A156620 Primes p such that p^2 - 2 is a 5-almost prime.

1201, 2999, 4001, 4273, 5009, 7151, 8467, 9769, 10427, 10937, 11701, 11897, 12011, 12113, 12323, 13339, 13681, 14087, 14563, 15187, 15277, 15809, 16139, 16699, 17209, 17383, 17483, 17623, 18757, 19051, 19267, 19697, 20107, 20129, 20297

Offset: 1

Author

Rick L. Shepherd, Feb 11 2009

Keywords

• nonn

Comments

Corresponding 5-almost primes are A156621.

This sequence is infinite: Ribenboim states that Rieger proved in 1969 that "there exist infinitely many primes p such that p^2 - 2 [is an element of] P_5", this being a particular case of a general theorem proved (also in 1969) by Richert: (again quoting Ribenboim) "Let f(X) be a polynomial with integral coefficients, positive leading coefficient, degree d >= 1 (and different from X). Assume that for every prime p, the number [rho](p) of solutions of f(X) = 0 (mod p) is less than p; moreover if p <= d+1 and p does not divide f(0) assume also that [rho](p) < p-1. Then, there exist infinitely many primes p such that f(p) is a (2d+1)-almost prime."

References

• H. Halberstam and H. E. Richert, Sieve Methods, Academic Press, NY, 1974.
• P. Ribenboim, The Little Book of Big Primes, Springer-Verlag, NY, 1991, p. 184.
• G. J. Rieger, On polynomials and almost-primes, Bull. Amer. Math. Soc., 75 (1969), 100-103.

Crossrefs

Cf. A156621, A014614.

Programs

Mathematica

Select[Prime[Range[5000]],PrimeOmega[#^2-2]==5&] (* Harvey P. Dale, Jul 11 2014 *)

PARI

forprime(p=2, prime(2500), if(bigomega(p^2-2)==5, print1(p,", ")))