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A260554
Primes p such that p = q^2 + 4*r^2 where q and r are also primes.
5
41, 61, 109, 137, 149, 157, 269, 317, 389, 397, 461, 509, 557, 653, 701, 773, 797, 857, 877, 941, 977, 997, 1013, 1061, 1181, 1277, 1453, 1493, 1613, 1637, 1697, 1733, 1877, 1949, 1973, 1997, 2141, 2237, 2309, 2333, 2357, 2477, 2693, 2837, 2909, 2957, 3373
OFFSET
1,1
COMMENTS
Green & Sawhney prove that this sequence is infinite, verifying a conjecture of Friedlander & Iwaniec. - Charles R Greathouse IV, Oct 08 2024
LINKS
John Friedlander and Henryk Iwaniec, Coordinate distribution of Gaussian primes, J. Eur. Math. Soc. (JEMS) 24 (2022), pp. 737-772. arXiv:1811.05507 [math.NT]
Ben Green and Mehtaab Sawhney, Primes of the form p^2 + nq^2, arXiv preprint (2024). arXiv:2410.04189 [math.NT]
EXAMPLE
149 is in the sequence because 149 = 7^2 + 4*5^2 and 149, 7 and 5 are all primes.
MATHEMATICA
Select[#1^2 + 4 #2^2 & @@ # & /@ Tuples[Prime@ Range@ 60, 2], PrimeQ] // Sort (* Michael De Vlieger, Jul 29 2015 *)
PROG
(PARI) list(lim)=my(v=List()); lim\=1; forprime(q=2, sqrtint((lim-9)\4), my(t=4*q^2); forprime(p=3, sqrtint(lim-t), my(r=t+p^2); if(isprime(r), listput(v, r)))); Set(v) \\ Charles R Greathouse IV, Oct 08 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Colin Barker, Jul 29 2015
STATUS
approved