Euclid’s argument can be developed further to obtain
other interesting facts about prime numbers. For exam-
ple, we can prove:
There are infinitely many primes that leave a
remainder of 3 when divided by 4. (That is,
there are infinitely many primes of the form
4n+ 3.)
Again, suppose to the contrary, that the list of such
primes is finite: 3, 7, 11,…,p, and this time consider
the quantity. N= 4 ×(3 ×7 ×11 ×…×p) –1. If this
number is prime, then we have found a new prime of
the required form. If it is not, then it factors into
primes: N= p1×p2×…×pk. Notice that since Nis not
divisible by 3, none of its prime factors is equal to 3. It
also cannot be the case either that all of the prime fac-
tors pileave a remainder of 1 when divided by 4 (for
then Nwould also leave a remainder of 1). Thus at
least one of these prime factors is a prime of the form
4n+ 3 not already in our list of such primes. It must
be the case then that the list of such primes goes on
forever.
In a similar way (though it is a little more compli-
cated) one can also prove:
There are infinitely many primes of the form
6n+5.
(Use N= 2 ×3 ×5 ×…×p– 1.) P
ETER
G
USTAV
L
EJEUNE
D
IRICHLET
(1805–59) generalized these results to prove
that if aand bare any two
RELATIVELY PRIME
whole
numbers, then there are infinitely many primes of the
form an + b.
See also
FUNDAMENTAL THEOREM OF ARITHMETIC
.
Eudoxus of Cnidus (ca. 408–355
B
.
C
.
E
.) Greek Geo-
metry, Number theory, Astronomy Born in Cnidus, in
Asia Minor (now Turkey), Eudoxus is remembered as
one of the greatest mathematicians of antiquity. All of
his original work is lost, but it is known from later writ-
ers that he was responsible for the material presented in
Book V of E
UCLID
’s famous treatise T
HE
E
LEMENTS
. In
his theory of proportions, Eudoxus developed a coher-
ent theory of
REAL NUMBERS
using absolute rigor and
precision. The full importance of this sophisticated
work came to light some two millennia later, when
scholars of the 19th century attempted to resolve some
fundamental difficulties with the theory of
CALCULUS
.
They discovered that Eudoxus had already anticipated
these fundamental problems and had made significant
steps toward resolving them. Eudoxus is also remem-
bered as the first to develop a “method of exhaustion”
for computing the
AREA
of curved figures.
As a young man Eudoxus traveled to Tarentum,
now in Italy, to study number theory, geometry, and
astronomy with A
RCHYTAS OF
T
ARENTUM
, a follower
of P
YTHAGORAS
. Both men worked to solve the famous
problem of
DUPLICATING THE CUBE
and, in fact,
Eudoxus came up with his own geometric solution to
the challenge using special curved lines as an aid.
(Although the problem calls for the use of nothing
more than a compass and a straight edge, this partial
solution was nonetheless a significant achievement.)
Eudoxus studied the theory of proportions. This
blend of
GEOMETRY
and
NUMBER THEORY
calls two
lengths aand b
COMMENSURABLE
if they are each a
whole-number multiple of some smaller length t: a = mt
and b= nt. In this approach, two ratios a : b and c : d
are said to be equal if they are the same multiples of
some fundamental lengths tand s: a = mt, b = nt and c
= ms, d= ns.
For a long time it was believed that all lengths were
commensurable and hence all ratios could be com-
pared. Consequently, the Pythagorean discovery of two
incommensurable lengths, namely 1 and √
–
2, the side
length and the diagonal of a unit square, caused a crisis
in the mathematical community. As H
IPPASUS OF
M
ETAPONTUM
(ca. 470
B
.
C
.
E
.) discovered, there is no
small value tsuch that 1 = mt and √
–
2 = nt. (This is
equivalent to the statement that the number √
–
2 cannot
be written as a fraction n/m.)
Eudoxus came to resolve the crisis of comparing
ratios even if they are not commensurable by avoiding
all use of a common length t. He defined ratios a : b
and c : d to be equal if, for every possible pairs of inte-
gers nand m:
i. ma < nb if mc < nd
ii. ma = nb if mc = nd
iii. ma > nb if mc > nd
With this formulation, Eudoxus was able to compare
lines of any length, either rational or irrational, and
obviate all philosophical difficulties associated with
incommensurable quantities. Mathematician J
ULIUS
W
ILHELM
R
ICHARD
D
EDEKIND
(1831–1916) based his
172 Eudoxus of Cnidus