Pythagoras and the Pythagoreans
10
It is known that there is an infinite number of primes, but there
is no way to find them. For example, it was only at the end of the
19
th
century that results were obtained that describe the asymptotic density
of the primes among the integers. They are relatively sparce as the
following formula
The number of primes
≤
n
∼
n
ln
n
shows.
14
Called the Prime Number Theorem, this celebrated results
was not even conjectured in its correct form until the late
18
th
century
and its proof uses mathematical machinery well beyond the scope of
the entirety of ancient Greek mathematical knowledge. The history of
this theorem is interesting in its own right and we will consider it in a
later chapter. For now we continue with the Pythagorean story.
The pair of numbers
a
and
b
are called
amicable
or
friendly
if
the divisors of
a
sum to
b
and if the divisors of
b
sum to
a
. The pair
220 and 284, were known to the Greeks. Iamblichus (C.300 -C.350
CE) attributes this discovery to Pythagoras by way of the anecdote of
Pythagoras upon being asked ‘what is a friend’ answered ‘
Alter ego‘
,
and on this thought applied the term directly to numbers pairs such as
220 and 284. Among other things it is not known if there is infinite set
of amicable pairs. Example: All primes are deficient. More interesting
that amicable numbers are perfect numbers, those numbers amicable to
themselves. Mathematically, a number
n
is
perfect
if the sum of its
divisors is itself.
Examples: ( 6, 28, 496, 8128, ...)
6 = 1 + 2 + 3
28 = 1 + 2 + 4 + 7 + 14
496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248
There are no direct references to the Pythagorean study of these
numbers, but in the comments on the Pythagorean study of amicable
numbers, they were almost certainly studied as well. In Euclid, we find
the following proposition.
Theorem
. (Euclid) If
2
p
−
1
is prime, then
(2
p
−
1)2
p
−
1
is perfect.
Proof.
The proof is straight forward. Suppose
2
p
−
1
is prime. We
identify all the factors of
(2
p
−
1)2
p
−
1
. They are
14
This asymptotic result if also expressed as follows. Let
P
(
n
) = The number of primes
≤
n
. Then lim
n
→∞
P
(
n
)
/
[
n
ln
n
] = 1.