Last Digit of Perfect Numbers Alternates between 6 and 8

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

• One Perfect Number for Each Number of Digits
• Perfect Number ends in 6 or 28 preceded by Odd Digit

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): $28$