I am currently skimming through Paolo Starni's Some Extensions to Touchard’s Theorem on Odd Perfect Numbers.
I shall start citing from Statement $1$ in page $1$:
Statement $1$ (Euler). If $n$ is an odd perfect number, then $n = \pi^{\alpha} M^2$, where $\pi$ is prime, $\gcd(\pi,M)=1$ and $\pi \equiv \alpha \equiv 1 \pmod 4$.
$\cdots$
On page $2$:
Note $1$ (factors of $n$ related to $\sigma(\pi^{\alpha})$). We can find factors of $n$ considering that: $$n = \pi^{\alpha} M^2 = \dfrac{\sigma(\pi^{\alpha})}{2}\cdot\sigma(M^2)$$ and $$\sigma(\pi^{\alpha}) = (\pi + 1)(1 + {\pi}^2 + + {\pi}^4 + \ldots + {\pi}^{\alpha - 1}).$$ Since $\gcd(\pi^{\alpha},\sigma(\pi^{\alpha}))=1$, we have that $(\pi + 1)/2 \mid M^2$. In particular, if $(\pi + 1)/2$ is squarefree, then $\bigg((\pi + 1)/2\bigg)^2 \mid M^2$.
My question is about the boldened portion as I have cited Note $1$ from page $2$:
QUESTION
Is it not the case that one should change the word squarefree to prime to make the implication mathematically correct? Or is the implication with the word squarefree mathematically correct as it is?
MY ATTEMPT
For example, when $\pi = 5$, then $(\pi + 1)/2 = 3 \mid M^2$, which correctly implies that $\bigg((\pi + 1)/2\bigg)^2 = 3^2 = 9 \mid M^2$.
I am, however, unable to come up with an example with prime $\pi \equiv 1 \pmod 4$ and squarefree but non-prime $(\pi + 1)/2$.
For an example of a prime $\pi \equiv 1 \pmod 4$ with squarefree but non-prime $(\pi + 1)/2$, just take $\pi = 29$, where $(\pi + 1)/2 = 15$.
I am therefore highly doubtful about the mathematical accuracy of using squarefree instead of prime in the implication indicated in the boldened portion above.
