Chapter 2. Arithmetic in Euclid’s Elements
We tend to think of Euclid’s Elements as a compendium of geometry, but, as we have
already noted, Books 7, 8 and 9 are devoted to elementary number theory. We will give
some indication of key ideas in these books, as they remain relevant to this day.
Book 7 begins with 22 definitions. We will not give all of them as some are not
especially interesting, but, in order to appreciate what they say, we must recall the subtle
distinction between number and magnitude that existed in Greek mathematics. As far as
these definitions are concerned, the word number means positive whole number or integer.
Furthermore, there is no number zero in Greek mathematics and the number 1 (called
here the unit) is not considered to be a number.
1. An unit is that by virtue of which each of the things that exist is called one.
2. A number is a multitude composed of units.
3. A number is a part of a number, the less of the greater, when it measures the
greater.
5. The greater number is a multiple of the less when it is measured by the less.
11. A prime number is that which is measured by an unit alone.
12. Numbers prime to one another are those which are measured by an unit alone as
a common measure.
13. A composite number is that which is measured by some number.
22. A perfect number is that which is equal to its own parts.
Here, if we say that a number measures another number, we mean that it divides
that number. The greatest common measure of two numbers is their gcd.
Proofs given in the arithmetic books of Euclid do not use symbols for numbers, as
we would today. The proofs are instead largely verbal or rhetorical. Euclid would often
represent a number as the length of a line segment and then argue using lengths to justify
his reasoning.
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Several of the fundamental ideas of arithmetic, such as we encounter in the first year
algebra course, are stated and proved by Euclid. We will describe a few of the highlights.
Proposition 1 of Book 7 is a means of deciding if two numbers are prime to each
other (in other words, if their gcd is 1).
Proposition 1. Two unequal numbers being set out, and the less being continually subtracted
in turn from the greater, if the number which is left never measures the one before it until
an unit is left, the original numbers will be prime to one another.
This proposition is really just a special case of Proposition 2, where the algorithm
to find the gcd of two numbers that are not prime to each other is described. We would
usually use make a single proposition suffice for both cases. Proposition 2 is a description
of what we know as Euclid’s algorithm. Euclid performs division processes by continual
subtraction.
The next twenty or so propositions are not particularly interesting from the modern
point of view, but are related to the Greek theory of ratio and proportion. Proposition 24
is more significant.
Proposition 24. If two numbers are prime to a given number, their product is also prime
to the number
Proposition 29 is also fundamental, although rather obvious to us.
Proposition 29. Any prime number is prime to any number which it does not measure.
This means that a prime number either divides another number exactly, or its gcd
with that number is 1.
Proposition 30 is also fundamental to our modern theories of congruence and finite
fields. Its proof uses Proposition 29.
Proposition 30. If two numbers by multiplying one another make some number, and any
prime number measure the product, it will also measure one of the original numbers.
This means that if a prime number divides the product of two numbers, it divides
at least one of the two numbers.
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In 1801, when he was just 24 years old, the famous German mathematician Gauss
published a lengthy treatise entitled Disquisitiones arithmeticae on what he called the
higher arithmetic. The work commenced with some elementary results in arithmetic,
including article 14, which is nothing other than this proposition. Gauss commented as
follows: “Euclid had already proved this theorem in his Elements (Book 7, No. 32).
However we do not wish to omit it because many modern authors have employed vague
computations in place of proof or have neglected the theorem completely, and because by
this very simple case we can more easily understand the nature of the method which will
be used later for solving much more difficult problems.”
Gauss’s comments suggest that, two thousand years after Euclid, many mathematicians had failed to understand the significance of Book VII, Proposition 30 (but note that
Gauss himself gave the wrong reference number for Euclid’s result!). In his article 16,
Gauss proceeded to prove what is sometimes called the fundamental theorem of arithmetic: a composite number can be resolved into prime factors in only one way. Euclid did
not mention this fairly straightforward consequence of his Proposition 30.
Proposition 32 states that a number is either a prime or is divisible by some prime.
We assume of course that the number is bigger than 1. This proposition is really just a
restatement of Proposition 31. This proposition can be used as the basis for the theorem
that a number can be factorized into its prime factors.
Book 8 generally contains less important theorems of arithmetic. It is partly inspired
by notions of geometry and deals with such concepts as square numbers and cube numbers.
Some of these themes are continued into Book 9. Probably the highlight of Book 9 is
Proposition 20: Prime numbers are more than any assigned multitude of prime numbers.
This means that there are infinitely many prime numbers. As is typical of Euclid’s
style, his proof appears to lack generality, but the fundamental idea is correct. He only
considers three primes A, B, and C and he notes that ABC + 1 is not divisible by any
of A, B or C. Since by Proposition 31 (or 32) of Book 7, ABC + 1 is divisible by some
prime, we obtain a new prime G, say, different from A, B and C. His argument ceases
at this point, but we would continue it and show that, given k primes, we can produce
another prime different from these.
There follow various propositions concerning odd and even numbers, which we would
consider fairly obvious nowadays, especially if we use algebraic notation to write our
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numbers. However, the last proposition of Book 9, namely, Proposition 36, is another
which has held the attention of mathematicians for more than two millennia. Stated in
Euclid’s own words, it is: If as many numbers as we please beginning from an unit be set
out continuously in double proportion, until the sum of all becomes prime, and if the sum
multiplied into the last make some number, the product will be perfect.
Definition 22 of Book 7 tells us what is a perfect number. From the modern point
of view, a number n > 1 is said to be perfect if the sum of all its positive integer divisors,
including 1 but excluding n itself, equals n. Thus, for example, 6 is perfect as its divisors
less than 6 are 1, 2 and 3 and 1 + 2 + 3 = 6. Similarly, 28 is perfect, as its divisors are 1,
2, 4, 7 and 14, whose sum is 28.
Euclid’s Proposition 36 provides us with a means of producing even perfect numbers.
Suppose that for some positive integer m, 2m −1 is a prime. Then the integer 2m−1 (2m −1)
is perfect. For, if we set 2m − 1 = p, where p is a prime, it is easy to see that the divisors
of n = 2m−1 p are
1, 2, 22 , · · · , 2m−1 , p, 2p, · · · , 2m−2 p.
Now
1 + 2 + 22 + · · · + 2m−1 = 2m − 1 = p
and
p + 2p + · · · + 2m−2 p = (2m−1 − 1)p
by the formula for summing a geometric progression. Thus it is clear that the sum of the
divisors of n is n in this case. We obtain the perfect numbers 6 by taking m = 2 and 28 by
taking m = 3. Further perfect numbers were described by the mathematician Nichomachus
of Gerasa, who lived around the year 100 CE. He wrote a treatise on number theory, known
in Latin as Introductio arithmeticae, of which two books have survived. Nichomachus was
influenced by the number mysticism of the Pythagoreans and the followers of Plato. The
Introductio gives the next two perfect numbers, which are 496 and 8,128. It is not clear if
the fifth largest perfect number, 212 (213 − 1) = 33, 550, 336 was known in antiquity but it
was certainly known by the 15th century CE. In 1732, the Swiss mathematician Leonhard
Euler proved that any even perfect number is necessarily one of the type described by
Euclid (of the form 2m−1 (2m − 1), with 2m − 1 a prime). The subject of perfect numbers
continues to fascinate people interested in mathematics, and not the least mysterious
aspect of the theory of perfect numbers is that no example of an odd perfect number is
known.
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