A351306 - OEIS

%I #34 Sep 26 2023 09:55:22

%S 1,1,1,10,2,2,2,4,6,4,2,2,4,8,1,1,1,1,2,2,2,2,10,2,2,2,2,10,10,2,1,1,

%T 1,2,2,2,2,8,2,2,2,2,2,10,2,2,1,1,5,1,1,4,10,10,2,2,6,10,4,4,2,4,1,3,

%U 1,1,1,10,2,2,2,2,2,2,4,2,2,2,2,1,1,1,1,2,2,2,2,4,10,2,2,4,6,6,1,1,1,1,5,2,2

%N Least positive integer m such that m^4*n = u^4 + v^4 - (x^4 + y^4) for some nonnegative integers u,v,x,y with x^4 + y^4 <= m^4*n^2.

%C Conjecture: Each n >= 0 can be written as u^4 + v^4 - (x^4 + y^4), where u,v,x,y are rational numbers with x^4 + y^4 <= n^2. In other words, a(n) exists for any nonnegative integer n.

%C A known result of R. Norrie states that any rational number can be written as u^4 + v^4 - (x^4 + y^4) with u,v,x,y rational numbers.

%H Zhi-Wei Sun, <a href="/A351306/b351306.txt">Table of n, a(n) for n = 0..800</a>

%H Tito Piezas III, <a href="https://sites.google.com/view/tpiezas/001b-assorted-identities-part-2-waring-like-problems">A Collection of Algebraic Identities, 001b: Assorted identities, Part 2 (Waring-like problems).

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/2010.05775">Sums of four rational squares with certain restrictions</a>, arXiv:2010.05775 [math.NT], 2020-2022.

%e a(3) = 10 with 10^4*3 = 8^4 + 13^4 - (4^4 + 7^4) and 4^4 + 7^4 <= 10^4*3^2.

%e a(242) = 15 with 15^4*242 = 73^4 + 153^4 - (36^4 + 154^4) and 36^4 + 154^4 <= 15^4*242^2.

%e a(248) = 28 with 28^4*248 = 95^4 + 270^4 - (52^4 + 269^4) and 52^4 + 269^4 <= 28^4*248^2.

%e a(313) = 30 with 30^4*313 = 37^4 + 128^4 - (7^4 + 64^4) and 7^4 + 64^4 <= 30^4*313^2.

%t QQ[n_]:=QQ[n]=IntegerQ[n^(1/4)];

%t tab={};Do[m=1;Label[bb];k=m^4;Do[If[QQ[k*n+x^4+y^4-z^4],tab=Append[tab,m];Goto[aa]],

%t {x,0,m*(n^2/2)^(1/4)},{y,x,(k*n^2-x^4)^(1/4)},{z,0,((k*n+x^4+y^4)/2)^(1/4)}];m=m+1;Goto[bb];Label[aa],{n,0,100}];Print[tab]

%Y Cf. A000583, A004831, A351312.

%K nonn

%O 0,4

%A _Zhi-Wei Sun_, Feb 06 2022