A256863 -id:A256863 - cp's OEIS frontend

A028916 Friedlander-Iwaniec primes: Primes of form a^2 + b^4.

2, 5, 17, 37, 41, 97, 101, 137, 181, 197, 241, 257, 277, 281, 337, 401, 457, 577, 617, 641, 661, 677, 757, 769, 821, 857, 881, 977, 1097, 1109, 1201, 1217, 1237, 1297, 1301, 1321, 1409, 1481, 1601, 1657, 1697, 1777, 2017, 2069, 2137, 2281, 2389, 2417, 2437

Offset: 1

Warut Roonguthai

nonn

John Friedlander and Henryk Iwaniec proved that there are infinitely many such primes.
A256852(A049084(a(n))) > 0. - Reinhard Zumkeller, Apr 11 2015
Primes in A111925. - Robert Israel, Oct 02 2015
Its intersection with A185086 is A262340, by the uniqueness part of Fermat's two-squares theorem. - Jonathan Sondow, Oct 05 2015
Cunningham calls these semi-quartan primes. - Charles R Greathouse IV, Aug 21 2017
Primes of the form (x^2 + y^2)/2, where x > y > 0, such that (x-y)/2 or (x+y)/2 is square. - Thomas Ordowski, Dec 04 2017
Named after the Canadian mathematician John Benjamin Friedlander (b. 1941) and the Polish-American mathematician Henryk Iwaniec (b. 1947). - Amiram Eldar, Jun 19 2021

			2 = 1^2 + 1^4.
5 = 2^2 + 1^4.
17 = 4^2 + 1^4 = 1^2 + 2^4.

T. D. Noe, Table of n, a(n) for n = 1..10000
Art of Problem Solving, Fermat's Two Squares Theorem.
A. J. C. Cunningham, High quartan factorisations and primes, Messenger of Mathematics, Vol. 36 (1907), pp. 145-174.
John Friedlander and Henryk Iwaniec, Using a parity-sensitive sieve to count prime values of a polynomial, Proc. Nat. Acad. Sci., Vol. 94 (1997), pp. 1054-1058.
J. Friedlander and H. Iwaniec, The polynomial x^2 + y^4 captures its primes, arXiv:math/9811185 [math.NT], 1998; Ann. of Math. 148 (1998), 945-1040.
Charles R Greathouse IV, Tables of special primes.
Wikipedia, Friedlander-Iwaniec theorem.

Cf. A078523, A111925.
Cf. A000290,  A000583, A000040, A256852, A256863 (complement), A002645 (subsequence), subsequence of A247857.
Primes of form n^2 + b^4, b fixed: A002496 (b = 1), A243451 (b = 2), A256775 (b = 3), A256776 (b = 4), A256777 (b = 5), A256834 (b = 6), A256835 (b = 7), A256836 (b = 8), A256837 (b = 9), A256838 (b = 10), A256839 (b = 11), A256840 (b = 12), A256841 (b = 13).

a028916 n = a028916_list !! (n-1)
a028916_list = map a000040 $ filter ((> 0) . a256852) [1..]
-- Reinhard Zumkeller, Apr 11 2015

N:= 10^5: # to get all terms <= N
S:= {seq(seq(a^2+b^4, a = 1 .. floor((N-b^4)^(1/2))),b=1..floor(N^(1/4)))}:
sort(convert(select(isprime,S),list)); # Robert Israel, Oct 02 2015

nn = 10000; t = {}; Do[n = a^2 + b^4; If[n <= nn && PrimeQ[n], AppendTo[t, n]], {a, Sqrt[nn]}, {b, nn^(1/4)}]; Union[t] (* T. D. Noe, Aug 06 2012 *)

list(lim)=my(v=List([2]),t);for(a=1,sqrt(lim\=1),forstep(b=a%2+1, sqrtint(sqrtint(lim-a^2)), 2, t=a^2+b^4;if(isprime(t),listput(v,t)))); vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Jun 12 2013

Title expanded by Jonathan Sondow, Oct 02 2015

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

Original entry on oeis.org

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

Reinhard Zumkeller, Apr 11 2015

nonn

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

			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 .

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

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

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'

cp's OEIS Frontend

A028916 Friedlander-Iwaniec primes: Primes of form a^2 + b^4.

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