A222008 Primes of the form 4^k + 1 for some k > 0.

5, 17, 257, 65537

Offset: 1

Author

Jonathan Sondow, Feb 04 2013

Keywords

• nonn

Comments

Same as Fermat primes 2^(2^m) + 1 for m >= 1. See A019434 for comments, etc.

Chebyshev showed that 3 is a primitive root of all primes of the form 2^(2*k) + 1 with k > 0. 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 = 3.

As a(n) is a Fermat prime > 3, by Pépin's test a(n) has primitive root 3.

Conjecture: let p a prime number, a(n) is not congruent to p mod (p^2-3)/2. - Vincenzo Librandi, Jun 15 2014

This conjecture is false when p = a(n), but may be true for primes p != a(n). - Jonathan Sondow, Jun 15 2014

Primes p with the property that k-th smallest divisor of its squares p^2, for all 1 <= k <= tau(p^2), contains exactly k "1" digits in its binary representation. Corresponding values of squares p^2: 25, 289, 66049, 4295098369. Example: p = 257, set of divisors of p^2 = 66049 in binary representation: {1, 100000001, 10000001000000001}. Subsequence of A255401. - Jaroslav Krizek, Dec 21 2016

Examples

			4^1 + 1 = 5 is prime, so a(1) = 5. Also, 3^k == 3, 4, 2, 1 (mod 5) for k = 1, 2, 3, 4, resp., so 3 is a primitive root for a(1).
		

References

• Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966, p. 102, nr. 3.
• P. L. Chebyshev, Theory of congruences. Elements of number theory, Chelsea, 1972, p. 306.

Links

• P. L. Chebyshev, Theorie der Congruenzen, Mayer & Mueller, 1889, p. 306-313.
• R. Fueter, Über primitive Wurzeln von Primzahlen, Comment. Math. Helv., 18 (1946), 217-223, p. 217.
• Wikipedia, Pépin's test
• Index entries for sequences related to Artin's conjecture
• Index entries for primes by primitive root

Crossrefs

Cf. A005596, A019334, A019434, A090866, A221981.

Programs

Mathematica

Select[Table[4^k + 1, {k, 10^3}], PrimeQ] (* Michael De Vlieger, Dec 22 2016 *)

Formula

a(n) = A019434(n+1) for n > 0.