square (prime+square)

[Richard Guy]

Such a square would be

  (k+a)^2 = k^2 + a(2k+a)

and  a(2k+a)  can only be prime if  a = 1.

R.

On Fri, 28 Mar 2003, Ralf Stephan wrote:

> Hello,
> the oeis, with my help, came up with the following conjecture. 
> Is it trivial?
> 
> Subject: COMMENT A075555
> From: ralf at ark.in-berlin.de
> 
> %I A075555
> %S A075555 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 A075555 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 A075555 11,67,89,113,7,23,5,61,3,2,37,17,79,103,257,13,31,11,29,97,71,7,181
> %N A075555 Smallest prime p such that p+n is a square, 0 if no such p seems to exist, 1 if it has been proven.
> %o A075555 (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,")))
> %C A075555 Conjecture: a(n)=0 iff n=A047845(i)^2, i.e. there are no squares of
> the form p+k^2, p prime, 2k+1 composite.
> 
> It has been checked for p up to 10^6 and 2k+1 up to 10^3.
> 
> Thanks!
> ralf
>