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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.

%I A282948 #38 Jul 13 2024 23:23:41
%S A282948 162401,2598416,13154481,41574656,101500625,210471696,389924801,
%T A282948 665194496,1065512961,1624010000,2377713041,3367547136,4638334961,
%U A282948 6238796816,8221550625,10643111936,13563893921,17048207376,21164260721,25984160000,31583908881,38043408656
%N 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.
%C A282948 All terms are composite.
%C A282948 If n is in this sequence, then n*k^4 with k > 0 is in this sequence.
%C A282948 Numbers n such that n and 2*n are both in A003336. - _Michel Marcus_, Feb 25 2017
%C A282948 The first term which is not a multiple of a(1) is a(84) = 8051889328801. - _Giovanni Resta_, Feb 25 2017
%C A282948 Based on _Giovanni Resta_'s b-file, the squarefree terms are 162401, 8051889328801, 9305528350081, 16778006844241, .... - _Altug Alkan_, Feb 26 2017
%C A282948 Izadi & Nabardi construct a collection of elliptic curves of rank >= 5 using (essentially) terms of this sequence. - _Charles R Greathouse IV_, Jul 13 2024
%H A282948 Giovanni Resta, <a href="/A282948/b282948.txt">Table of n, a(n) for n = 1..513</a> (terms < 10^16)
%H A282948 Farzali Izadi and Kamran Nabardi, <a href="https://arxiv.org/abs/1501.03809">A Family of Elliptic Curves With Rank >= 5</a>, arXiv preprint (2015). arXiv:1501.03809 [math.NT]
%e A282948 (19^4 + 21^4)/2 = 7^4 + 20^4 = 162401.
%o A282948 (PARI) isA003336(n) = for(k=1, sqrtnint(n\2, 4), ispower(n-k^4, 4) && return(1));
%o A282948 is(n) = isA003336(n) && isA003336(2*n);
%o A282948 (PARI) T=thueinit('x^4+1, 1);
%o A282948 has(n)=#thue(T, n)>0 && !issquare(n)
%o A282948 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
%Y A282948 Cf. A003336, A191345.
%K A282948 nonn
%O A282948 1,1
%A A282948 _Altug Alkan_ and _Thomas Ordowski_, Feb 25 2017
%E A282948 a(10)-a(22) from _Giovanni Resta_, Feb 25 2017