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%I #21 Sep 08 2022 08:45:38
%S 5,17,101,197,5477,8837,16901,17957,21317,25601,52901,65537,106277,
%T 115601,122501,164837,184901,193601,220901,341057,401957,470597,
%U 490001,495617,614657,739601,846401,972197,1110917,1144901,1336337,1464101
%N Lower twin primes p1 such that p1-1 is a square.
%C 3 is the only lower twin prime for which p1+1 is a square. This follows from the fact that lower twin primes > 3 are of the form 3n+2 and adding 1 we get a number of the form 3m. Then 3m = k^2 implies k = 3r and 3m = 9r^2. Subtracting 1 we have 3m = (3r-1)(3r+1) not prime contradicting 3m-1 is prime.
%C Conjecture: There exist an infinite number of primes of this form.
%C a(n) = A080149(n)^2 + 1. - _Zak Seidov_, Oct 21 2008
%H Zak Seidov, <a href="/A145824/b145824.txt">Table of n, a(n) for n=1..4663, a(n)<10^12</a>
%e p1 = 5 is a lower twin prime. 5-1 = 4 is a square.
%t lst={}; Do[p=Prime@n; If[PrimeQ@(p+2)&&Sqrt[p-1]==IntegerPart[Sqrt[p-1]],AppendTo[lst,p]],{n,9!}]; lst (* _Vladimir Joseph Stephan Orlovsky_, Aug 11 2009 *)
%o (PARI) g(n) = for(x=1,n,y=twinl(x)-1;if(issquare(y),print1(y+1",")))
%o twinl(n) = local(c, x); c=0;x=1;while(c<n,if(ispseudoprime(prime(x)+2),c++);
%o x++;);return(prime(x-1))
%o (Magma) [p: p in PrimesUpTo(2000000) | IsSquare(p-1) and IsPrime(p+2)]; // _Vincenzo Librandi_, Nov 08 2014
%Y Cf. A080149. - _Zak Seidov_, Oct 21 2008
%Y Subsequence of A002496 (Primes of form n^2 + 1). - _Zak Seidov_, Nov 25 2011
%K nonn
%O 1,1
%A _Cino Hilliard_, Oct 20 2008
%E More terms from _Zak Seidov_, Oct 21 2008
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