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A069003
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Smallest integer d such that n^2 + d^2 is a prime number.
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19
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1, 1, 2, 1, 2, 1, 2, 3, 4, 1, 4, 7, 2, 1, 2, 1, 2, 5, 6, 1, 4, 5, 8, 1, 4, 1, 2, 5, 4, 11, 4, 3, 2, 5, 2, 1, 2, 3, 10, 1, 4, 5, 8, 9, 2, 5, 2, 13, 4, 7, 4, 3, 10, 1, 4, 1, 2, 3, 6, 13, 10, 3, 32, 9, 2, 1, 2, 5, 10, 3, 6, 5, 2, 1, 4, 5, 10, 7, 4, 7, 4, 3, 18, 1, 2, 9, 2, 3, 4, 1, 4, 7, 8, 1, 2, 5, 2, 3, 4, 3
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OFFSET
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1,3
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COMMENTS
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With i being the imaginary unit, n + di is the smallest Gaussian prime with real part n and a positive imaginary part. Likewise for n - di. See A002145 for Gaussian primes with imaginary part 0. - Alonso del Arte, Feb 07 2011
Conjecture: a(n) does not exceed 4*sqrt(n+1) for any positive integer n. - Zhi-Wei Sun, Apr 15 2013
Conjecture holds for the first 15*10^6 terms. - Joerg Arndt, Aug 19 2014
Infinitely many d exist such that n^2 + d^2 is prime, under Schinzel's Hypothesis H; see Sierpinski (1988), p. 221. - Jonathan Sondow, Nov 09 2015
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REFERENCES
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W. Sierpinski, Elementary Theory of Numbers, 2nd English edition, revised and enlarged by A. Schinzel, Elsevier, 1988.
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LINKS
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EXAMPLE
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a(5)=2 because 2 is the smallest integer d such that 5^2+d^2 is a prime number.
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MAPLE
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f:= proc(n) local d;
for d from 1+(n mod 2) by 2 do
if isprime(n^2+d^2) then return d fi
od
end proc:
f(1):= 1:
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MATHEMATICA
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imP4P[n_] := Module[{k = 1}, While[Not[PrimeQ[n^2 + k^2]], k++]; k]; Table[imP4P[n], {n, 50}] (* Alonso del Arte, Feb 07 2011 *)
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PROG
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CROSSREFS
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Cf. A068486 (lists the prime numbers n^2 + d^2).
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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