|
%I #7 Jul 23 2022 22:04:10
%S 520,975,2040,2080,3567,3900,4680,7215,7800,8160,8320,8775,9840,13000,
%T 13920,14268,15600,18360,18720,19680,24375,25480,28860,30160,31200,
%U 32103,32640,33280,35100,39360,40545,42120,47775,51000,52000,53040
%N Positive integers n such that n^2 = (x^4 - y^4)*(z^4 - t^4) where the pairs of integers (x,y) and (z,t) are not proportional.
%C Positive integers n such that n^2 = s^4*A147858(m)*A147858(k) for positive integers s and k<m. If n belongs to this sequence then so does n*s^2 for any positive integer s. Primitive elements of this sequence are given by A147856.
%C Euler proved that if n^2 = (x^4 - y^4)*(z^4 - t^4) then a,b,c (if n is even) or 4a,4b,4c (if n is odd) form a triple of integers with all pairwise sums and differences being squares, where a=(x^4+y^4)*(z^4+t^4)/2, b=(n^2+(2xyzt)^2)/2 and c=(n^2-(2xyzt)^2)/2. Note that a,b,c are pairwise distinct if and only if (x,y) and (z,t) are not proportional.
%C 4*A196289(n) = 4*(n^9 - n) belong to this sequence since (4*(n^9 - n))^2 = ((n^4+2*n^2-1)^4 - (n^4-2*n^2-1)^4) * (n^4 - 1).
%Y Cf. A147856, A147857, A147858.
%K nonn
%O 1,1
%A _Max Alekseyev_, Nov 17 2008, Nov 19 2008
|
