From: steiner@math.bgsu.edu (ray steiner)
Subject: Re: Integer equations
Date: Wed, 03 Nov 1999 16:29:05 -0500
Newsgroups: sci.math
Keywords: triangular number which is also a square pyramidal number

In article <7vodr0$4kn$1@nnrp1.deja.com>, Ron Winther 
<rwinther@my-deja.com> wrote:

> In article <m0vumpocls41@forum.swarthmore.edu>,
>   stuart@northage.netlineuk.net (Stuart M Northage) wrote:
> > Does anybody know how to find the integer solutions to an equation of
> > the form :
> >
> > x = sqrt(3)*(sqrt (8y^3+12y^2+4y+3)-sqrt(3))
> >     ---------------------------------------
> >                       6
> >
> > I've been struggling for weeks on this one!
> >
> 
> Assuming that x and y are nonnegative, this is equivalent to solving
> 
>   x(x+1)/2 = y*(y+1)*(2y+1)/6
> 
> If x and y are integers, the left side is the sum of the integers
> from 1 to x and the right side is the sum of the squares of the
> integers from 1 to y. I don't know if that makes the problem any
> easier to approach (or maybe that's where it came from in the
> first place??)
> 
> I've resorted to searching, and the solutions I found are
> 
>   x=1, y=1;   x=10, y=5;   x=13, y=6;   x=645, y=85
> 
> There are no other solutions for y < 100000
> 
> I realize this doesn't answer your question, but it's all I can
> offer at the moment.
> 

There are exactly two other integer solutions:
y= -1 and y=0.
This is the problem "triangular number= square pyramidal number".
For the complete solution see:
Uchiyama, S. Solution of a Diophantine problem, Tsukuba J. Math 8(1984), 
131-137.
There is also an earlier solution by Avanesov(in Russian) in
 Volz. Mat. Sb. Vyp.,(8)(1971), 3-6.
This problem is also mentioned in section D3 of Guy's book UNSOLVED PROBLEMS IN
NUMBER THEORY. 
Regards,
Ray Steiner

-- 
steiner@math.bgsu.edu
==============================================================================
[The substitutions  x=Y/18-1/2, y=X/6-1/2  render this equation into normal
 form  Y^2 = X^3 - 9 X + 81 .  This is a curve with no torsion and rank 2,
 two independent generators being the points [X,Y] = [+-3, 9], i.e.
 [x,y]=[0,0] and [0,-1]. --djr]
=============================================================================