A282948 Numbers n such that (u^4 + v^4)/2 = x^4 + y^4 = n has a solution in positive integers u,v,x,y.

162401, 2598416, 13154481, 41574656, 101500625, 210471696, 389924801, 665194496, 1065512961, 1624010000, 2377713041, 3367547136, 4638334961, 6238796816, 8221550625, 10643111936, 13563893921, 17048207376, 21164260721, 25984160000, 31583908881, 38043408656

Offset: 1

Author

Altug Alkan and Thomas Ordowski, Feb 25 2017

Keywords

• nonn

Comments

All terms are composite.

If n is in this sequence, then n*k^4 with k > 0 is in this sequence.

Numbers n such that n and 2*n are both in A003336. - Michel Marcus, Feb 25 2017

The first term which is not a multiple of a(1) is a(84) = 8051889328801. - Giovanni Resta, Feb 25 2017

Based on Giovanni Resta's b-file, the squarefree terms are 162401, 8051889328801, 9305528350081, 16778006844241, .... - Altug Alkan, Feb 26 2017

Izadi & Nabardi construct a collection of elliptic curves of rank >= 5 using (essentially) terms of this sequence. - Charles R Greathouse IV, Jul 13 2024

Examples

			(19^4 + 21^4)/2 = 7^4 + 20^4 = 162401.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..513 (terms < 10^16)
• Farzali Izadi and Kamran Nabardi, A Family of Elliptic Curves With Rank >= 5, arXiv preprint (2015). arXiv:1501.03809 [math.NT]

Crossrefs

Cf. A003336, A191345.

Programs

PARI

isA003336(n) = for(k=1, sqrtnint(n\2, 4), ispower(n-k^4, 4) && return(1));
is(n) = isA003336(n) && isA003336(2*n);

PARI

T=thueinit('x^4+1, 1);
has(n)=#thue(T, n)>0 && !issquare(n)
list(lim)=my(v=List(),x4,t); for(x=1,sqrtnint(lim\=1,4), x4=x^4; for(y=1,min(sqrtnint(lim-x4,4),x), t=x4+y^4; if(has(2*t), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 26 2017

Extensions

a(10)-a(22) from Giovanni Resta, Feb 25 2017