Pythagoras and the Pythagoreans
9
•
A
prime
number is rectilinear, meaning that it can only be set out
in one dimension. The number 2 was not originally regarded as a
prime number, or even as a number at all.
•
A
composite
number is that which is measured by (has a factor)
some number. (Euclid)
•
Two numbers are
prime to one another
or
composite to one
another
if their greatest common divisor
11
is one or greater than
one, respectively. Again, as with even and odd numbers there were
numerous alternative classifications, which also failed to survive
as viable concepts.
12
For prime numbers, we have from Euclid the following theorem, whose
proof is considered by many mathematicians as the quintessentially most
elegant of all mathematical proofs.
Proposition.
There are an infinite number of primes.
Proof.
(Euclid) Suppose that there exist only finitely many primes
p
1
< p
2
< ... < p
r
. Let
N
= (
p
1
)(
p
2
)
...
(
p
r
)
>
2
. The integer
N
−
1
,
being a product of primes, has a prime divisor
p
i
in common with
N
;
so,
p
i
divides
N
−
(
N
−
1) = 1
, which is absurd!
The search for primes goes on.
Eratsothenes
(276 B.C. - 197 B.C.)
13
,
who worked in Alexandria, devised a
sieve
for determining primes.
This sieve is based on a simple concept:
Lay off all the numbers, then mark of all the multiples of 2, then
3, then 5, and so on. A prime is determined when a number is not
marked out. So, 3 is uncovered after the multiples of two are marked
out; 5 is uncovered after the multiples of two and three are marked out.
Although it is not possible to determine large primes in this fashion,
the sieve was used to determine early tables of primes. (This makes a
wonderful exercise in the discovery of primes for young students.)
11
in modern terms
12
We have
—
prime and incomposite
– ordinary primes excluding 2,
—
secondary and composite
– ordinary composite with prime factors only,
—
relatively prime
– two composite numbers but prime and incomposite to another num-
ber, e.g. 9 and 25. Actually the third category is wholly subsumed by the second.
13
Eratsothenes will be studied in somewhat more detail later, was gifted in almost every
intellectual endeavor.
His admirers call him the second Plato and some called him
beta
,
indicating that he was the second of the wise men of antiquity.