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

1,2

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

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?

Example

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

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

Crossrefs

Cf. A152044.

Sequence in context: A127955 A047738 A079323 * A204529 A211239 A180464

Adjacent sequences: A228757 A228758 A228759 * A228761 A228762 A228763

Keyword

nonn

Author

Robert Israel, Sep 02 2013

Status

approved