A378964 a(n) is the least prime of the form p^2 + n*q^2 where p and q are primes, or -1 if there are none.

13, 17, 31, 41, 29, 73, 37, 41, 61, 89, 53, 73, 61, 151, 109, 73, 157, 97, 101, 89, 109, 97, 101, 241, 109, 113, 157, 137, -1, 241, 149, 137, 157, 257, 149, 193, 157, 367, 181, 281, 173, 193, 181, 421, 229, 193, 197, 241, 317, 499, 229, 233, -1, 241, 229, 233, 277, 241, -1, 409, 269, 257, 277

Offset: 1

Author

Robert Israel, Dec 12 2024

Keywords

• sign
• look

Comments

Green and Sawhney (see link) proved that there are infinitely many such primes if n == 0 or 4 (mod 6).

If n is odd, p or q must be 2.

If n == 2 (mod 3), p or q must be 3.

Thus if n == 5 (mod 6), the only possible primes of this form are 9 + 4 n and 4 + 9 n.

Examples

			a(6) = 73 because 73 = 7^2 + 6 * 2^2 is the least prime of the form p^2 + 6 * q^2.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• B. Green and M. Sawhney, Primes of the form p^2 + n q^2, arXiv:2410.04189 [math.NT], 2024.

Crossrefs

Cf. A111199 (a(n) = 9+4*n).

Programs

Maple

f:= proc(n) uses priqueue;
    local pq, p, t;
    if n mod 6 = 5 then
      if isprime(3^2 + n * 2^2) then return 3^2 + n*2^2
      elif isprime(2^2 + n*3^2) then return 2^2 + n*3^2
      else return -1
    fi fi;
    initialize(pq);
    insert([-9 - 4*n,3,2],pq);
    insert([-4 - 9*n,2,3],pq);
    if n mod 3 = 2 then
      do
        t:= extract(pq);
        if isprime(-t[1]) then return -t[1] fi;
        if t[2] = 3 then p:= nextprime(t[3]); if p = 3 then p:= 5 fi; insert([-9 - n*p^2,3,p],pq) fi;
        if t[3] = 3 then p:= nextprime(t[2]); if p = 3 then p:= 5 fi; insert([-p^2 - n*9,p,3],pq) fi;
      od
    else
      do
        t:= extract(pq);
        if isprime(-t[1]) then return -t[1] fi;
        if t[3] = 2  then p:= nextprime(t[2]); insert([-p^2 - n*4  , p, 2],pq) fi;
        if t[2] = 2 or n::even then
          p:= nextprime(t[3]); insert([-t[2]^2 - n*p^2,t[2],p],pq)
        fi;
      od
    fi
end proc:
map(f, [$1..200]);