integral
(3.29). As a consequence, one expects it to be possible to contract the
holomorphic disk by deforming the cycle without changing the central charge.
In the first and simplest example, coming from the finite web in fig. 14 with
two green lines of different type, this is indeed possible. The resulting
finite web is shown in fig. 35. Here, the green line necessarily crosses the
branch cut, where $(++)$ and $(--)$ get exchanged. This would not be
compatible with the two green lines initially being of the same type.

Figure 35: Ghost web for obstructed network from fig. 14.

While these finite webs cannot be obtained by small surgery from our basic
constituents (in particular, they do not coalesce onto one D4 plus 3 D0-branes
in the limit $\vartheta_{\rm D4}\to 0$), they are in fact part of the double
wall network $\mathcal{W}_{c}$, and it seems reasonable that they will
contribute via (3.28) to canceling our obstructed webs.

The same mechanism also works for the finite web in fig. 34 corresponding to a
non-zero vev for $B_{3}$, where contracting the holomorphic disk leads to the
finite web shown in fig. 36.

Figure 36: Ghost web for obstructed network from fig. 34.

Thus we see that at least for $n=3$, each obstructed finite web appears to be
accompanied by another “ghost web” that results from contracting the
uncanceled holomorphic disk. The formal interpretation could be that the
finite “critical” webs are in fact organized in some kind of graded complex
whose degree $0$ piece is generated by webs corresponding to stable modules
for (1.2), though possibly violating the F-terms, and the differential comes
from (the dual of) contracting holomorphic disks. In an ideal world this
complex would have cohomology concentrated in degree $0$ with $\dim
H^{0}=\Omega(\gamma)$. Although this would be a very satisfying result, at
this stage we have little to offer beyond the case $n=3$ that we just recorded
(while the cases $n=1$ and $n=2$ are trivial).

As a small piece of evidence for $n=4$, we give an account of the ghost webs
for the $6$ obstructed webs obtained from fig. 37 by mixing green lines of
different type.

Figure 37: Finite web corresponding to $(4)$ and $(1,1,1,1)$ partition.

By initially contracting the two (potential) disks visible in the figure, we
find the $2$ webs shown in fig. 38, each of which supports $4$ choices of
labels for a total of $8$ ghost webs.

Figure 38: Ghosts for the obstructed finite webs in fig. 37.

This clearly is an over-counting as a result of the finite webs with label
sequence $(++)\,(--)\,(++)$ and $(--)\,(++)\,(--)$ being canceled twice, and
should have a familiar resolution from “ghosts for ghosts”. Indeed we find two
such, as shown in fig. 39, such that our final count could again be balanced.

Figure 39: “Ghost for ghosts” web for fig. 38.

In conclusion, the combinatorics of these purported ghost networks is already
quite complicated for $n=4$ since the webs in fig. 37 can have $2$ non-
canceling holomorphic disks, and it is not immediately clear how this will
generalize. In particular, we have observed that also canceling holomorphic
disks can be contracted to give ghost webs and “ghosts for ghosts”. For
example, contracting both holomorphic disks in the finite web for the $(2,2)$
partition in figure 16 gives rise to the “ghost for ghosts” web shown in
figure 40. The elucidation of these ghost networks and their possible role in
the relation between finite webs and non-abelianization as presented in
section 3.2 is high on the list for future investigation.

Figure 40: “Ghost for ghosts” for the finite web in fig. 16.

## Acknowledgments

We would like to thank P. Longhi, L. Merkens, B. Pozzetti, V. Tatitscheff for
valuable discussion and collaboration on related projects, and all and any
available ADHM enthusiasts, including in particular T. Dimofte, M. Kontsevich,
N. Nekrasov, I. Saberi, for much-needed encouragement. S.B. thanks Max Planck
Institute for Mathematics, Bonn where part of the work was done, and
acknowledges Heidelberg University for providing excellent working conditions
at the late stages of this work. R.S. expresses his sincere gratitude to the
Institut des Hautes Études Scientifiques (IHES) in Bures-sur-Yvette, for their
gracious hospitality. J.W. thanks the organizers of the TSIMF conference on
“Homological Algebra of the Infrared”, where this work was presented, and YMSC
for kind hospitality during finalization of the paper. M.R. acknowledges
support from the National Key Research and Development Program of China, grant
No. 2020YFA0713000, the Research Fund for International Young Scientists, NSFC
grant No. 1195041050. MR also acknowledges Heidelberg University, IHES, SCGP,
Steklov Institute, Moscow for hospitality at the final stages of this work.
The work of S.B. has been supported in part by the ERC-SyG project “Recursive
and Exact New Quantum Theory” (ReNewQuantum), which received funding from the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation program, grant agreement No. 810573. J.W.’s partial
funding statement reads: This work is funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy EXC 2181/1 — 390900948 (the Heidelberg STRUCTURES
Excellence Cluster)

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