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.
Wikipedia, Schinzel's Hypothesis H.
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
KEYWORD
easy,nonn
AUTHOR
T. D. Noe, Apr 02 2002
STATUS
approved