The 'Factor chain' concept

Question

I have been working on Odd Perfect Numbers for a while now. When I started to go through recent publications, I saw that a large number of papers have been proving results using the Factor Chain argument and computation. I am unable to understand the arguments that have been stated in those papers. Please explain the basics of the concept. If possible, provide links where the concept is explained in detail but in a basic way(because I am new to this idea).

Answer 1

The following MSE question contains such an example of a factor chain argument.

Basically, the factor chain argument makes use of the simple observation that $$\sigma(q^k)\sigma(n^2)=\sigma(q^k n^2)=2m=2 q^k n^2,$$ if $m = q^k n^2$ is an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.

Therefore, if we assume that a squared prime $p^2 \parallel n^2$ (where $p^2 \parallel n^2$ means $p^2 \mid n^2$ but $p^3 \nmid n^2$), then via factor/sigma chains, $\sigma(p^2)=p^2+p+1 \mid n^2$.

The primes $p$ obtained in this manner are called assumed factors, while the divisors of $\sigma(p^2)$ at each step of the factor/sigma chain approach are called consequent factors.

Note that necessarily we have $p \neq q$, since $\gcd(q,n)=1$.

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Factor-chain based algorithms are also discussed in pages $96$ to $97$ of Sorli's thesis, available online here.