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A069003
Smallest integer d such that n^2 + d^2 is a prime number.
19
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
OFFSET
1,3
COMMENTS
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
REFERENCES
W. Sierpinski, Elementary Theory of Numbers, 2nd English edition, revised and enlarged by A. Schinzel, Elsevier, 1988.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Eric Weisstein's World of Mathematics, Gaussian Prime.
EXAMPLE
a(5)=2 because 2 is the smallest integer d such that 5^2+d^2 is a prime number.
MAPLE
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:
map(f, [$1..1000]); # Robert Israel, Jul 06 2015
MATHEMATICA
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 *)
PROG
(PARI) a(n)=my(k); while(!isprime(n^2+k++^2), ); k \\ Charles R Greathouse IV, Mar 20 2013
CROSSREFS
Cf. A068486 (lists the prime numbers n^2 + d^2).
Cf. A239388, A239389 (record values).
Cf. A053000.
Sequence in context: A177803 A274080 A074641 * A087855 A083409 A344779
KEYWORD
easy,nonn
AUTHOR
T. D. Noe, Apr 02 2002
STATUS
approved