A364654 - OEIS

%I #15 Sep 03 2023 10:25:04

%S 0,1,2,127,128,129,256,2059,2186,2187,2188,2315,4374,14197,16256,

%T 16383,16384,16385,16512,18571,32768,61741,75938,77997,78124,78125,

%U 78126,78253,80312,94509,156250,201811,263552,277749,279808,279935,279936,279937,280064,282123,296320

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

%C 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.

%H Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/4743944/are-all-bivariate-polynomials-of-degree-7-non-injective-on-rational-numbers">Are all bivariate polynomials of degree < 7 non-injective on rational numbers?</a>

%H Bjorn Poonen, <a href="https://arxiv.org/abs/0902.3961">Multivariable polynomial injections on rational numbers</a>, arXiv:0902.3961 [math.NT], 2009-2010; Acta Arith. 145 (2010), no. 2, 123-127.

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

%o (PARI) T=thueinit('z^7+1);

%o is(n) = (n==0) || (#thue(T, n)>0); \\ _Michel Marcus_, Aug 01 2023

%Y Cf. A001015, A247099.

%K nonn

%O 1,3

%A _Geoffrey Caveney_, Jul 31 2023