Definition·D031
Prime number
A number above one whose only divisors are one and itself.
In words
p is prime when p is greater than one and no number divides p except one and itself.
Rests onno axioms yet
Never needed: F02 · F03 · F04 · F05 · F06 · F08 · F09 · F10 · F11 · F12 · F13 · F14 · A01 · A02 · A03 · A04 · A05 · A06 · A07 · A08 (computed from the citation graph, not asserted).
Remarks
The first few:
. That
is prime: its divisors are bounded by
(L19 (vii)), leaving only
and
as candidates (
divides only
). Excluding
from the primes is a deliberate normalization:
is a unit (it divides everything), and admitting it would wreck every uniqueness statement about factorizations. Primes are the multiplicative atoms of
: every number above
has a prime divisor (L20), and there are infinitely many primes (T11).
Used by
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