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

1,1

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.

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.

Example

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

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]);

Crossrefs

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

Sequence in context: A108388 A083983 A129338 * A138535 A116671 A062338

Adjacent sequences: A378954 A378957 A378958 * A378967 A378968 A378969

Keyword

sign,look,new

Author

Robert Israel, Dec 12 2024

Status

approved