Is Euclid's proof of the innitude of prime numbers tautological? ZeeshanMahmud January 31, 2011 Abstract Euclid's classic proof about the innitude of prime numbers has been a standard model of reasoning in student textbooks and books of elementary number theory. It has withstood scrutiny for over 2000 years but we shall prove that despite the deceptive appearance of its analytical reasoning it is tautological in nature. We shall argue that the proof is more of an observation about the general property of a prime numbers than an expository style of natural deduction of the proof of their innitude. A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the rst man whom you meet on the street. David Hilbert 1 Introduction There exist many versions of the proof and we will inspect three such instances of them. 1.1 Ribenboim's statement of Euclid's proof Theorem. There are innitely many primes. Proof. Suppose that p1=2 < p2 = 3 < ... < pr are all of the primes. Let P = p1p2...pr+1 and let p be a prime dividing P; then p can not be any of p1, p2, ..., pr, otherwise p would divide the dierence Pp1p2...pr=1, which is impossible. So this prime p is still another prime, and p1, p2, ..., pr would not be all of the primes. 1.2 David Joyce's English translation of Euclid's theorem Theorem. Prime numbers are more than any assigned multitude of prime numbers. 1 Proof. Let A, B, and C be the assigned prime numbers. I say that there are more prime numbers than A, B, and C. Take the least number DE measured by A, B, and C. Add the unit DF to DE. Then EF is either prime or not. First, let it be prime. Then the prime numbers A, B, C, and EF have been found which are more than A, B, and C.Next, let EF not be prime. Therefore it is measured by some prime number. Let it be measured by the prime number G. I say that G is not the same with any of the numbers A, B, and C. If possible, let it be so. Now A, B, and C measure DE, therefore G also measures DE. But it also measures EF. Therefore G, being a number, measures the remainder, the unit DF, which is absurd. Therefore G is not the same with any one of the numbers A, B, and C. And by hypothesis it is prime. Therefore the prime numbers A, B, C, and G have been found which are more than the assigned multitude of A, B, and C. Therefore, prime numbers are more than any assigned multitude of prime numbers. and, 1.3 Reformulation in modern terms Theorem. There are more primes than found in any nite list of primes. Proof. Call the primes in our nite list p1, p2, ..., pr. Let P be any common multiple of these primes plus one (for example, P = p1p2...pr+1). Now P is either prime or it is not. If it is prime, then P is a prime that was not in our list. If P is not prime, then it is divisible by some prime, call it p. Notice p can not be any of p1, p2, ..., pr, otherwise p would divide 1, which is impossible. So this prime p is some prime that was not in our original list. Either way, the original list was incomplete. 1.4 Formal statement of Euclid's theorem Theorem(Euclid). There exists an innite set of prime numbers for any set of natural numbers. ` (N ∈ N → ∃j∈N(N < j ∧ ∀k∈N((j/k) ∈ N → (k = 1 ∨ k = j)))) 2 In words: There exist innitely many prime numbers: for any natural number N, there exists a prime number j greater than N. 2 Where Euclid erred We take note of Euclid's denition of prime from Book VII, Denition 11: Denition 2.1. A prime number is that which is measured by a unit alone. It is understood that Euclid had a very dierent notion of innite in terms of measure and thus he wrote the theorem in an informal language. It was not until Cantor that the theory of innite was made into a science and for sake of lucidity we shall examine Theorem 1.2 rst. Euclid begins the proof by stating: Call the primes in our nite list p1, p2, ..., pr. The moment we start o with a nite list of primes, we are assuming that there is a nite list to begin with! Intuitively suppose we live in an universe where time moved in super-slow motion where our thinking took place at snail pace giving us miniscule intelligence. Let us then take a set of rst two primes {2,3}. How do we know that the next prime in the series is 5? The reason is we have at our disposal two already 'created' numbers less than 5 with which we can test to see if the next number is prime. What if we lived in another 'lowdimensional' universe per se where we could not even determine if 3 is a prime? Or what if we truncate the set to include 2 only? Now if we have reached a cul-de-sac as we have dened 2 as the rst prime. And therein lies the problem. For there to exist innitely many primes there must exist at least one prime. For that prime to exist we must dene it. Thus if we dene it we are proving a property of something that we already dened. The moment one denes a prime one pollutes the 'pure' set of numerals with a structure like a droplet of ink in a clear liquid. One causes a 'dent' so to speak. It is almost analoguous to 'Don't think of an elephant!' dilemma as it requires you to think of a structure with an assumption that a structure of that object exist. Euclid then proceeds with his proof which we shall enumerate for sake of reference: 1. Let A, B, and C be the assigned prime numbers. 2. I say that there are more prime numbers than A, B, and C. 3. Take the least number DE measured by A, B, and C. Add the unit DF to DE. 4. Then EF is either prime or not. 5. First, let it be prime. Then the prime numbers A, B, C, and EF have been found which are more than A, B, and C.Next, let EF not be prime. Therefore it is measured by some prime number. Let it be measured by the 3 prime number G. I say that G is not the same with any of the numbers A, B, and C. 6. If possible, let it be so. 7. Now A, B, and C measure DE, therefore G also measures DE. But it also measures EF. Therefore G, being a number, measures the remainder, the unit DF, which is absurd. 8. Therefore G is not the same with any one of the numbers A, B, and C. And by hypothesis it is prime. Therefore the prime numbers A, B, C, and G have been found which are more than the assigned multitude of A, B, and C. Therefore, prime numbers are more than any assigned multitude of prime numbers. Although Euclid's classical proof is mistakenly identied as using reductio ad absurdum, it is actually constructive in nature. With a nite list of primes in arsenal Euclid then constructs a case such that there would be more beasts of the same essence. By multiplying the primes and adding 1 to the resultant Euclid is surreptiously creating a constraint for the prime. Assuming a list of nite well-ordered primes, multiply them 2 X 3 X 5 X 7 X 11... and add 1 to get EF as in Steps 1-3, proceed to test for primality. But the Steps 1-3 themselves constitute a test for primality! If we lived in our hypothetical universe where time was slowed down then we would not be privy to access the list of initial nite list of primes. There are innitely many primes implies existence of at least one prime. But how do we know that the prime exists? Simple because we invent it! In order to bootstrap our way out of the recurisveness we dene the rst prime such as the number 2which invariantly dilutes our number system, as mentioned above, and thus just by creating a prime, we are creating innite primes! 3 Field of natural number embeds prime number Note we are obviously not denying the existence of innitude of primes. Denition 3.1. A countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers. A set that is not countable is called uncountable. Theorem 3.1. The set of prime numbers is countable. Proof. Let X be the inductive set of natural numbers. By axiom of innity, X is innite. Let Y be the set of prime numbers where by prime we mean a number y such that it is divisible by y or 1 only. Does it follow that if X is innite then Y is innite? Assume no. But what if the elements of Y "builds up" X or rather, Y encodes the genes of X. Hence, X: 1 2 3 4 5 6 7 8 9... Y: 2 3 5 7 11 13 17... 4 Every subset of a countable set is countable. In particular, every innite subset of a countably innite set is countably innite. For example, the set of prime numbers is countable, by mapping the n-th prime number to n: 2 maps to 1 3 maps to 2 5 maps to 3 7 maps to 4 11 maps to 5 13 maps to 6 17 maps to 7 19 maps to 8 23 maps to 9 and so on... Either the set of the prime numbers can be put in one-to-one correspondence with set of natural numbers or not. Assume that it cannot be. Since, the cardinality of the natural numbers is א0 if the prime numbers dry out then it has a nite cardinality C where C < א0. However, when C < א0 by mathematical induction it is possible to construct a number such that it is greater than C. This new number (C + ∆). For example, assume that in the set S of {2,3,5,7,11,13,17...} the primes stop at 17. This nite list has a cardinality 7 which is obviously smaller than א0. But it is possible to conceive of a number 7 + ∆, where ∆could be, say 4, and thus we get the new cardinal 11 which means that 4 is the cardinality of the successive non-primes. But if we extend this notion assuming that primes dried out then we can divide the set of natural numbers into two sets the rst one being the set of primes and the last one being that of non primes. The set of nonprimes must be innite with cardinality א0. Thus we get C + א0< א0 which is a contradiction proving our assumption wrong. Thus the set of primes must be innite or countable. 4 Prime numbers are innite by virtue of their denition Theorem 4.1. S is tautological where S:'There are innitely many primes'. Informally, can we create innite number of cars without creating innite number of chasis when the very basic core of a car is the chasis? Or for the same matter the reason why it is impossible to create innite amount of cars without any color. Innite list of such implies innite list of suchness ipso facto. The preceeding proof in Section 3 of the innitude of prime numbers highlights that prime number verication requires the set Y to be innite. If it follows that if X is innite then Y must be innite. But X encodes Y and X is innite. So by stating that X is innite we automatically have to assume certain characteristics of X. Primeness is a virtue of that characteristic. And by 5 assuming primeness we are stating in circular tongue that : If X is innite then a subset of X must be innite when we know that the denition of niteness or measure depends upon the function acting upon itself. Therefore it is redundant and tautological to 'prove' the innitude of prime numbers. Consider the statement S: There are inntely many primes. S is equivalent to S' : There are innitely many prime numbers in a set of natural numbers. S' is equivalent to S: There are innitely many prime numbers in an innite set of natural numbers. S is equivalent to S ': There are innitely many prime numbers in an innite set of natural numbers containing prime numbers (nite or innite). Or, if there exist a set N such that N is innite and contains set P such that every element of N can be expressed in terms of elements of P, then P must be innite. 5 Implications to Riemann Hypothesis I invite the reader to venture a journey with the rigor of an atheist reading text of Bible. When confronted with the question: Does God exist? it immediately begs the question the denition of God but then by doing so we are concocting a predened image in mind. Similarly when we begin a proof by assuming that there are primes to begin with we have certain mental construct of primeness. We have a taste for particular pattern. We prefer that particular pattern and because of our best interest we begin a search to look for them. We dene prime with the characteristic of primeness. If we dene them we create them and once they are created with as minimum as one ur-element then it is enough to guarantee innitude of it. A number system cannot exist without primes just like nouns cannot exist without adjectives. As stated by Don Zagier the following: "There are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The rst is that, despite their simple denitions and role as the building blocks of the natural numbers, the prime numbers belong to the most arbitrary and ornery objects studied by mathematicians: they grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behavior, and that they obey these laws with almost military precision." The First 50 Million Prime Numbers If we acknowledge the tautological nature of Euclid's proof than it immediately becomes obvious to us that the reason for the exhibition of the strange behavior 6 is simply because we designed it to be that way! Or rather a non-technical answer would be: Duh! Not only we dene the prime number but we set it forth like Prime Mover a collection of sinusoidal strings each with unique wavelength where a set begins rather 'awkwardly' with 2,3,5,7 where 2 is the only even prime number. When confronted with the question if Riemann Hypothesis is true, the catchall, silver bullet response would be: Depends! with the understanding in mind that it must be either true, false or formally undecidable. Whatever frame of reference we use the answer would vary accordingly. If an 'observer' designs a set of numbers explicitly with a framework of design in mind and asks if it extends indenitely even with such specic questions as the Montgomery Pair Correlation conjecture then it necessiates that we start o with a solid foundation of what we is it that we are explicitly seeking. 6 Conclusion The above informal proof has been a process. By actively participating in the process we came to nd out about the conclusion hence it is empirical in nature. Thus proof of innitude of prime numbers can only be found from empiricism which itself is based on reason and opens up many avenues for philosophical implications of the nature of proof theory or the psychology of the matter as there is no independent form out there contrary to what Erdos would have us believe with the idealistic existence of The Book.