Odd perfect numbers

Question

In this other question, somebody mentions that in a letter to Mersenne dated November 15, 1638, Descartes showed that $D=3^2⋅7^2⋅11^2⋅13^2⋅22021=198585576189$ would be an odd perfect number if $22021$ were prime.

As far as I can see, the sum of the divisors of $D$ equals:

$$ S(D) = (1+3+3^2) \cdot (1+7+7^2) \cdot (1+11+11^2) \cdot (1+13+13^2) + (1+22021)$$

Doing some calculation, this becomes:

$$S(D) = 13 \cdot 57 \cdot 133 \cdot 183 \cdot 22022$$

That second factor has $19$ as a prime factor ($57=3 \cdot 19$ and even $133=7 \cdot 19$), but $D$ does not have $19$ as a prime factor. As far as I can judge, when a number is perfect, then the sum of its divisors is the double of that number, so it can't have any other new prime factors (except for $2$, obviously).

So, even if $22021$ were a prime number, $D$ would still not be a perfect number.

Am I making a mistake here?

Answer 1

There is no mistake (apart from assuming that $22021$ is a prime, of course), the primes $19$ and $61$ will not be in this "prime factorization" because $19^2\cdot 61=22021$ and we pretend $22021$ is already a prime, so no new prime numbers except $2$ are introduced as expected. In short $S(D)=2\cdot 3^2 \cdot 7^2 \cdot 11^2 \cdot 13^2 \cdot 22021=2D$. Of course you could just calculate both $S(D)$ and $2D$ as you did, multiply everything out and compare they are in fact equal.

For more info see also https://en.wikipedia.org/wiki/Descartes_number, it has the full factorization written out.

Answer 2

$1+13+13^2\ne 182$. It should be $183$.

Answer 3

@Dominique: I invite you to read Odd, spoof perfect factorizations by the BYU Computational Number Theory Group, led by Pace Nielsen, who is also one of the number theory experts who work on odd perfect numbers.

Here is a portion of the abstract:

"We show that the structure of odd, spoof perfect factorizations is extremely rich, and there are multiple infinite families of them. This implies that certain approaches to the odd perfect number problem that use only the multiplicative nature of the sum-of-divisors function are unworkable. On the other hand, we prove that there are only finitely many nontrivial, odd, primitive spoof perfect factorizations with a fixed number of bases."