Pythagoras and the Pythagoreans
15
What about odd perfect numbers? As we have seen Euler char-
acterized all even perfect numbers. But nothing is known about odd
perfect numbers except these few facts:
•
If
n
is an odd perfect number, then it must have the form
n
=
q
2
·
p
2
k
+1
,
where
p
is prime,
q
is an odd integer and k is a nonnegative integer.
•
It has at least 8 different prime factors and at least 29 prime factors.
•
It has at least 300 decimal digits.
Truly a challenge, finding an odd perfect number, or proving there are
none will resolve the one of the last open problems considered by the
Greeks.
5
Figurate Numbers.
Numbers geometrically constructed had a particular importance to the
Pythagoreans.
Triangular numbers.
These numbers are 1, 3, 6, 10, ... . The
general form is the familiar
1 + 2 + 3 +
. . .
+
n
=
n
(
n
+ 1)
2
.
Triangular Numbers
