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

*To*: mathgroup at smc.vnet.net*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