|
| |
|
|
A087856
|
|
Primes of the form 16m^2+25, m=1,3,5, ...
|
|
1
| |
|
|
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
(list; graph; refs; listen; history; internal format)
|
|
|
|
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=4n+1 and b = 4m. From this it follows that p = 16(m^2+n^2) + 8n +1. When n=1 we have p=16m^2 + 25. If we let k=16m then the arithmetic progression km + 25 has an infinite number of primes from Dirichlet's theorem.
Primes of the form 64*n^2+64*n+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] (* From 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, n) = { forstep(x=1, m, 2, y=16*(x^2+n^2)+8*n+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
|
|
|
CROSSREFS
| Cf. A087857, A087861, A087862.
Sequence in context: A200914 A167737 A125551 * A010957 A161662 A162178
Adjacent sequences: A087853 A087854 A087855 * A087857 A087858 A087859
|
|
|
KEYWORD
| nonn,easy
|
|
|
AUTHOR
| Cino Hilliard (hillcino368(AT)gmail.com), Oct 09 2003
|
| |
|
|
