Wilson’s theorem for prime powers
For every natural number n, let (n!¯)p denote the product
of numbers 1≤m≤n with gcd(m,p)=1.
For prime p and s∈ℕ
Proof: We pair up all factors of the product into those
numbers where and those where this is not
the case. So is congruent (modulo ) to the product of those
numbers where .
Now let and . Then
Since
we have
For , but for
Title | Wilson’s theorem for prime powers |
---|---|
Canonical name | WilsonsTheoremForPrimePowers |
Date of creation | 2013-03-22 13:22:14 |
Last modified on | 2013-03-22 13:22:14 |
Owner | Thomas Heye (1234) |
Last modified by | Thomas Heye (1234) |
Numerical id | 8 |
Author | Thomas Heye (1234) |
Entry type | Theorem |
Classification | msc 11A07 |
Classification | msc 11A41 |