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

Jack Brennen jfb at brennen.net
Thu Jan 22 22:59:28 CET 2009


It would seem that it contains an infinite number of multiples of 1001.

Maybe someone can determine what property of the integer Z leads to
(1001*Z)^2 == X^2 + 40*Y^2 having other solutions than the 13
primitive solutions multiplied by Z?

The values of Z which do lead to extra solutions include all multiples
of:

7, 11, 13, 19, 23, 37, 41, 47, 53, 59, 89, 103, 127, 131, 139, ...

(Which doesn't appear to be a sequence in the OEIS.)

zak seidov wrote:
> Dear seqfans,
> Is this sequence finite? 
> 
> %I A000001
> %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. 
> %C A000001 Is the sequence finite? 
> %e A000001 n=1001: {x,y}={{169,156},{231,154},{279,152},{371,147},{429,143},{681,116},{749,105},{781,99},{819,91},{871,78},{921,62},{979,33},{999,10}};
> %e A000001 n=1463: {x,y}={{77,231},{247,228},{627,209},{653,207},{777,196},{787,195},{903,182},{1197,133},{1273,114},{1287,110},{1353,88},{1427,51},{1453,27}}. 
> %Y A000001 A107145 Primes of the form x^2+40y^2. 
> %K A000001 nonn
> %O A000001 1,1
> %A A000001 Zak Seidov (zakseidov(AT)yahoo.com), Jan 22 2009
> 
> 
> 
> 
>       
> 
> 
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