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A075555 Smallest prime p such that p+n is a square, or 0 if no such p exists. 4


%S 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,

%T 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,

%U 11,67,89,113,7,23,5,61,3,2,37,17,79,103,257,13,31,11,29,97,71,7,181,5

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

%C 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.

%H T. D. Noe, <a href="/A075555/b075555.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SchinzelsHypothesis.html">Schinzel's Hypothesis.</a>

%e a(8) = 17 because 8 + 17 is the first square that can be made by adding a prime to 8.

%e a(16) = 0 because 16 + p cannot be x^2, since then p = x^2 - 16 = (x-4)(x+4).

%t 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]]]]

%o (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,")))

%Y Cf. A075556.

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

%K nonn

%O 1,1

%A _Amarnath Murthy_, Sep 23 2002

%E More terms from _Ralf Stephan_, Mar 28 2003

%E Edited by _Dean Hickerson_, Mar 31 2003

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Last modified August 10 01:45 EDT 2020. Contains 336363 sequences. (Running on oeis4.)