OFFSET
1,1
COMMENTS
A number k has this property iff every field satisfying x^k = x is a prime field.
Equivalently, every ring satisfying x^k = x is additively generated by idempotents.
These numbers have been called "simple" (see MathOverflow link).
The natural density of the set appears to be approximately 0.462118.
LINKS
Martin Brandenburg, Equational proofs of Jacobson's Theorem, arXiv:2310.05301 [math.RA], 2023.
MathOverflow, Natural density of the set of simple numbers
PROG
(SageMath)
def n_powers(n):
"""Computes the prime powers q with q-1 | n-1"""
return [x+1 for x in divisors(n-1) if (x+1).is_prime_power()]
def is_simple(n):
"""Checks if n is a simple number"""
return n > 1 and all(q.is_prime() for q in n_powers(n))
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Martin Brandenburg, Oct 07 2023
STATUS
approved