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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.)