Primes thin out as you count upward, so it is natural to wonder whether they eventually stop. You cannot settle that by listing them, since any list you finish leaves the question open. By the time Euclid’s Elements appeared, around 300 BC, several important results about primes had been proved.
By assuming the opposite and watching it fail. In Book IX of the Elements, Euclid proves there are infinitely many prime numbers, and MacTutor calls it one of the first known proofs to use the to establish a result. The same book gives the .
Greek mathematicians were interested in perfect and amicable numbers before they were interested in primes for their own sake. A equals the sum of its proper divisors, like 6 and 28, and an amicable pair like 220 and 284 each sums to the other.
By the time the Elements appeared in about 300 BC, several important results about primes had been proved. In Book IX, Euclid proves that there are infinitely many prime numbers, and this is one of the first proofs known to use the method of contradiction to establish a result.
The same work carries two more results that still matter. Euclid proves the Fundamental Theorem of Arithmetic, that every integer can be written as a product of primes in an essentially unique way, and he connects primes to perfect numbers.
Eratosthenes devised his for calculating primes about a century later, and then the history goes quiet for a long stretch. The next important developments came from Fermat at the beginning of the seventeenth century, which is nearly two thousand years of waiting.
wat. Assume the primes run out, and the assumption builds a prime that is not on your list.
wat wat
A finite check can never answer a question about every number, and this is where mathematics learned to answer one anyway.
How do we know?
The proof survives in the text of the Elements itself, and the history of what was known before and after it comes from the record of Greek mathematics assembled in this account.
What is still unknown
Attributing a proof in the Elements to Euclid himself is a convention of the field, since the work compiles results from earlier mathematicians as well.
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Where to next
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