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A111252 Primes p such that the difference between the closest squares surrounding p is prime.

%I A111252 #23 Feb 16 2025 08:32:58
%S A111252 2,3,5,7,11,13,29,31,37,41,43,47,67,71,73,79,83,89,97,127,131,137,139,
%T A111252 197,199,211,223,227,229,233,239,241,251,331,337,347,349,353,359,401,
%U A111252 409,419,421,431,433,439,443,449,457,461,463,467,479,541,547,557,563
%N A111252 Primes p such that the difference between the closest squares surrounding p is prime.
%C A111252 Conjecture: The number of terms in this sequence is infinite.
%C A111252 That there are infinitely many terms in this sequence would follow from the Legendre conjecture (one of the Landau problems - see the Weisstein link) that there is always a prime between n^2 and (n+1)^2. This is still an open problem. - _Max Alekseyev_, Apr 20 2006
%H A111252 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LandausProblems.html">Landau's Problems</a>
%F A111252 Let p be a prime number and r = floor(sqrt(p)). Then the closest surrounding squares of p are r^2 and (r+1)^2. So d = (r+1)^2 - r^2 = 2r+1. If d is prime then list p.
%e A111252 29 is a prime number. 5^2 and 6^2 are the closest squares surrounding 29. Now the difference 36-25 = 11 is prime so 29 is in the table.
%t A111252 Clear[f,lst,p,n]; f[n_]:=IntegerPart[Sqrt[n]]; lst={};Do[p=Prime[n];If[PrimeQ[a=(f[p]+1)^2-f[p]^2],AppendTo[lst,p]],{n,6!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Aug 05 2009 *)
%t A111252 Select[Prime[Range@103],PrimeQ[2*Floor[Sqrt[#]]+1]&] (* _Ivan N. Ianakiev_, Jul 30 2015 *)
%o A111252 (PARI) surrsqpr(n) = { local(x,y,j,r,d); forprime(x=2,n, r=floor(sqrt(x)); d=r+r+1; if(isprime(d), print1(x, ",") ) ) }
%K A111252 easy,nonn
%O A111252 1,1
%A A111252 _Cino Hilliard_, Nov 12 2005