Existence of Number to Power of Prime Minus 1 less 1 divisible by Prime Squared/Examples/5
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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:
- $n^{5 - 1} \equiv 1 \pmod {5^2}$
is $7$.
Proof
Only positive integers coprime to $5$ need be checked.
We have that $5^2 = 25$.
Thus:
| \(\ds 2^4\) | \(=\) | \(\, \ds 16 \, \) | \(\, \ds \equiv \, \) | \(\ds 16\) | \(\ds \pmod {25}\) | |||||||||
| \(\ds 3^4\) | \(=\) | \(\, \ds 81 \, \) | \(\, \ds \equiv \, \) | \(\ds 6\) | \(\ds \pmod {25}\) | |||||||||
| \(\ds 4^4\) | \(=\) | \(\, \ds 250 \, \) | \(\, \ds \equiv \, \) | \(\ds 6\) | \(\ds \pmod {25}\) | |||||||||
| \(\ds 6^4\) | \(=\) | \(\, \ds 1296 \, \) | \(\, \ds \equiv \, \) | \(\ds 21\) | \(\ds \pmod {25}\) | |||||||||
| \(\ds 7^4\) | \(=\) | \(\, \ds 2401 \, \) | \(\, \ds \equiv \, \) | \(\ds 1\) | \(\ds \pmod {25}\) |
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $64$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $64$