A266314 Least positive integer x such that n + x^7 = y^2 + z^2 for some positive integers y and z, or 0 if no such x exists.

2, 1, 2, 13, 1, 3, 7, 1, 2, 1, 3, 15, 1, 6, 11, 11, 1, 1, 2, 1, 2, 2, 7, 3, 1, 1, 3, 5, 1, 2, 7, 1, 2, 1, 2, 5, 1, 4, 3, 1, 1, 2, 2, 7, 1, 2, 7, 3, 5, 1, 2, 1, 1, 2, 11, 21, 5, 1, 3, 5, 1, 3, 3, 3, 1, 2, 2, 1, 4, 2, 3

Offset: 0

Author

Zhi-Wei Sun, Dec 27 2015

Keywords

• nonn

Comments

The general conjecture in A266277 implies that for each odd prime p and any integer m there are positive integers x, y and z such that m + x^p = y^2 + z^2.

For k = 4,6,8,... and any integer m == 6 (mod 8), there are no integers x, y and z with m + x^k = y^2 + z^2 since m + x^k with x an integer is congruent to 6 or 7 modulo 8.

As 2j+1 = (j+1)^2 - j^2, if m - z^k is odd with |m - z^k| > 1 then m + x^2 = y^2 + z^k for some positive integers x and y.

Examples

			a(2) = 2 since 2 + 2^7 = 3^2 + 11^2.
a(3) = 13 since 3 + 13^7 = 554^2 + 7902^2.
a(5) = 3 since 5 + 3^7 = 16^2 + 44^2.
a(6) = 7 since 6 + 7^7 = 30^2 + 907^2.
a(462) = 71 since 462 + 71^7 = 456497^2 + 2981062^2.
		

Links

• Zhi-Wei Sun and Chai Wah Wu, Table of n, a(n) for n = 0..10000 n = 0..2000 from Zhi-Wei Sun
• Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34 (2017), no.2, 97-120. (Cf. Section 5.)

Crossrefs

Cf. A000290, A001015, A266152, A266153, A266230, A266231, A266277.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[x=1;Label[bb];Do[If[SQ[n+x^7-y^2],Print[n," ",x];Goto[aa]],{y,1,Sqrt[(n+x^7)/2]}];x=x+1;Goto[bb];Label[aa];Continue,{n,1,70}]
(* second program: *)
xmax = 100; r[n_, x_] := Reduce[y>0 && z>0 && n+x^7 == y^2+z^2, {y, z}, Integers]; a[n_] := For[x=1, x <= xmax, x++, If[r[n, x] =!= False, Return[x]]] /. Null -> 0; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Dec 27 2015 *)