A244290 Smallest prime a(n) = x^2 + y^2 such that c^2 + d^2 = A002313(n) and c*x + d*y = 1, where c,d,x,y are integers.

5, 2, 2, 53, 5, 173, 2, 17, 2, 29, 13, 5, 1697, 53, 2, 73, 13, 5, 37, 2, 389, 733, 2753, 89, 17, 1093, 773, 13, 397, 1789, 2, 41, 821, 53, 5, 29, 193, 281, 6257, 173, 2, 149, 593, 701, 5, 1289, 157, 5, 7993, 13, 2213, 449, 877, 2, 61, 37, 389, 17, 5, 24061

Offset: 1

Author

Thomas Ordowski, Jun 27 2014

Keywords

• nonn

Comments

Let c^2 + d^2 = p be a prime, A002313(n). Then x^2 + y^2 = q is the smallest prime, a(n), such that cx + dy = 1 (Bézout's identity), where c,d,x,y are integers. We have pq = m^2 + 1 at m = cy - dx.

a(n) is the smallest prime q such that q*A002313(n)-1 is a square. - Thomas Ordowski, Sep 13 2015

Conjecture: a(n) < A002313(n)^2 for n > 1. - Thomas Ordowski, Dec 28 2017

Examples

			For prime 2 = 1^2 + 1^2 is 1*2 + 1*(-1) = 1 and 2^2 + (-1)^2 = 5 is prime, so a(1) = 5. For A002313(2) = 5 is vice versa so a(2) = 2.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..2910

Crossrefs

Cf. A002313, A002330, A002331.

Programs

Maple

N:= 10^6: # to get all a(n) before the first one > N
P:= select(isprime, [2,seq(4*i+1, i=1..floor((N-1)/4))]):
f:= proc(p) local i;
  for i from 1 to nops(P) do
   if issqr(p*P[i]-1) then return P[i] fi
od:
  -1
end proc:
for i from 1 to nops(P) do
  v:= f(P[i]);
if v = -1 then break fi;
A[i]:= v;
od:
seq(A[j],j=1..i-1); # Robert Israel, Sep 13 2015

PARI

\\ cs should contain terms from A002330
\\ ds should contain terms from A002331
a244290(cs, ds) = {
  vector(#cs, i,
    c=cs[i]; d=ds[i]; [u,v]=gcdext(c, d);
    x=u; y=v; while(!isprime(x^2+y^2), x+=d; y-=c); e=x^2+y^2;
    x=u; y=v; while(!isprime(x^2+y^2), x-=d; y+=c); f=x^2+y^2;
    min(e, f)
  )
} \\ Colin Barker, Jul 06 2014

Extensions

More terms from Colin Barker, Jul 06 2014