Existence of Number to Power of Prime Minus 1 less 1 divisible by Prime Squared/Mistake

Source Work

1986: David Wells: Curious and Interesting Numbers:

The Dictionary

$64$

1997: David Wells: Curious and Interesting Numbers (2nd ed.):

The Dictionary

$64$

Mistake

For every prime $p$, there are values of $a$ such that $a^{p - 1} = 1$ is actually divisible by ${p^2}$. The smallest such value for $p = 3$ is $8^2 = 64$: $64 - 1$ is divisible by $3^2 = 9$.

But for any prime $p$, the positive integer $1$ fits the bill trivially:

$1^{p - 1} - 1 = 0$

which is divisible by all integers, by Integer Divides Zero, and so of course in particular by $p^2$.

Hence the statement needs to be amended to:

For every prime $p$, there are values of $a$ greater than $1$ such that $a^{p - 1} = 1$ is actually divisible by ${p^2}$.

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $64$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $64$