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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 (list; graph; refs; listen; history; text; internal format)
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
CROSSREFS
Sequence in context: A172075 A042644 A263126 * A139997 A103565 A098117
KEYWORD
nonn
AUTHOR
Jonathan Sondow, Feb 02 2013
STATUS
approved

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Last modified April 23 08:11 EDT 2024. Contains 371905 sequences. (Running on oeis4.)