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

1,1

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

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]

Example

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

Prog

(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

Crossrefs

Cf. A003336, A191345.

Sequence in context: A244348 A083633 A043649 * A230925 A204653 A031677

Adjacent sequences: A282945 A282946 A282947 * A282949 A282950 A282951

Keyword

nonn

Author

Altug Alkan and Thomas Ordowski, Feb 25 2017

Extensions

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

Status

approved