A364654 Numbers which are the sum or difference of two seventh powers.

0, 1, 2, 127, 128, 129, 256, 2059, 2186, 2187, 2188, 2315, 4374, 14197, 16256, 16383, 16384, 16385, 16512, 18571, 32768, 61741, 75938, 77997, 78124, 78125, 78126, 78253, 80312, 94509, 156250, 201811, 263552, 277749, 279808, 279935, 279936, 279937, 280064, 282123, 296320

Offset

1,3

Comments

Don Zagier's conjecture that the polynomial x^7 + 3y^7 is injective on rational numbers is equivalent to the non-existence of any term in this sequence that is exactly 3 times another term in this sequence.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Mathematics Stack Exchange, Are all bivariate polynomials of degree < 7 non-injective on rational numbers?

Bjorn Poonen, Multivariable polynomial injections on rational numbers, arXiv:0902.3961 [math.NT], 2009-2010; Acta Arith. 145 (2010), no. 2, 123-127.

Example

2059 = 3^7 - 2^7, 2315 = 3^7 + 2^7, 358061 = 6^7 + 5^7, 543607 = 7^7 - 6^7.

Maple

N:= 10^7: # for terms <= N
S:= {}:
for n from 0 while n^7 - (n-1)^7 <= N do
  for j from 0 to n do
    s:= n^7 + j^7;
    if s > N then break fi;
    S:= S union {s}
  od;
  for j from n-1 to 1 by -1 do
    s:= n^7 - j^7;
    if s > N then break fi;
    S:= S union {s}
  od
od:
sort(convert(S, list)); # Robert Israel, Jan 05 2026

Prog

(PARI) T=thueinit('z^7+1);
is(n) = (n==0) || (#thue(T, n)>0); \\ Michel Marcus, Aug 01 2023

Crossrefs

Cf. A001015, A247099.

Sequence in context: A157070 A064070 A266993 * A139904 A167414 A065381

Adjacent sequences: A364651 A364652 A364653 * A364655 A364656 A364657

Keyword

nonn

Author

Geoffrey Caveney, Jul 31 2023

Status

approved