2026 4 13 http://ozark.hendrix.edu/~yorgey/forest/00EG/ 00EG /~yorgey/forest/00EG/ Fundamental Theorem of Arithmetic Theorem Every positive integer is equal to a product of zero or more primes, known as its prime factorization. The product is unique up to reordering; say, if the primes are listed in order from smallest to largest.There are two things to prove: first, that it is always possible to write any positive integer as a product of primes, and second, that the product is always unique. At this point, we will only prove that the product exists. 2026413Proof We will prove via strong induction that every positive integer is equal to a product of primes. 2026413Subproof can be written as an empty product, i.e. a product of zero primes. 2026413Subproof Now let be aribitrary, and suppose that every number is equal to a product of primes. We must show this for as well. Since , it is either prime or composite. 2026413Subproof If is prime, then we are done: it is equal to a "product" of just one prime (itself). 2026413Subproof Otherwise, is composite, which means there must be positive integers such that . Since and , by the induction hypothesis we know that they are both equal to a product of primes. Hence so is : the product of two products of primes is itself a product of primes. References Context 2026 4 13 http://ozark.hendrix.edu/~yorgey/forest/00EC/ 00EC /~yorgey/forest/00EC/ Primes You're probably familiar with prime numbers and prime factorization from elementary school. Let's study these concepts formally, using the framework that we've built so far. 2026 4 14 http://ozark.hendrix.edu/~yorgey/forest/00EN/ 00EN /~yorgey/forest/00EN/ Trivial divisors Definition Notice that for any natural number we always have (since ) and (since ). and are called trivial divisors of . 2026 4 13 http://ozark.hendrix.edu/~yorgey/forest/00ED/ 00ED /~yorgey/forest/00ED/ Prime Definition An integer 1]]> is prime iff it has no nontrivial divisors. That is, 1) \land (\forall {d}: {\mathbb {N}}.\;{(1 < d < p) \to d \nmid p}).]]> 2026 4 13 http://ozark.hendrix.edu/~yorgey/forest/00EF/ 00EF /~yorgey/forest/00EF/ Example , , , and are prime, which we can easily check by trying all possibilities: is prime because there aren't even any numbers between and which could be divisors. is prime because . is not prime, because . is prime, because , , and . is not prime, because (and ). is prime, because , , , , and . 2026 4 13 http://ozark.hendrix.edu/~yorgey/forest/00EH/ 00EH /~yorgey/forest/00EH/ Composite Definition Integers 1]]> that are not prime are called composite. That is, a composite number is one which has at least one divisor with . Note that if , then by definition for some integer , which means is also a divisor of ; divisors come in pairs (except for the special case when ). 2026 4 13 http://ozark.hendrix.edu/~yorgey/forest/00EE/ 00EE /~yorgey/forest/00EE/ Remark Notice that according to the definition, and are neither prime nor composite! They are special because they are the multiplicative annihilator and identity, respectively. We can now state one of the most fundamental results in number theory, which several ancient civilizations discovered independently, and you probably learned in elementary school. 2026 4 13 http://ozark.hendrix.edu/~yorgey/forest/00EG/ 00EG /~yorgey/forest/00EG/ Fundamental Theorem of Arithmetic Theorem Every positive integer is equal to a product of zero or more primes, known as its prime factorization. The product is unique up to reordering; say, if the primes are listed in order from smallest to largest.There are two things to prove: first, that it is always possible to write any positive integer as a product of primes, and second, that the product is always unique. At this point, we will only prove that the product exists. 2026413Proof We will prove via strong induction that every positive integer is equal to a product of primes. 2026413Subproof can be written as an empty product, i.e. a product of zero primes. 2026413Subproof Now let be aribitrary, and suppose that every number is equal to a product of primes. We must show this for as well. Since , it is either prime or composite. 2026413Subproof If is prime, then we are done: it is equal to a "product" of just one prime (itself). 2026413Subproof Otherwise, is composite, which means there must be positive integers such that . Since and , by the induction hypothesis we know that they are both equal to a product of primes. Hence so is : the product of two products of primes is itself a product of primes. 2026 4 13 http://ozark.hendrix.edu/~yorgey/forest/00EI/ 00EI /~yorgey/forest/00EI/ Composites have a divisor Theorem If is composite, it has a prime divisor .2026413http://ozark.hendrix.edu/~yorgey/forest/00EJ/00EJ/~yorgey/forest/00EJ/RemarkIt’s worth spelling out how to write this in formal predicate logic: The phrase “it has a prime divisor ” actually expands into an with a conjunction of three propositions! Each of “prime”, “divisor”, and “” corresponds to a separate property of the number that is claimed to exist. 2026413Proof Suppose is composite. That means there exist such that . If both \sqrt n]]> and \sqrt n]]>, then their product would be n]]>, which is a contradiction—hence either or . Suppose (the proof is essentially the same if ). We are almost done but not quite— is a divisor of , and , but we don’t necessarily know that it is prime! However, by the Fundamental Theorem of Arithmetic, there is some prime which divides , and by transitivity of divisibility, , and also . 2026 4 13 http://ozark.hendrix.edu/~yorgey/forest/00EK/ 00EK /~yorgey/forest/00EK/ Sieve of Eratosthenes Example The sieve of Eratosthenes is an ancient Greek method for finding all the primes up to some limit; consider as an example finding all the primes up to 100. We start by circling as the first prime, then crossing out all multiples of . Since the is not crossed out, it must be prime too. We circle it and cross out all multiples of . The is already crossed out now, but is not, which means it must be prime—if it were composite it would have some smaller prime divisor, but we have already crossed out all multiples of smaller primes. So we circle and cross out all its multiples; finally we circle and cross out all its multiples. But now we can stop! We have found all the primes up to . The rest of the numbers up to are already crossed out; by the previous theorem, composite numbers up to must have a prime divisor less than or equal to , but we have already crossed out all multiples of primes less than . So all the remaining numbers must be prime and we can simply circle them. Here is another classic piece of mathematics from ancient Greece: 2026 4 13 http://ozark.hendrix.edu/~yorgey/forest/00EL/ 00EL /~yorgey/forest/00EL/ Infinitude of primes Theorem There are infinitely many primes. 2026413Proof Pick any prime , and let be one more than the product of all the primes up to , that is, By the Fundamental Theorem of Arithmetic, must be divisible by at least one prime, call it . However, notice that cannot be divisible by any of the primes up to , since , , , and so on for for each prime up to . Therefore, must be a prime larger than . But was arbitrary: since for any prime we can always find a larger one, there must be infinitely many primes. (Question to ponder: is the in this proof always a prime itself?) 2026 4 13 http://ozark.hendrix.edu/~yorgey/forest/00EM/ 00EM /~yorgey/forest/00EM/ The largest known prime So there is no largest prime; but the currently (as of April 13, 2026) largest known prime—that is, the largest number we can specifically point to and say for certain it is prime—is This number has decimal digits (pop quiz: how many binary digits does it have?), and was found by the Great Internet Mersenne Prime Search. If it were printed in a book, with, say, digits per line and lines of digits per page, it would require around pages—probably a 20-volume set of books taking up a whole shelf! Backlinks 2026 4 13 http://ozark.hendrix.edu/~yorgey/forest/00EL/ 00EL /~yorgey/forest/00EL/ Infinitude of primes Theorem There are infinitely many primes. 2026413Proof Pick any prime , and let be one more than the product of all the primes up to , that is, By the Fundamental Theorem of Arithmetic, must be divisible by at least one prime, call it . However, notice that cannot be divisible by any of the primes up to , since , , , and so on for for each prime up to . Therefore, must be a prime larger than . But was arbitrary: since for any prime we can always find a larger one, there must be infinitely many primes. (Question to ponder: is the in this proof always a prime itself?) 2026 4 13 http://ozark.hendrix.edu/~yorgey/forest/00EI/ 00EI /~yorgey/forest/00EI/ Composites have a divisor Theorem If is composite, it has a prime divisor .2026413http://ozark.hendrix.edu/~yorgey/forest/00EJ/00EJ/~yorgey/forest/00EJ/RemarkIt’s worth spelling out how to write this in formal predicate logic: The phrase “it has a prime divisor ” actually expands into an with a conjunction of three propositions! Each of “prime”, “divisor”, and “” corresponds to a separate property of the number that is claimed to exist. 2026413Proof Suppose is composite. That means there exist such that . If both \sqrt n]]> and \sqrt n]]>, then their product would be n]]>, which is a contradiction—hence either or . Suppose (the proof is essentially the same if ). We are almost done but not quite— is a divisor of , and , but we don’t necessarily know that it is prime! However, by the Fundamental Theorem of Arithmetic, there is some prime which divides , and by transitivity of divisibility, , and also . http://ozark.hendrix.edu/~yorgey/forest/00FY/ 00FY /~yorgey/forest/00FY/ Agda proof of the Fundamental Theorem of Arithmetic On Monday, I was idly brainstorming potential final projects for my Functional Programming students. Having just taught my Discrete Math students the Fundamental Theorem of Arithmetic, I wondered whether formalizing it in Agda could make a nice project. (I mean formalizing it from scratch: of course, the FTA must already in the Agda standard library.) Then I... kind of fell down the rabbit hole. Several hours and 560 lines of Agda later, I'm very close: all I have left is to prove that the result of the division algorithm is unique (which is actually kinda tricky). Well... actually, I've only formalized half of the FTA so far, namely, the part that says any number can be factored into a product of primes. I haven't yet attempted to formalize the claim that the resulting factorization is unique up to permutation. We'll see if I decide to add that. In any case, this has been a fun and pleasant exercise, requiring a bunch of facts about addition, multiplication, and inequality; well-founded induction; encoding loops from a start value up to an end value; and various other things. I've been doing everything completely from scratch/memory, without peeking at any Agda stdlib code. Once I finish it I might clean it up and turn it into a blog post. Formalizing the proof has led me to rethink a few of the wys I present things in Discrete Math. For example, I had been defining a number to be prime if "all divisors of are trivial", that is, formally, But now I actually think it is better to define it as " has no nontrivial divisors", that is, Since primality is decidable, these are constructively equivalent, but... I like the second definition better now, for reasons I can't quite put my finger on. Related Contributions