Definition:Perfect Number/Historical Note

Historical Note on Perfect Number

The first $4$ perfect numbers:

$6, 18, 496, 8128$

were known to the ancient Greeks.

All were listed by both Nicomachus of Gerasa and Iamblichus Chalcidensis.

Nicomachus made the following conjectures:

One Perfect Number for Each Number of Digits

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

both of which are seen to be incorrect from the next few instances in the sequence:

$6, 28, 496, 8128, 33 \, 550 \, 336, 8 \, 589 \, 869 \, 056, \ldots$

A manuscript of $1456$ correctly gives the $5$th perfect number as $33 \, 550 \, 536$.

Sources

• 1919: Leonard Eugene Dickson: History of the Theory of Numbers: Volume $\text { I }$ ... (previous) ... (next): Preface
• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $28$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $28$