A228760 Least positive integer x such that x and n*x are both differences of fourth powers.

1, 179727600, 80, 1040, 16, 2320, 4080, 236187120, 76960, 240, 17680, 76960, 80, 1040, 1, 1, 15, 65520, 4851120, 224991600, 100880, 1728480, 27120, 1389920, 19578624, 1048560, 240, 2986560, 80, 80, 2465, 11232975, 65, 16, 80, 2320, 12240, 707200, 16, 6560

Offset: 1

Author

Robert Israel, Sep 02 2013

Keywords

• nonn

Comments

It's not obvious that a(n) exists for all n.

a(967) > 8*10^15 (if it exists). - Donovan Johnson, Sep 04 2013

Examples

			For n = 3, 80 = 3^4 - 1^4 and 3*80 = 4^4 - 2^4.
		

References

• A. Choudhry, Indian J. pure appl. Math. 26(11) (1995), 1057-1061

Links

• Robert Israel and Donovan Johnson, Table of n, a(n) for n = 1..966 (first 205 terms from Robert Israel)
• Tito Piezas II, Is the quartic diophantine equation a^4+n*b^4 = c^4+n*d^4 solvable for any integer n?

Crossrefs

Cf. A152044.

Programs

Maple

T:= 10^12; N:= 100;  # to get solutions with n*a(n)<=T and n <= N
cmax := floor(fsolve('c'^4 - ('c'-1)^4 = T));
S:= {seq(seq(c^4 - a^4, a = ceil((max(0,c^4 - T))^(1/4))..c-1),c=1..cmax)}:
for n from 1 to N do
  B:= S intersect map(`*`,S,n);
  if B <> {} then
    A[n]:= min(B)/n;
    printf("a[%d] = %d\n",n,A[n]);
  end if
end do:  # Robert Israel, Sep 02 2013