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A156620 Primes p such that p^2 - 2 is a 5-almost prime. 1


%S 1201,2999,4001,4273,5009,7151,8467,9769,10427,10937,11701,11897,

%T 12011,12113,12323,13339,13681,14087,14563,15187,15277,15809,16139,

%U 16699,17209,17383,17483,17623,18757,19051,19267,19697,20107,20129,20297

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

%C Corresponding 5-almost primes are A156621.

%C 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."

%D H. Halberstam and H. E. Richert, Sieve Methods, Academic Press, NY, 1974.

%D P. Ribenboim, The Little Book of Big Primes, Springer-Verlag, NY, 1991, p. 184.

%D G. J. Rieger, On polynomials and almost-primes, Bull. Amer. Math. Soc., 75 (1969), 100-103.

%t Select[Prime[Range[5000]],PrimeOmega[#^2-2]==5&] (* _Harvey P. Dale_, Jul 11 2014 *)

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

%Y Cf. A156621, A014614.

%K nonn

%O 1,1

%A _Rick L. Shepherd_, Feb 11 2009

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Last modified February 7 09:08 EST 2023. Contains 360115 sequences. (Running on oeis4.)