A075555 Smallest prime p such that p+n is a square, or 0 if no such p exists.

3, 2, 13, 5, 11, 3, 2, 17, 7, 71, 5, 13, 3, 2, 181, 0, 19, 7, 17, 5, 43, 3, 2, 97, 11, 23, 37, 53, 7, 19, 5, 17, 3, 2, 29, 13, 107, 11, 61, 41, 23, 7, 101, 5, 19, 3, 2, 73, 0, 31, 13, 29, 11, 67, 89, 113, 7, 23, 5, 61, 3, 2, 37, 17, 79, 103, 257, 13, 31, 11, 29, 97, 71, 7, 181, 5

Offset: 1

Author

Amarnath Murthy, Sep 23 2002

Keywords

• nonn

Comments

If n=A047845(i)^2 for some i, i.e. if n has the form ((k-1)/2)^2 with k odd but not prime, then a(n)=0. It is conjectured that these are the only values of n for which a(n)=0; this would follow from Schinzel's hypothesis.

Examples

			a(8) = 17 because 8 + 17 is the first square that can be made by adding a prime to 8.
a(16) = 0 because 16 + p cannot be x^2, since then p = x^2 - 16 = (x-4)(x+4).
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Schinzel's Hypothesis.

Crossrefs

Cf. A075556.

a(n) = A105016(n)^2 - n, if a(n) exists.

Programs

Mathematica

a[n_] := If[IntegerQ[s=Sqrt[n]]&&!PrimeQ[2s+1], 0, For[x=Ceiling[s], True, x++, If[PrimeQ[x^2-n], Return[x^2-n]]]]

PARI

for(n=1,100,f=0:forprime(p=2,10^7,if(issquare(p+n),f=p:break)): if(f,print1(f","),print1("0,")))

Extensions

More terms from Ralf Stephan, Mar 28 2003

Edited by Dean Hickerson, Mar 31 2003