Supervisor: Dr. W. B¨ohm Working title: Prime Time for a Prime Number Keywords: elementary number theory, computational number theory, prime numbers, modular arithmetic, public key encryption, internet data security A Gentle Invitation. On 20 January 2016, BBC News headlined: Largest known prime number dis- covered in Missouri! Immediately many other TV-stations and newspapers followed and posted similar messages. For instance, the New York Times on 21 January: New Biggest Prime Number = 2 to the 74 Mil ... Uh, It's Big! What is this beast, let us call it bigP, that received such a wide media response? Well, here it is: bigP = 224 207 281 − 1 In decimal notation this number has 22 338 618 digits. It has been found on January 7 by a team of the Great Internet Mersenne Prime Search Project (GIMPS). Ok, people tell us that bigP is a prime number. But what is a prime? May be, you recall the definition of a prime number. For definiteness here it is: A natural number p is prime, if it has exactly two distinct divisors, 1 and p. The list of primes starts with 2; 3; 5; 7; 11; 13; 17; 19; 23; 29; 31; 37; 41; 43; 47;::: A number1 that is not prime is called composite. Note that 1 is not considered a prime. There are very good reasons for excluding 1 from the set of primes. Stated in simple terms, if we let 1 be a prime, most non-trivial theorems about primes would be rendered false. So, 1 is not a prime. Probably you also know that there a infinitely many primes, and you may have heard that primes are in a certain sense the building blocks of all natural numbers because any number can be factored into primes. This is the statement of the Fundamental Theorem of Arithmetic. E.g., 15624 = 23 · 32 · 7 · 31 So, primes are very special numbers. You may have already guessed it: this thesis should be about prime numbers. More precisely, it should be about the elementary theory of prime numbers 1In the sequel when the term number occurs, it always means a natural number. 1 and it should also be a coverage of interesting computational issues related to primes. At this point you may stop reading further this description and argue: 1. I have never learned anything about primes except for some really basic facts, neither at high school nor during my university studies at WU. 2. So, to master this thesis, I will have to learn quite a lot about number theory. 3. Frankly speaking, number theory is quite an esoteric part of mathematics, only real nerds are working in this field. And most importantly, as I am studying economics at WU, when I take a thesis topic with a mathemat- ical flavor, then it must be useful mathematics. But number theory and especially prime numbers seem to be pretty useless stuff! Let me briefly comment on these objections: Ad (1): True. Ad (2): Very true, but see below. Ad (3): Completely wrong! Give me a chance to explain why. Very likely you are customer of one or the other internet trading site, e.g. Amazon. After having made your choice and put some stuff into your virtual shopping basket you will have to pay. Usually you will have to enter your credit card number or other sensible information about your bank account. And if you do so, don't you worry that this information may be stolen, may fall into the wrong hands? After all, sending a message over the internet is no more secure then sending a postcard. In principle, everybody can read it. Of course, the solution is to encrypt sensible information. For this purpose modern web browsers communicate with web servers using a special protocol called Secure Socket Layer. Whenever SSL is used you can see this in the command line of your browser. The web address you are communicating with is preceded by the word https. Data sent over SSL is encrypted by some encryption system and the receiver of the secret message deciphers it. But there is a problem. Deciphering an encoded messages requires a key. Until the 1970s it was necessary to exchange a key, e.g. during a secret meeting. But did you ever meet an agent of Amazon at a clandestine location and fix a key word, say yellowsubmarine ? No, you won't do that, because: • This procedure is potentially unsafe for several reasons. Exchanging the key may be intercepted, but more importantly, using the same key word for a longer time makes it very likely that your code will be broken. • After all, this procedure is totally impractical! No system of e-commerce or e-banking with many thousands of customers would be possible on this basis. 2 But in the early 1970s public-key cryptography has been invented and now prime numbers come into play! This goes as follows (basically the so-called RSA cryptosystem): • The security server at Amazon generates two large primes p and q. Here large means some several hundred decimal digits. • The server then multiplies p and q to give a large composite number m = p · q. • The number m is sent to the web client, e.g. your browser. This uses m to encrypt sensible information and sends this back to the security server. • Amazon's server decrypts the secret message and now knows about your account information, credit card number, etc. How does this methods work? Why is it considered secure? The crucial point is: Given a large composite number m, finding its prime factors is an extremely difficult task. Until now no algorithms are known which can do that job in reasonable time on computer hardware currently available for civil purposes. Thus Amazon can send the number m to your server, your client browser sends back an encrpyted message using m as key, say t(m), and although the unau- thorized eavesdropper can read m and the encrypted message t(m) he will not be able to decipher t(m) in reasonable time because finding factors of large numbers is so difficult. But how can Amazon decipher t(m)? This is possible because number the- ory provides so marvelous tools like modular arithmetic and Fermat's Little Theorem for us. We shall have to say more about these in a few minutes. Let us pause for a moment here. If you have read the introduction up to this point and your are still uninterested, then ok! I don't worry. If you are still in two minds about this thesis topic, then please read on. I am going now to discuss some interesting points regarding bigP in special and primes in general. It seems that bigP is a really big number, isn't it? Well, this depends. Clearly, compared to numbers we are usually dealing with in economic applications, bigP appears to be really big. After all, it has about 22 mill. decimal digits! So, let us have a look at physics, that science which con- nects the micro cosmos of quantum world to the macro cosmos of our universe. In 1938 Arthur Stanley Eddington argued that the total number of protons in the whole universe is about 1080, plus/minus a few percent. Note that one percent of 1080 is 1078! Still quite a big number, but nevertheless ridiculously small compared to bigP. Coming back to number theory, in a very real sense, there are no big numbers: any explicit number can be said to be small. Indeed, no matter how many digits or towers of exponents you write down there are only finitely many numbers smaller than your candidate and infinitely many that are larger (Crandall et al., 3 2005, p. 2). This is a rather general reservation. But even if we leave it aside, bigP is not really a record holder regarding size. 34 Very impressive in size is the Skews Number 101010 which in the 1950s played some role in prime number theory. This number is so big that if we could manage to somehow materialize each of its decimal digits to a subatomic particle our whole universe would be too small to hold it. The strange numbers googol = 10100 and googolplex = 10googol are both smaller than the Skews number - but: googol < bigP < googolplex The number bigP is a Mersenne number And this is no exception: among the largest known primes the first 12 are all Mersenne numbers. So, what are these special numbers? For any number n ≥ 0 the n-th Mersenne number is defined by n Mn = 2 − 1; n = 0; 1; 2;::: These numbers are named after the french monk Marin Mersenne who studied them in the 17th century. Mn can be prime only if n is prime. For, if n = ab is composite, then 2ab − 1 is divisible my 2a − 1 and 2b − 1. It is easy to see that: just use the well known formula for the sum of a finite geometric progression. Let us have a look at a few Mersenne numbers with prime exponent: n 2 3 5 7 11 13 17 19 23 n Mn = 2 − 1 3 7 31 127 2047 8191 131071 524287 8388607 This table shows us that primality of n does not guarantee primality of Mn. Indeed, in the second row of this table there are two composite numbers: M11 = 2047 = 23 · 90;M23 = 8388607 = 47 · 178481; as you can check easily with your pocket calculator. But the fact that among the record holders only with a few exceptions all primes have been Mersenne numbers may give rise to the suspicion that Mersenne numbers have some strong affinity to being prime.