Existence of Number to Power of Prime Minus 1 less 1 divisible by Prime Squared
Jump to navigation
Jump to search
Theorem
Let $p$ be a prime number.
Then there exists at least one positive integer $n$ greater than $1$ such that:
- $n^{p - 1} \equiv 1 \pmod {p^2}$
Proof
\(\ds p^2\) | \(\equiv\) | \(\ds 0\) | \(\ds \pmod {p^2}\) | |||||||||||
\(\ds 1\) | \(\equiv\) | \(\ds 1\) | \(\ds \pmod {p^2}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds p^2 + 1\) | \(\equiv\) | \(\ds 0 + 1\) | \(\ds \pmod {p^2}\) | Modulo Addition is Well-Defined | |||||||||
\(\ds \) | \(\equiv\) | \(\ds 1\) | \(\ds \pmod {p^2}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {p^2 + 1}^{p - 1}\) | \(\equiv\) | \(\ds 1^{p - 1}\) | \(\ds \pmod {p^2}\) | Congruence of Powers | |||||||||
\(\ds \) | \(\equiv\) | \(\ds 1\) | \(\ds \pmod {p^2}\) |
Hence $p^2 + 1$ fulfils the conditions for the value of $n$ whose existence was required to be demonstrated.
$\blacksquare$
Examples
$p = 3$
The smallest positive integer $n$ greater than $1$ such that:
- $n^{3 - 1} \equiv 1 \pmod {3^2}$
is $8$.
$p = 5$
The smallest positive integer $n$ greater than $1$ such that:
- $n^{5 - 1} \equiv 1 \pmod {5^2}$
is $7$.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $64$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $64$