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Primes p such that p = q^2 + 10*r^2 where q and r are also primes.
5

%I #12 Oct 08 2024 14:53:41

%S 89,139,211,379,401,419,499,569,619,659,881,1019,1051,1091,1259,1409,

%T 1451,1459,1499,1571,1619,1699,1721,1811,1889,1931,1979,2099,2339,

%U 2459,2531,2579,2699,2939,3011,3251,3299,3371,3539,3571,3659,3761,3779,3851,4019

%N Primes p such that p = q^2 + 10*r^2 where q and r are also primes.

%C Green & Sawhney prove that this sequence is infinite. - _Charles R Greathouse IV_, Oct 08 2024

%H Colin Barker, <a href="/A260557/b260557.txt">Table of n, a(n) for n = 1..1750</a>

%H Ben Green and Mehtaab Sawhney, <a href="https://arxiv.org/abs/2410.04189">Primes of the form p^2 + nq^2</a>, arXiv preprint (2024). arXiv:2410.04189 [math.NT]

%e 419 is in the sequence because 419 = 13^2 + 10*5^2 and 419, 13 and 5 are all primes.

%t Select[#1^2 + 10 #2^2 & @@ # & /@ Tuples[Prime@ Range@ 36, 2], PrimeQ] // Sort (* _Michael De Vlieger_, Jul 29 2015 *)

%o (PARI) list(lim)=my(v=List()); lim\=1; forprime(q=2, sqrtint((lim-9)\10), my(t=10*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

%Y Cf. A260553, A260554, A260555, A260556.

%K nonn

%O 1,1

%A _Colin Barker_, Jul 29 2015