       Help with an "Equal Sums of Like Powers" and a square problem?

• To: mathgroup at smc.vnet.net
• Subject: [mg117385] Help with an "Equal Sums of Like Powers" and a square problem?
• From: TPiezas <tpiezas at gmail.com>
• Date: Wed, 16 Mar 2011 06:31:29 -0500 (EST)

```Hello all,

Does anyone know of an efficient code to solve this:

"Problem:  Find,

a^k+b^k+c^k+d^k = e^k+f^k+g^k+h^k,   for both k = 2 & 4,

such that, for either n = -7, or n = -21, (note the negative signs,
pls) then,

n(a^6+b^6+c^6+d^6-e^6-f^6-g^6-h^6)(a^8+b^8+c^8+d^8-e^8-f^8-g^8-h^8) =
y^2   (eq.1)

for some NON-ZERO square integer y."

The only solution known is when n = -21,

{a,b,c,d} = {240, 63, 197, 122}
{e,f,g,h} = {167, -10,  243, 168}

found by Jarek Wroblewski, though he found this indirectly and didn't
realized it satisfied eq.1. (The sign of -10 does not matter, but I
included it to highlight the curious fact that a-b = e-f, c-d = g-h.)
The context of this problem is that it leads to a 9th deg multi-grade
equality valid for k = {1,2,3,4,5,9}.

I managed to make some code, but it is good only for the range below
120 (less than Wroblewski's solution). Is there an efficient way to
solve this kind of system for a higher range?

Thanks.

- Tito

```

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