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

5, 17, 257, 65537

Offset

1,1

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

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

Table of n, a(n) for n=1..4.

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

Formula

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

Example

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).

Mathematica

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

Crossrefs

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

Sequence in context: A273948 A271657 A273999 * A274000 A093428 A274002

Adjacent sequences: A222005 A222006 A222007 * A222009 A222010 A222011

Keyword

nonn

Author

Jonathan Sondow, Feb 04 2013

Status

approved