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Numbers n such that n^12 + 488669 is prime.
2

%I #43 Jan 12 2025 10:49:59

%S 616980,764400,933660,1051050,1730820,1758120,2467920,3093090,3164070,

%T 3461640,3696420,3890250,3923010,3950310,4051320,4075890,4196010,

%U 4286100,4324320,4337970,4389840,4556370,4561830,4720170,4793880

%N Numbers n such that n^12 + 488669 is prime.

%C When n is odd the polynomial produces even values, so it is trivially composite. When n ends in 2, 4, 6, or 8 then n^12 ends in 6, so the polynomial produces values ending in 5, again composite. Hence all the values in this sequence are divisible by 10. - _Dmitry Kamenetsky_, Nov 08 2016

%C By Fermat's Little Theorem the polynomial is divisible by 2, 3, 5, 7 and 13 when n is not divisible by these primes. Hence all the values in the sequence are divisible by the product of these primes, 2730. - _Chun Lok Yiu_, Mar 14 2022

%D Paulo Ribenboim, The Little Book of Big Primes, Springer Verlag, 1996.

%H Alois P. Heinz, <a href="/A122131/b122131.txt">Table of n, a(n) for n = 1..10000</a> (first 138 terms from Vincenzo Librandi)

%H Dan Ismailescu and Yunkyu James Lee, <a href="https://arxiv.org/abs/2501.04851">Polynomially growing integer sequences all whose terms are composite</a>, arXiv:2501.04851 [math.NT], 2025. See p. 2.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BouniakowskyConjecture.html">Bouniakowsky Conjecture</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">Prime-Generating Polynomial</a>

%t a = {}; Do[If[PrimeQ[x^12 + 488669], AppendTo[a, x]], {x, 616979, 10000000}]; a (* _Artur Jasinski_, Dec 31 2006 *)

%t Select[Range[10^7], PrimeQ[#^12 + 488669] &] (* _Vincenzo Librandi_, May 03 2014 *)

%o (Magma) [n: n in [0..10^7] | IsPrime(n^12+488669)]; // _Vincenzo Librandi_, May 03 2014

%o (PARI) is(n)=isprime(n^12+488669) \\ _Charles R Greathouse IV_, Jun 06 2017

%Y Cf. A066386, A126893, A126894, A126897.

%K nonn,easy

%O 1,1

%A _Eric W. Weisstein_, Aug 21 2006

%E Edited by _N. J. A. Sloane_, May 07 2007