Existence of Number to Power of Prime Minus 1 less 1 divisible by Prime Squared/Examples/3
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Example of Existence of Number to Power of Prime Minus 1 less 1 divisible by Prime Squared
The smallest positive integer $n$ greater than $1$ such that:
- $n^{3 - 1} \equiv 1 \pmod {3^2}$
is $8$.
Proof
Only positive integers coprime to $3$ need be checked.
We have that $3^2 = 9$.
Thus:
| \(\ds 2^2\) | \(=\) | \(\, \ds 4 \, \) | \(\, \ds \equiv \, \) | \(\ds 4\) | \(\ds \pmod 9\) | |||||||||
| \(\ds 4^2\) | \(=\) | \(\, \ds 16 \, \) | \(\, \ds \equiv \, \) | \(\ds 7\) | \(\ds \pmod 9\) | |||||||||
| \(\ds 5^2\) | \(=\) | \(\, \ds 25 \, \) | \(\, \ds \equiv \, \) | \(\ds 7\) | \(\ds \pmod 9\) | |||||||||
| \(\ds 7^2\) | \(=\) | \(\, \ds 49 \, \) | \(\, \ds \equiv \, \) | \(\ds 4\) | \(\ds \pmod 9\) | |||||||||
| \(\ds 8^2\) | \(=\) | \(\, \ds 64 \, \) | \(\, \ds \equiv \, \) | \(\ds 1\) | \(\ds \pmod 9\) |
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $64$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $64$