Euclid · Mathematics
Euclid’s proof that there are infinitely many prime numbers (Book IX, Proposition 20) - one of the most beautiful and durable arguments ever devised, and a window into the number theory hidden inside the <em>Elements</em>.
The Elements is famous for geometry, but its seventh, eighth, and ninth books turn to number theory - the study of the whole numbers - and there Euclid proves some of his most beautiful results. The central objects are the prime numbers: a prime is a whole number greater than 1 that has no divisors except 1 and itself. The first few are 2, 3, 5, 7, 11, 13, 17, 19, 23 … Numbers that are not prime - like 6 (= 2 × 3) or 12 (= 2 × 2 × 3) - are called composite, because they can be built by multiplying smaller numbers. The primes are the indivisible atoms of arithmetic: every whole number greater than 1 is either prime itself or can be written as a product of primes, in essentially one way (this is the ‘fundamental theorem of arithmetic’, also rooted in Euclid’s number books). Just as every molecule is built from atoms, every number is built from primes. This makes the primes the most fundamental objects in arithmetic - and immediately raises a question that a child can ask but that goes to the heart of mathematics: do the primes ever run out? As you go further along the number line, the primes grow sparser. Is there a largest prime, after which there are none? Or do they continue forever?
Euclid’s proof works by a brilliant strategy: assume you have any finite list of primes, and show that there must always be a prime not on your list. Since this works for any finite list, no finite list can contain all the primes - so the primes are infinite. Here is how. Suppose someone hands you a finite collection of primes - say 2, 3, and 5. Multiply them all together: 2 × 3 × 5 = 30. Now add one: 30 + 1 = 31. Consider this new number, 31. Either it is prime, or it is not. If it is prime, then we have found a prime (31) that was not in our list - done. If it is not prime, then it must be divisible by some prime. But here is the key: it cannot be divisible by 2, or 3, or 5 - by any prime in our original list - because dividing 31 by any of those leaves a remainder of exactly 1 (we added 1 on purpose). So whatever prime does divide 31 must be a prime outside our original list. Either way - whether 31 is prime itself, or has a prime factor not in the list - we have produced a prime that was not in our collection. And since we started with a completely arbitrary finite list, the same trick produces a new prime no matter what list we begin with. Therefore no finite list can be complete: the primes go on forever.
Euclid’s proof of the infinitude of primes is celebrated not only for what it establishes but for how it establishes it. It is a model of mathematical reasoning at its purest: from the simplest possible assumptions, by a short and transparent argument, it reaches an absolutely certain conclusion about an infinite totality - that there are infinitely many primes - without ever examining infinitely many cases. This is the magic of proof: a finite argument that settles an infinite question. The English mathematician G. H. Hardy, in his famous defence of pure mathematics, singled out this very proof as an example of mathematical beauty - ‘as fresh and significant as when it was discovered - two thousand years have not written a wrinkle on it’. The proof also illustrates the technique of reductio ad absurdum in one of its common forms: assume the opposite of what you want (a complete finite list of all primes), and derive a contradiction (a prime not on the complete list). And it opened the door to number theory - the study of the whole numbers - which from Euclid through Fermat, Euler, and Gauss to the present day has been a source of the deepest and most difficult problems in mathematics, problems easy to state and fiendishly hard to solve. The primes, those simple atoms of arithmetic, hold secrets that the greatest mathematicians have spent lifetimes pursuing, and many remain unsolved to this day.
This is the opening of the lesson. The rest — the dialogue, the primary source, and the recall — is in the app.
You learned Euclid’s proof that there are infinitely many primes (Book IX, Proposition 20). Reconstruct the argument and explain why it works.
Leads to Gauss.
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