A perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. The first five perfect numbers are 6, 28, 496, 8128, and 33550336.

Nicomachus made several statements about perfect numbers c. 100 AD. It is common to interpret one of these (eg, see the MacTutor entry on perfect numbers) as stating that (in modern terminology)

The nth perfect number has n digits.

As we see, this is incorrect because the 5th perfect number has 8 digits. Nearly a millennium later c. 980 AD, Al-Baghdadi wrote instead:

He who affirms that there is only one perfect number in each power of 10 is wrong; there is no perfect number between ten thousand and one hundred thousand.

This is actually relatively easy to prove — assuming that all perfect numbers are even (which both Nicomachus and Al-Baghdadi also stated) and given the Euclid-Euler theorem characterizing even perfect numbers.

However, in order to demonstrate that the fifth perfect number does not have five digits, it is also strictly-speaking necessary to demonstrate that there is no odd perfect number with five digits. Who was the first to provide a sufficiently-precise analysis of odd perfect numbers to demonstrate that there does not exist a perfect number between 10,000 and 100,000, as asserted by Al-Baghdadi? Or more importantly, when?

Remark: The history of perfect numbers is littered with false claims as well as bold, still-unproven assertions. And of course the standards of proof have evolved considerably over the millennia. However, as a minimum bar for "sufficiently-precise", I would like there to be any analysis of odd perfect numbers whatsoever beyond the claim that they don't exist.



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