One Perfect Number for Each Number of Digits

Conjecture

There is one perfect number for each number of digits:

$1$ digit: $6$

$2$ digits: $28$

$3$ digits: $496$

$4$ digits: $8128$

Refutation

The $5$th perfect number is $33 \, 550 \, 336$, which clearly does not have $5$ digits.

$\blacksquare$

Also see

• Last Digit of Perfect Numbers Alternates between $6$ and $8$

Historical Note

The conjecture that there is One Perfect Number for Each Number of Digits was made by Nicomachus of Gerasa in his Introduction to Arithmetic, published some time around the $2$nd century.

It was a simple extrapolation from the knowledge of the perfect numbers at the time.

Some sources suggest that Iamblichus Chalcidensis made these conjectures, but, as it is believed that Iamblichus already knew the $5$th perfect number to be $33 \, 550 \, 336$, this cannot be correct.

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $28$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $28$