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Numbers k such that every prime power q with q-1 | k-1 is a prime number.
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%I #23 Oct 11 2023 04:00:05

%S 2,3,5,6,11,12,14,18,20,21,23,24,26,30,35,38,39,42,44,45,47,48,51,54,

%T 56,59,60,62,66,68,69,72,74,75,77,80,83,84,86,87,90,93,95,96,98,101,

%U 102,104,107,108,110,111,114,116,117,119,122,123,126,132,135,138,140,143,144,146,147,149,150

%N Numbers k such that every prime power q with q-1 | k-1 is a prime number.

%C A number k has this property iff every field satisfying x^k = x is a prime field.

%C Equivalently, every ring satisfying x^k = x is additively generated by idempotents.

%C These numbers have been called "simple" (see MathOverflow link).

%C The natural density of the set appears to be approximately 0.462118.

%H Martin Brandenburg, <a href="https://arxiv.org/abs/2310.05301">Equational proofs of Jacobson's Theorem</a>, arXiv:2310.05301 [math.RA], 2023.

%H MathOverflow, <a href="https://mathoverflow.net/questions/455979">Natural density of the set of simple numbers</a>

%o (SageMath)

%o def n_powers(n):

%o """Computes the prime powers q with q-1 | n-1"""

%o return [x+1 for x in divisors(n-1) if (x+1).is_prime_power()]

%o def is_simple(n):

%o """Checks if n is a simple number"""

%o return n > 1 and all(q.is_prime() for q in n_powers(n))

%o (PARI) isok(k) = if (k>1, fordiv(k-1, d, if (isprimepower(d+1) && !isprime(d+1), return(0));); return(1)); \\ _Michel Marcus_, Oct 07 2023

%K nonn

%O 1,1

%A _Martin Brandenburg_, Oct 07 2023