A221981 Primes q = 4*p+1, where p == 2 (mod 5) is also prime.

29, 149, 269, 389, 509, 1109, 1229, 1949, 2309, 2909, 3989, 4349, 5189, 5309, 6269, 6389, 7109, 7949, 8069, 9749, 10589, 10709, 11069, 11549, 12149, 12269, 13229, 13829, 14549, 15629, 16229, 17189, 17789, 18269, 19949, 20789, 22109, 22229, 24029, 24989, 25349, 25469, 25589, 26189, 26309, 28109, 28229, 28949, 29669, 30029, 30869, 31469, 32069, 33149, 34589, 34949, 36269, 36629, 36749, 37589

Offset: 1

Author

Jonathan Sondow, Feb 02 2013

Keywords

• nonn

Comments

Moree (2012) says that Chebyshev observed that if q = 4p + 1 is prime, with prime p == 2 (mod 5), then 10 is a primitive root modulo q.

If the sequence is infinite, then Artin's conjecture ("every nonsquare integer n != -1 is a primitive root of infinitely many primes q") is true for n = 10.

The corresponding primes p are A221982.

The sequence is infinite under Dickson's conjecture, thus Dickson's conjecture implies Artin's conjecture for n = 10. - Charles R Greathouse IV, Apr 18 2013

Two conjectures: (a) These primes have primitive root 40; (b) if a(n)*8 + 1 is prime then it has primitive root 10. - Davide Rotondo, Dec 31 2024

Examples

			29 is a member because 29 = 4*7 + 1 and 7 == 2 (mod 5) are prime.
		

References

• P. L. Chebyshev, Theory of congruences, Elements of number theory, Chelsea, 1972, p. 306.
• Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section F9, pp. 377-380.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• P. L. Chebyshev, Theorie der Congruenzen, Mayer & Mueller, 1889, pp. 306-313.
• Pieter Moree, Artin's primitive root conjecture - a survey, arXiv:math/0412262 [math.NT], 2004, revised 2012, p. 1.
• Index entries for sequences related to Artin's conjecture
• Index entries for primes by primitive root

Crossrefs

Cf. A001913, A005596, A006883, A045380, A106849, A221982, A222008.

Programs

Maple

A221981:=n->`if`(isprime(4*n+1) and isprime(n) and n mod 5 = 2, 4*n+1, NULL): seq(A221981(n), n=1..10^4); # Wesley Ivan Hurt, Dec 11 2015

Mathematica

Select[ Prime[ Range[4000]], Mod[(# - 1)/4, 5] == 2 && PrimeQ[(# - 1)/4] &]

PARI

is(n)=n%20==9 && isprime(n) && isprime(n\4) \\ Charles R Greathouse IV, Apr 18 2013

Formula

a(n) = 4*A221982(n) + 1.

a(n) >> n log^2 n. - Charles R Greathouse IV, Dec 30 2024