# Last Digit of Perfect Numbers Alternates between 6 and 8

Jump to navigation
Jump to search

## Conjecture

The last digit of the sequence of perfect numbers alternates between $6$ and $8$:

- $6$
- $28$
- $496$
- $8128$

## Refutation

The sequence continues:

- $33 \, 550 \, 336$
- $8 \, 589 \, 869 \, 056$

... two consecutive perfect numbers ending in $6$.

$\blacksquare$

## Also see

## Historical Note

The conjecture that there is Last Digit of Perfect Numbers Alternates between $6$ and $8$ 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 this appears to be incorrect.

## Sources

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