A256852 Number of ways to write prime(n) = a^2 + b^4.

1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Reinhard Zumkeller, Apr 11 2015

Keywords

• nonn

Comments

a(A049084(A028916(n))) > 0; a(A049084(A256863(n))) = 0;

Conjecture: a(n) <= 2, empirically checked for the first 10^6 primes.

The conjecture is true, because by the uniqueness part of Fermat's two-squares theorem, at most one duplicate of a^2 + b^4 can exist. Namely, if a is a square, say a = B^2, then a^2 + b^4 = A^2 + B^4 where A = b^2. - Jonathan Sondow, Oct 03 2015

Friedlander and Iwaniec proved that a(n) > 0 infinitely often. - Jonathan Sondow, Oct 05 2015

Examples

			First numbers n, such that a(n) > 0:
.   k |  n |   prime(n)                    | a(n)
. ----+----+-------------------------------+-----
.   1 |  1 |    2 = 1^2 + 1^4              |   1
.   2 |  3 |    5 = 2^2 + 1^4              |   1
.   3 |  7 |   17 = 1^2 + 2^4 = 4^2 + 1^4  |   2
.   4 | 12 |   37 = 6^2 + 1^4              |   1
.   5 | 13 |   41 = 5^2 + 2^4              |   1
.   6 | 25 |   97 = 4^2 + 3^4 = 9^2 + 2^4  |   2
.   7 | 33 |  101 = 10^2 + 1^4             |   1
.   8 | 42 |  181 = 10^2 + 3^4             |   1
.   9 | 45 |  197 = 14^2 + 1^4             |   1
.  10 | 53 |  241 = 15^2 + 2^4             |   1
.  11 | 55 |  257 = 1^2 + 4^4 = 16^2 + 1^4 |   2
.  12 | 59 |  277 = 14^2 + 3^4             |   1
.  13 | 60 |  281 = 5^2 + 4^4              |   1
.  14 | 68 |  337 = 9^2 + 4^4 = 16^2 + 3^4 |   2
.  15 | 79 |  401 = 20^2 + 1^4             |   1
.  16 | 88 |  457 = 21^2 + 2^4             |   1 .
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Art of Problem Solving, Fermat's Two Squares Theorem
• John Friedlander and Henryk Iwaniec, Using a parity-sensitive sieve to count prime values of a polynomial, PNAS, vol. 94 no. 4, pp. 1054-1058.
• Wikipedia, Friedlander-Iwaniec theorem

Crossrefs

Cf. A000040, A000290, A000583, A010052, A002645, A028916, A256863.

Programs

Haskell

a256852 n = a256852_list !! (n-1)
a256852_list = f a000040_list [] $ tail a000583_list where
   f ps'@(p:ps) us vs'@(v:vs)
     | p > v     = f ps' (v:us) vs
     | otherwise = (sum $ map (a010052 . (p -)) us) : f ps us vs'