Re: a^k+b^k+c^k = d^k+e^k+f^k, for k = 2,4, with a constraint

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• Subject: [mg102445] Re: a^k+b^k+c^k = d^k+e^k+f^k, for k = 2,4, with a constraint
• From: TPiezas <tpiezas at gmail.com>
• Date: Sun, 9 Aug 2009 18:19:48 -0400 (EDT)
• References: <h5m6ut\$fe5\$1@smc.vnet.net>

```On Aug 9, 4:04 am, TPiezas <tpie... at gmail.com> wrote:
> Hello all,
>
> It's easy to find small and non-trivial solns to the multi-grade eqn,
>
> a^k+b^k+c^k = d^k+e^k+f^k, for k = 2,4.
>
> However, I need the additional constraint that,
>
> a^2+b^2+c^2 = y^2.
>
> What's an efficient and _fast_ code in Mathematica to find solns to
> this?  Any help will be appreciated.  Thanks.
>
> P.S. This system came up in a certain 6-8-7 identity similar to
> Ramanujan's 6-10-8.
>
> - Titus

I also posted this problem in sci.math.symbolic and someone (Martin)
gave the single soln with terms < 1000 as,

{u,v,w,x,y,z} = {2, 289, 610, 170, 223, 614}

Here is the relevant math. Let,

u^k+v^k+w^k = x^k+y^k+z^k,  for k = 2,4

and u^2+v^2+w^2 = t^2. Define,

{a,b,c,d,e,f} = {t+u, t-u, t+v, t-v, t+w, t-w}
{g,h,i, j,k,l}  = {t+x, t-x, t+y, t-y, t+z, t-z}

then,

49 (a^6+b^6+c^6+d^6+e^6+f^6-g^6-h^6-i^6-j^6-k^6-l^6)
(a^8+b^8+c^8+d^8+e^8+f^8-g^8-h^8-i^8-j^8-k^8-l^8) = (88/3)
(a^7+b^7+c^7+d^7+e^7+f^7-g^7-h^7-i^7-j^7-k^7-l^7)^2

Given the soln and t = 675, we get,

{a,b,c,d,e,f} = {677, 673, 964, 386, 1285, 65}
{g,h,i, j,k,l}  = {845, 505, 898, 452, 1289, 61}

Note that a,b,...l is also an "ideal solution" of the Prouhett- Tarry-
Escott problem up to 5th powers.

a^n+b^n+c^n+d^n+e^n+f^n = g^n+h^n+i^n+j^n+k^n+l^n, for n = 1,2,3,4,5.

Nice, hm?

P.S. Is there an efficient and fast way to find other solns using
Mathematica?

- Titus

```

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