A070756 Numbers m such that the Diophantine equation x^4+y^5 = m^4 has solutions.

30, 85, 195, 240, 525, 700, 960, 1020, 1170, 1845, 2720, 3045, 3120, 4026, 4818, 5005, 5655, 6240, 7290, 7680, 7735, 7770, 9717, 11349, 11570, 12432, 13560, 15015, 16240, 16695, 16800, 17476, 20655, 22185, 22400, 22605, 23730, 27768, 30720, 32120, 32640, 32760, 34390, 37440, 40885, 42105, 43095

Offset

1,1

Comments

From Robert Israel, Oct 29 2019: (Start)

The first number that occurs in more than one way is 1845, as 205^4 + 410^5 = 1476^4 + 369^5 = 1845^4.

If m is a term then so is x^5*m for any x.

Includes s*(s^4-t^4)/u^5 if s>t>0 and s^4 == t^4 (mod u^5), as x = t*(s^4-t^4)/u^5, y = (s^4-t^4)/u^4, m = s*(s^4-t^4)/u^5 is a solution. (End)

An infinite subfamily is given by m = k*(k+1)*(k^2+1)/4 for k != 2 (mod 4). Corresponding solutions x = k*(k-1)*(k^2+1)/4 and y = k*(k^2+1)/2. - Rayhan Ahmed, Apr 18 2026

A generalized infinite subfamily: m = n*(n^4 - d^4) with x = d*(n^4 - d^4) and y = n^4 - d^4, where n > d >= 1. The case d = 1 yields A061167(n) for n >= 2. - Rayhan Ahmed, May 05 2026

Links

Chai Wah Wu, Table of n, a(n) for n = 1..4069 (terms 1..100 from Robert Israel)

Maple

filter:= proc(n) local n4, y, s;
  n4:= n^4;
  for y from ceil((n4-(n-1)^4)^(1/5)) do
    s:= n4-y^5;
    if s <= 0 then return false fi;
    if issqr(s) and issqr(sqrt(s)) then return true fi
  od
end proc:
select(filter, [$1..10000]); # Robert Israel, Oct 29 2019

Mathematica

filterQ[n_] := Module[{n4 = n^4, y, s}, For[y = Ceiling[(n4 - (n-1)^4 )^(1/5)], True, y++, s = n4 - y^5; If[s <= 0, Return[False]]; If[ IntegerQ[Sqrt[s]] && IntegerQ[Sqrt[Sqrt[s]]], Return[True]]]];
Reap[Do[If[filterQ[n], Print[n]; Sow[n]], {n, 10000}]][[2, 1]] (* Jean-François Alcover, Aug 16 2020, after Maple *)

Prog

(PARI) for(n=0, 500, if(sum(i=1, n, sum(j=1, i, if(i^4+j^5-n^4, 0, 1)))>0, print1(n, ", ")))

Crossrefs

Cf. A061167.

Sequence in context: A326838 A098996 A130862 * A058903 A254474 A103906

Adjacent sequences: A070753 A070754 A070755 * A070757 A070758 A070759

Keyword

nonn

Author

Benoit Cloitre, May 14 2002

Extensions

More terms from John W. Layman, Nov 25 2002

More terms from Robert Israel, Oct 29 2019

Status

approved