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A244290
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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.
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1
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MAPLE
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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:
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PROG
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(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)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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