triangle/triangle

[Richard Guy]

The equation can be written as a Baskhara
(aka Pell) equation

      (2n+1)^2 - c(2m+1)^2 = 1-c

I believe that this is solvable just if  c
is not a non-trivial square, though the
(infinitely many) solutions don't always
come from the convergents to the continued
fraction for  root c

E.g. If we try  c = 7, we get the convergents

0  1  2  3  5  8  37  45  82  127  590
-  -  -  -  -  -  --  --  --  ---  --- ...
1  0  1  1  2  3  14  17  31   48  223

none of which work, but the mediants

0+1      5+8             82+127
---      ---             ------     ...
1+0      2+3              31+48

give (m,n) =

(0,0)   (2,6)           (39,104)

and the next one is  (629,1665)

Best,   R.

On Wed, 21 Nov 2001, David W. Wilson wrote:

> 
> Let c = a/b where a and b are positive triangulars and c is an integer.
> Empirically, it looks as if c can take on any positive integer value
> except a square >= 4.  Is this the case?
>