

A145824


Lower twin primes p1 such that p11 is a square.


3



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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 = (3r1)(3r+1) not prime contradicting 3m1 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. 51 = 4 is a square.


MATHEMATICA

lst={}; Do[p=Prime@n; If[PrimeQ@(p+2)&&Sqrt[p1]==IntegerPart[Sqrt[p1]], 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(x1))
(Magma) [p: p in PrimesUpTo(2000000)  IsSquare(p1) 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



