A145824 Lower twin primes p1 such that p1-1 is a square.

5, 17, 101, 197, 5477, 8837, 16901, 17957, 21317, 25601, 52901, 65537, 106277, 115601, 122501, 164837, 184901, 193601, 220901, 341057, 401957, 470597, 490001, 495617, 614657, 739601, 846401, 972197, 1110917, 1144901, 1336337, 1464101

Offset

1,1

Comments

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.

Conjecture: There exist an infinite number of primes of this form.

a(n) = A080149(n)^2 + 1. - Zak Seidov, Oct 21 2008

Links

Zak Seidov, Table of n, a(n) for n=1..4663, a(n)<10^12

Example

p1 = 5 is a lower twin prime. 5-1 = 4 is a square.

Mathematica

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 *)

Prog

(PARI) g(n) = for(x=1, n, y=twinl(x)-1; if(issquare(y), print1(y+1", ")))
twinl(n) = local(c, x); c=0; x=1; while(c<n, if(ispseudoprime(prime(x)+2), c++);
x++; ); return(prime(x-1))
(Magma) [p: p in PrimesUpTo(2000000) | IsSquare(p-1) and IsPrime(p+2)]; // Vincenzo Librandi, Nov 08 2014

Crossrefs

Cf. A080149. - Zak Seidov, Oct 21 2008

Subsequence of A002496 (Primes of form n^2 + 1). - Zak Seidov, Nov 25 2011

Sequence in context: A287842 A254759 A139390 * A076516 A145986 A200992

Adjacent sequences: A145821 A145822 A145823 * A145825 A145826 A145827

Keyword

nonn

Author

Cino Hilliard, Oct 20 2008

Extensions

More terms from Zak Seidov, Oct 21 2008

Status

approved