Originally Posted by

**alexmahone** ....On second thoughts, this doesn't seem to work because if

are odd,

and

are even.

Perhaps you can patch up your proof with minor alternations. I'll be looking to do this myself since this problem interests me. FWIW, I computed the results for N up to 30 using the C program below. Taking it beyond 30 will require a Bignum library, and while I'm looking for one so as not to have to reinvent the wheel, so far I haven't had any luck. What I've found seems rather obscure since it's not well documented, and I'm only interested in integer arithmetic—not floating point. Besides, if I implement one myself I'll have a working knowledge of how to extend it in various ways that may be more useful to its application re number theory.

I've noticed that there is only one solution for every case of N < 30. Perhaps there is always a unique solution. It would be interesting to show that. Furthermore, if this is true, the property of integers could prove useful in cryptography, since it would assign a unique pair of odd numbers to every power of 2 >= 8. I'm not saying how it would be useful, I'm just speculating that it might be.

The program:

Code:

#include <math.h>
#include <stdio.h>
int main()
{
int N;
unsigned long a, b, L = 4, M, P;
for (N = 3; N < 31; N++)
{
L *= 2; //note: L = 2^N
M = sqrt((double)L);
for (b = 1; b < M; b += 2)
{
P = L - b*b;
if (!(P % 7))
{
a = sqrt((double)(P = P/7));
if (P == a*a) printf("N = %i, 2^N = %i, a = %i, b = %i\n", N, L, a, b);
}
}
}
}

The output:

N = 3, 2^N = 8, a = 1, b = 1

N = 4, 2^N = 16, a = 1, b = 3

N = 6, 2^N = 64, a = 3, b = 1

N = 8, 2^N = 256, a = 5, b = 9

N = 9, 2^N = 512, a = 7, b = 13

N = 10, 2^N = 1024, a = 3, b = 31

N = 11, 2^N = 2048, a = 17, b = 5

N = 12, 2^N = 4096, a = 11, b = 57

N = 13, 2^N = 8192, a = 23, b = 67

N = 14, 2^N = 16384, a = 45, b = 47

N = 16, 2^N = 65536, a = 91, b = 87

N = 17, 2^N = 131072, a = 89, b = 275

N = 18, 2^N = 262144, a = 93, b = 449

N = 19, 2^N = 524288, a = 271, b = 101

N = 20, 2^N = 1048576, a = 85, b = 999

N = 21, 2^N = 2097152, a = 457, b = 797

N = 22, 2^N = 4194304, a = 627, b = 1201

N = 23, 2^N = 8388608, a = 287, b = 2795

N = 24, 2^N = 16777216, a = 1541, b = 393

N = 25, 2^N = 33554432, a = 967, b = 5197

N = 26, 2^N = 67108864, a = 2115, b = 5983

N = 27, 2^N = 134217728, a = 4049, b = 4411

N = 28, 2^N = 268435456, a = 181, b = 16377

N = 29, 2^N = 536870912, a = 8279, b = 7555

N = 30, 2^N = 1073741824, a = 7917, b = 25199