OFFSET
3,1
COMMENTS
Provided that the quadruple (x,y,z,t) satisfies x^2+x*y+y^2 = z^2+z*t+t^2, define E(k,x,y,z,t) = (x^k+y^k+(x+y)^k) - (z^k+t^k+(z+t)^k). When k is odd, the only thing that can be said is that E(k,x,y,z,t) is even, and when k=4, E(k,x,y,z,t) equals 0. So the sequence will concern only k even and greater than 4, hence offset is 3 with even values m=2n only.
LINKS
Étienne Ghys, Le plaisir de l’amateur, Images des Mathématiques, CNRS, 2009
Vincent Thill, Curiosités Arithmétiques, Editions Le Sanctuaire, 2008.
EXAMPLE
The quadruple [5,6,1,9] satisfies (x^2+x*y+y^2) = (z^2+z*t+t^2). Then, for m=6, the expression is 302400 = 7*43200.
PROG
(PARI) q(x, y, n) = x^n+y^n+(x+y)^n;
expr(n, rv) = q(rv[1], rv[2], n) - q(rv[3], rv[4], n);
findgcd(n, v) = gcd(vector(#v, i, expr(n, v[i])));
lista(nn) = {vsols = [[5, 6, 1, 9], [4, 9, 1, 11], [7, 7, 2, 11], [8, 9, 3, 13], [7, 11, 3, 14], [5, 13, 2, 15], [9, 11, 4, 15], [8, 11, 1, 16], [10, 13, 5, 17], [7, 14, 1, 18]]; for (i=3, nn, print1(findgcd(2*i, vsols), ", "); ); }
\\ Michel Marcus, Mar 02 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Marcus, Mar 01 2014
EXTENSIONS
New name from Charles R Greathouse IV, Mar 03 2014
STATUS
approved