A087856 Primes of the form 16*m^2 + 25, m=1,3,5,...

41, 809, 1321, 2729, 4649, 5801, 11689, 15401, 17449, 21929, 26921, 41641, 52009, 55721, 59561, 71849, 80681, 94889, 99881, 126761, 156841, 169769, 190121, 197161, 204329, 226601, 234281, 266281, 327209, 345769, 394409, 457001, 467881, 524201

Offset

1,1

Comments

This is a special case of the theorem that all prime numbers of the form 4k+1 can be expressed as the sum of two squares. Let p = a^2 + b^2 then a=4k+1 and b = 4m. From this it follows that p = 16(m^2 + k^2) + 8k + 1. When k=1 we have p = 16m^2 + 25. If we let j=16m then the arithmetic progression j*m + 25 has an infinite number of primes by Dirichlet's theorem.

Primes of the form 64*k^2 + 64*k + 41. - Vincenzo Librandi, Dec 11 2011

References

H. Rademacher, Lectures on Elementary Number Theory, 1964, pp. 121-136.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Mathematica

Select[Table[16m^2 + 25, {m, 1, 201, 2}], PrimeQ] (* Harvey P. Dale, Jan 24 2011 *)
Select[Table[64n^2+64n+41, {n, 0, 4000}], PrimeQ] (* Vincenzo Librandi, Dec 11 2011 *)

Prog

(PARI) fourmp1(m, k=1) = { forstep(x=1, m, 2, y=16*(x^2+k^2)+8*k+1; if(isprime(y), print1(y", ")) ) }
(Magma) [a: n in [0..100] | IsPrime(a) where a is 64*n^2+64*n+41]; // Vincenzo Librandi, Dec 11 2011
(GAP) Filtered(List([1, 3..201], m->16*m^2+25), IsPrime); # Muniru A Asiru, Nov 24 2018

Crossrefs

Cf. A087857, A087861, A087862.

Sequence in context: A167737 A268993 A125551 * A010957 A346981 A299332

Adjacent sequences: A087853 A087854 A087855 * A087857 A087858 A087859

Keyword

nonn,easy

Author

Cino Hilliard, Oct 09 2003

Status

approved