# [seqfan] Re: Squares of the form x^2+40y^2 .

Edwin Clark eclark at math.usf.edu
Fri Jan 23 00:09:34 CET 2009

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On Thu, 22 Jan 2009, zak seidov wrote:

> Dear seqfans,
> Is this sequence finite?
> %S A000001 1001,1463,1729,1771,2002,2093,2717,2849,2926,3003,3059,3157
> %N A000001 Numbers n with property that n^2 has exactly 13 representations as x^2+40y^2 with positive x, y.

If Maple's isolve is correct there are lots more n than those in your
list:

Checking n only up to 10,000 I get:

a:=NULL:
for n from 1 to 10000 do
X:=remove(hastype,{isolve(x^2+40*y^2=n^2)},nonposint );
if nops(X) = 13 then a:=a,n; fi;
od:
a;
1001, 1463, 1729, 1771, 2002, 2093, 2717, 2849, 2926, 3003, 3059, 3157,
3289,

3367, 3458, 3542, 3619, 3731, 4004, 4081, 4186, 4277, 4389, 4543, 4807,

4823, 4921, 5005, 5187, 5291, 5313, 5369, 5434, 5453, 5681, 5698, 5852,

5863, 5957, 6006, 6118, 6251, 6279, 6314, 6578, 6601, 6721, 6734, 6853,

6916, 7049, 7084, 7238, 7315, 7462, 7567, 7579, 7733, 7847, 7931, 8008,

8099, 8151, 8162, 8372, 8437, 8533, 8547, 8554, 8569, 8645, 8778, 8855,

9009, 9086, 9139, 9177, 9361, 9373, 9471, 9499, 9614, 9646, 9779, 9823,

9842, 9867

```