A242517 List of primes p for which p^n - 2 is prime for n = 1, 3, and 5.

31, 619, 2791, 4801, 15331, 33829, 40129, 63421, 69151, 98731, 127291, 142789, 143569, 149971, 151849, 176599, 184969, 201829, 210601, 225289, 231841, 243589, 250951, 271279, 273271, 277549, 280591, 392269, 405439, 441799, 472711, 510709, 530599, 568441, 578689

Offset: 1

Author

Abhiram R Devesh, May 17 2014

Keywords

• nonn
• easy

Examples

			31 is in the sequence because
p = 31 (prime),
p - 2 = 29 (prime),
p^3 - 2 = 29789 (prime), and
p^5 - 2 = 28629149 (prime).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..250 from Abhiram R Devesh)

Crossrefs

Intersection of A006512, A178251 and A154834, hence, intersection of A240126 and A154834.

Cf. A001359.

Programs

Mathematica

Select[Range[600000], PrimeQ[#] && AllTrue[#^{1, 3, 5} - 2, PrimeQ] &] (* Amiram Eldar, Apr 06 2020 *)

PARI

isok(p) = isprime(p) && isprime(p-2) && isprime(p^3-2) && isprime(p^5-2); \\ Michel Marcus, Apr 06 2020

PARI

list(lim)=my(v=List(),p=29); forprime(q=31,lim, if(q-p==2 && isprime(q^3-2) && isprime(q^5-2), listput(v,q)); p=q); Vec(v) \\ Charles R Greathouse IV, Apr 06 2020

Python

import sympy
n=2
while n>1:
    n1=n-2
    n2=((n**3)-2)
    n3=((n**5)-2)
    ##Check if n1, n2 and n3 are also primes.
    if sympy.ntheory.isprime(n1)== True and sympy.ntheory.isprime(n2)== True and sympy.ntheory.isprime(n3)== True:
        print(n, " , " , n1, " , ", n2, " , ", n3)
    n=sympy.ntheory.nextprime(n)