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A351312 Least positive integer m such that m^4*n = x^4 + y^4 - z^2 for some nonnegative integers x,y,z with z <= m^2*n.

%I A351312 #22 Feb 22 2022 09:05:03
%S A351312 1,1,1,2,20,2,2,1,1,24,6,2,1,1,15,1,1,1,1,20,2,2,2,1,5,2,2,4,1,17,1,1,
%T A351312 1,1,2,2,2,2,6,2,1,1,2,2,13,1,1,1,1,1,1,4,2,2,10,2,1,1,2,10,1,1,1,2,
%U A351312 10,1,1,2,2,20,6,1,1,1,12,6,1,1,1,4,1,1,1,2,6,2,2,1,1,5,2,2,1,1,1,2,1,1,1,4,2
%N A351312 Least positive integer m such that m^4*n = x^4 + y^4 - z^2 for some nonnegative integers x,y,z with z <= m^2*n.
%C A351312 Conjecture: Any integer m can be written as x^4 + y^4 - z^2, where x,y,z are rational numbers with z <= |m|.
%C A351312 This implies the existence of a(n) for all n >= 0. As a/b = (a*b^3)/b^4 for any integer a and nonzero integer b, the conjecture also implies that any rational number can be written as x^4 + y^4 - z^2 with x,y,z rational numbers.
%H A351312 Chai Wah Wu, <a href="/A351312/b351312.txt">Table of n, a(n) for n = 0..2648</a> (terms 0..200 from Zhi-Wei Sun)
%H A351312 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 A351312 a(4) = 20 with 20^4*4 = 15^4 + 28^4 - 159^2 and 159 < 20^2*4.
%e A351312 a(9) = 24 with 24^4*9 = 20^4 + 45^4 - 1129^2 and 1129 < 24^2*9.
%e A351312 a(164) = 30 with 30^4*164 = 66^4 + 185^4 - 32519^2 and 32519 < 30^2*164.
%e A351312 From _Chai Wah Wu_, Feb 21 2022: (Start)
%e A351312 a(244) = 50 with 50^4*244 = 455^4 + 504^4 - 325359^2 and 325359 < 50^2*244.
%e A351312 a(329) = 46 with 46^4*329 = 90^4 + 195^4 - 6199^2 and 6199 < 46^2*329.
%e A351312 a(414) = 21 with 21^4*414 = 135^4 + 415^4 - 172954^2 and 172954 < 21^2*414.
%e A351312 a(554) = 74 with 74^4*554 = 475^4 + 710^4 - 537039^2 and 537039 < 74^2*554.
%e A351312 (End)
%t A351312 QQ[n_]:=QQ[n]=IntegerQ[n^(1/4)];
%t A351312 tab={};Do[m=1; Label[bb]; k=m^4; Do[If[QQ[k*n+z^2-x^4],
%t A351312 tab=Append[tab,m];Goto[aa]],{z,0,m^2*n},{x,0,((k*n+z^2)/2)^(1/4)}]; m=m+1; Goto[bb];Label[aa],{n,0,100}];Print[tab]
%Y A351312 Cf. A000290, A000583, A351306.
%K A351312 nonn
%O A351312 0,4
%A A351312 _Zhi-Wei Sun_, Feb 06 2022