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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. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Let c^2 + d^2 = p be a prime, A002313(n). Then x^2 + y^2 = q is smallest prime, a(n), such that cx + dy = 1 (Bezout'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

LINKS

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

EXAMPLE

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.

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

PROG

(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

CROSSREFS

Cf. A002313, A002330, A002331.

Sequence in context: A201328 A199189 A145438 * A175232 A074640 A089261

Adjacent sequences:  A244287 A244288 A244289 * A244291 A244292 A244293

KEYWORD

nonn

AUTHOR

Thomas Ordowski, Jun 27 2014

EXTENSIONS

More terms from Colin Barker, Jul 06 2014

STATUS

approved

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Last modified February 26 18:29 EST 2020. Contains 332293 sequences. (Running on oeis4.)