A185086 Fouvry-Iwaniec primes: Primes of the form k^2 + p^2 where p is a prime.

5, 13, 29, 41, 53, 61, 73, 89, 109, 113, 137, 149, 157, 173, 193, 229, 233, 269, 281, 293, 313, 317, 349, 353, 373, 389, 397, 409, 433, 449, 461, 509, 521, 557, 569, 593, 601, 613, 617, 653, 673, 701, 733, 761, 773, 797, 809, 853, 857, 877, 929, 937, 941, 953

Offset: 1

Author

Charles R Greathouse IV, Feb 18 2011

Keywords

• nonn
• nice

Comments

Sequence is infinite, see Fouvry & Iwaniec.

Its intersection with A028916 is A262340, by the uniqueness part of Fermat's two-squares theorem. - Jonathan Sondow, Oct 05 2015

Named after the French mathematician Étienne Fouvry (b. 1953) and the Polish-American mathematician Henryk Iwaniec (b. 1947). - Amiram Eldar, Jun 20 2021

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Étienne Fouvry and Henryk Iwaniec, Gaussian primes, Acta Arithmetica, Vol. 79, No. 3 (1997), pp. 249-287.
• Lasse Grimmelt, Vinogradov's Theorem with Fouvry-Iwaniec Primes, arXiv:1809.10008 [math.NT], 2018.
• Art of Problem Solving, Fermat's Two Squares Theorem.

Crossrefs

Subsequence of A002144 and hence of A002313.

The positive terms of A240130 form a subsequence.

Cf. A010052, A001248, A000040, A028916, A262340.

Programs

Haskell

a185086 n = a185086_list !! (n-1)
a185086_list = filter (\p -> any ((== 1) . a010052) $
               map (p -) $ takeWhile (<= p) a001248_list) a000040_list
-- Reinhard Zumkeller, Mar 17 2013

Mathematica

nn = 1000; Union[Reap[Do[n = k^2 + p^2; If[n <= nn && PrimeQ[n], Sow[n]], {k, Sqrt[nn]}, {p, Prime[Range[PrimePi[Sqrt[nn]]]]}]][[2, 1]]]

PARI

is(n)=forprime(p=2,sqrtint(n),if(issquare(n-p^2),return(isprime(n))));0

PARI

list(lim)=my(v=List(),N,t);forprime(p=2,sqrt(lim), N=p^2; for(n=1,sqrt(lim-N), if(ispseudoprime(t=N+n^2), listput(v,t)))); v=vecsort(Vec(v),,8); v