A216280 - OEIS

%I #22 Feb 22 2019 05:17:27

%S 1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,

%T 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,

%U 0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0

%N Number of nonnegative solutions to the equation x^4 + y^4 = n.

%C The first n with a(n) > 1 is 635318657 = 41 * 113 * 241 * 569, with a(635318657) = 2. Izadi, Khoshnam, & Nabardi show that for any n with a(n) > 1, the elliptic curve y^2 = x^3 - nx has rank at least 3. According to gp, y^2 = x^3 - 635318657x has analytic rank 4 (and first nonzero derivative around 35741.7839). - _Charles R Greathouse IV_, Jan 12 2017

%H Antti Karttunen, <a href="/A216280/b216280.txt">Table of n, a(n) for n = 1..16561</a>

%H F. A. Izadi, F. Khoshnam, K. Nabardi, <a href="https://arxiv.org/abs/1202.5676">Sums of two biquadrates and elliptic curves of rank ≥4</a> (2012). arXiv:1202.5676 [math.NT], 2012.

%H F. A. Izadi, F. Khoshnam, K. Nabardi, <a href="http://www.math.okayama-u.ac.jp/mjou/mjou56/_05_izadi.pdf">Sums of two biquadrates and elliptic curves of rank ≥ 4</a>, Math. J. Okayama Univ. 56 (2014), 51-63.

%t Reap[For[n = 1, n <= 1000, n++, r = Reduce[0 <= x <= y && x^4 + y^4 == n, {x, y}, Integers]; sols = Which[r === False, 0, r[[0]] == And, 1, r[[0]] == Or, Length[r], True, Print[n, " ", r]]; If[sols != 0, Print[n, " ", sols, " ", r]]; Sow[sols]]][[2, 1]] (* _Jean-François Alcover_, Feb 22 2019 *)

%o (PARI) a(n)=my(t=thue(thueinit('x^4+1,1),n)); sum(i=1,#t, t[i][1]>=0 && t[i][2]>=t[i][1]) \\ _Charles R Greathouse IV_, Jan 12 2017

%o (PARI) first(n)=my(T=thueinit('x^4+1,1),v=vector(n),t); for(k=1,n, t=thue(T,k); v[k]=sum(i=1,#t, t[i][1]>=0 && t[i][2]>=t[i][1])); v \\ _Charles R Greathouse IV_, Jan 12 2017

%Y Cf. A025455, A025446, A000161, A025426, A216284.

%Y Cf. A004831 (positions of nonzero terms).

%K nonn

%O 1,635318657

%A _V. Raman_, Sep 03 2012

%E Offset added by _Charles R Greathouse IV_, Jan 12 2017