The Euclid-Euler theorem, sometimes called the Euler-Euclid theorem, states that an even positive integer 
is a perfect
number iff
 |
where 
is an integer and 
is a Mersenne prime.
The forward implication is Proposition IX.36 of Euclid's Elements
(Dunham 1990, p. 75; Dickson 2005, p. 3) when the implication is read from
the displayed form to perfectness. Euler proved the converse in a paper published
posthumously in 1849 (Dickson 2005, p. 19). A video exposition is given by Veritasium
(2024).
For Euclid's implication, set 
and 
. Since 
and 
are relatively prime
and the divisor function is multiplicative,
 |
so 
is a perfect number.
Conversely, write an even perfect number as 
,
where 
is odd. Multiplicativity and 
give 
. Since 
and 
are relatively prime,
divides 
.
Write 
,
so that 
.
Both 
and 
are positive divisors of 
, and therefore 
. Equality forces 
and leaves 1 and 
as the only positive divisors
of 
.
Thus 
is prime, proving the converse (McDaniel 1975).
