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A221981
Primes q = 4*p+1, where p == 2 (mod 5) is also prime.
4
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
1,1
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
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
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.
FORMULA
a(n) = 4*A221982(n) + 1.
EXAMPLE
29 is a member because 29 = 4*7 + 1 and 7 == 2 (mod 5) are prime.
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] &]
PROG
(PARI) is(n)=n%20==9 && isprime(n) && isprime(n\4) \\ Charles R Greathouse IV, Apr 18 2013
KEYWORD
nonn
AUTHOR
Jonathan Sondow, Feb 02 2013
STATUS
approved