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Primes of the form 4n^3 + 1.
9

%I #32 Sep 08 2022 08:46:00

%S 5,109,257,1373,2917,4001,27437,62501,157217,202613,237277,296353,

%T 470597,629857,665501,1492993,1556069,1898209,2456501,2634013,3217429,

%U 3322337,4244833,5038849,5180117,6572129,10512289,11453153,12706093

%N Primes of the form 4n^3 + 1.

%C Dirichlet's theorem on primes in arithmetic progressions tells us, for example, that there are infinitely many primes of the form 4n+1. For primes represented by polynomials of degree greater than 1, the Bateman-Horn paper gives a conjecture on the density.

%H Vincenzo Librandi, <a href="/A199307/b199307.txt">Table of n, a(n) for n = 1..1000</a>

%H P. Bateman and R. A. Horn, <a href="https://doi.org/10.1090/S0025-5718-1962-0148632-7">A heuristic asymptotic formula concerning the distribution of prime numbers</a>, Mathematics of Computation, 16 (1962), 363-367.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Bateman%E2%80%93Horn_conjecture">Bateman-Horn Conjecture</a>

%t Select[Table[4n^3+1,{n,1100}],PrimeQ] (* _Vincenzo Librandi_, Aug 01 2012 *)

%o (Magma) [ a: n in [0..200] | IsPrime(a) where a is 4*n^3+1 ]; // _Vincenzo Librandi_, Nov 08 2011

%o (PARI) for(n=1,1e3,if(isprime(t=4*n^3+1),print1(t", "))) \\ _Charles R Greathouse IV_, Nov 21 2011

%Y Cf. A115104, A001912, A002496, A005574.

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_, Nov 05 2011