(I apologize in advance if this question is unsuitable for MO. If so, please let me know and I will migrate it to MSE.)

Let $\sigma(M)$ be the sum of the divisors of the positive integer $M$. For example, $\sigma(6)=1+2+3+6=12$.

A number $N$ is called *perfect* if $\sigma(N)=2N$.

Euler proved that an *odd* perfect number $N$ must have the form $N=q^k n^2$ where $q$ is prime with $q \equiv k \equiv 1 \pmod 4$. We call $q$ the *Euler prime* of $N$.

Here are my questions:

(1)Who is credited with making the conjecture that $q$ is the largest prime factor of $N$?

(2)Additionally, to whom should I attribute the conjecture that $k=1$? Should it be Descartes, Frenicle, or Sorli?

(**Added September 28 2016** For **(2)**: In an edit to this MO post, it is stated that (per Beasley), "Dris $\ldots$ refers to Descartes’ and Frenicle’s claim (that $k=1$) as Sorli’s conjecture; Dickson has documented Descartes’s conjecture as occurring in a letter to Marin Mersenne in 1638, with Frenicle’s subsequent observation occurring in 1657". As commented by Gerry, one would need to double-check Dickson's *History of Number Theory* to verify Beasley's statements.)