A147854 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.

520, 975, 2040, 2080, 3567, 3900, 4680, 7215, 7800, 8160, 8320, 8775, 9840, 13000, 13920, 14268, 15600, 18360, 18720, 19680, 24375, 25480, 28860, 30160, 31200, 32103, 32640, 33280, 35100, 39360, 40545, 42120, 47775, 51000, 52000, 53040

Offset

1,1

Comments

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.

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.

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).

Links

Table of n, a(n) for n=1..36.

Crossrefs

Cf. A147856, A147857, A147858.

Sequence in context: A345596 A345855 A235274 * A147856 A094903 A250693

Adjacent sequences: A147851 A147852 A147853 * A147855 A147856 A147857

Keyword

nonn

Author

Max Alekseyev, Nov 17 2008, Nov 19 2008

Status

approved