Pythagoras and the Pythagoreans
11
1
,
2
,
4
, . . . ,
2
p
−
1
,
and
1
·
(2
p
−
1
−
1)
,
2
·
(2
p
−
1
−
1)
,
4
·
(2
p
−
1
−
1)
, , . . . ,
2
p
−
2
·
(2
p
−
1
−
1)
Adding we have
15
p
−
1
X
n
=0
2
n
+ (2
p
−
1
−
1)
p
−
2
X
n
=0
2
n
= 2
p
−
1 + (2
p
−
1)(2
p
−
1
−
1)
= (2
p
−
1)2
p
−
1
and the proof is complete.
(Try,
p
= 2
,
3
,
5
, and 7 to get the numbers above.) There is just
something about the word “perfect”. The search for perfect numbers
continues to this day. By Euclid’s theorem, this means the search is for
primes of the form
(2
p
−
1)
, where
p
is a prime. The story of and search
for perfect numbers is far from over. First of all, it is not known if there
are an infinite number of perfect numbers. However, as we shall soon
see, this hasn’t been for a lack of trying. Completing this concept of
describing of numbers according to the sum of their divisors, the number
a
is classified as
abundant
or
deficient
16
according as their divisors
sums greater or less than
a
, respectively. Example: The divisors of 12
are: 6,4,3,2,1 — Their sum is 16. So, 12 is abundant. Clearly all prime
numbers, with only one divisor (namely, 1) are deficient.
In about 1736, one of history’s greatest mathematicians, Leonhard
Euler (1707 - 1783) showed that all even perfect numbers must have the
form given in Euclid’s theorem. This theorem stated below is singularly
remarkable in that the individual contributions span more than two
millenia. Even more remarkable is that Euler’s proof could have been
discovered with known methods from the time of Euclid. The proof
below is particularly elementary.
Theorem
(Euclid - Euler) An even number is perfect if and only if it
has the form
(2
p
−
1)2
p
−
1
where
2
p
−
1
is prime.
Proof.
The sufficiency has been already proved. We turn to the neces-
sity. The slight change that Euler brings to the description of perfect
numbers is that he includes the number itself as a divisor. Thus a per-
fect is one whose divisors add to twice the number. We use this new
definition below. Suppose that
m
is an even perfect number. Factor
m
15
Recall, the geometric series
P
N
n
=0
r
n
=
r
N
+1
−
1
r
−
1
. This was also well known in antiquity
and is in Euclid,
The Elements
.
16
Other terms used were
over-perfect
and
defective
respectively for these concepts.