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Mathematical Ideas that Shaped the World Prime numbers

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Plan for this class Why are prime numbers interesting? What is the Prime Number Theorem? How could prime numbers win you a million dollars? How does public key cryptography work? What food did you bring for the maths picnic?!

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But first... Some mathematical mind reading!

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A prime number is a number which is only divisible by itself and 1. Prime numbers 76

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The building blocks of numbers Primes are often known as the building blocks of numbers, since they generate all other numbers. The Fundamental Theorem of Arithmetic states that every number can be written uniquely as a product of primes. 28 = 2 2 x7, 60 = 2 2 x3x5

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Sieve of Eratosthenes This is an ancient Greek method for finding all prime numbers.

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Pattern of the primes Prime numbers seem to occur randomly. Sometimes they come in pairs, e.g. (11,13), (29,31), (59,61)… …and other times there are long gaps between them, e.g. (113, 127) There is no formula that will predict where the next prime number will be.

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Sequences of primes

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Fermat primes

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Mersenne primes

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Mersenne primes

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The Ulam Spiral In 1963 the mathematician Stanislaw Ulam was doodling during a boring meeting… 37363534333231 38171615141330 39185431229 40196121128 41207891027 42212223242526 43444546474849

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The Ulam Spiral 200x200 Ulam spiralSacks spiral

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Distribution of the primes Rather than trying to find patterns in primes, mathematicians started looking at the general distribution of primes among the numbers. For example, if you pick a random number in the range of 0 to N, what is the chance that this number is prime?

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The Prime Number Theorem

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Prime counting

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The prime number theorem always a bit less than (x)

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Enter Riemann In 1859 Riemann gave an explicit formula for (x). There was another function, called the Riemann zeta function, which controlled how far away the primes were from their expected positions. (s) Get back over here!!

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Bernhard Riemann (1826 – 1866) Born near Hanover to a poor family. Was shy, had frequent nervous breakdowns and a fear of public speaking. Trained to become a pastor but kept getting distracted by maths.

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Bernhard Riemann (1826 – 1866) While studying theology in Göttingen he met Gauss, and was persuaded to switch to maths. Founded Riemannian geometry – the cornerstone of Einstein’s theory of relativity. Died of TB at age 40.

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The zeta function

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The Riemann hypothesis

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The Riemann Hypothesis

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The position of the zeros of the zeta function dictate the positions of the prime numbers. The Riemann Hypothesis was Problem 8 of Hilbert’s 1900 list of unsolved problems. It is now one of 6 remaining Clay Institute Millennium Prize Problems worth $1 million.

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The Riemann Hypothesis So far 10,000,000,000,000 zeros have been checked and they all satisfy the conjecture.

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Are prime numbers useless? "I have never done anything 'useful'. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.“ G.H.Hardy

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Modular arithmetic What time will it be 28 hours from now? What day of the week will it be in 30 days? What is the last digit of 4538729 x 9957397632? To answer these questions, you have used modular arithmetic!

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Modular arithmetic In modular arithmetic we have a clock with n numbers on. Whenever we go around we start again at zero. We say we are working modulo n, or mod n. 0 1 2 3 4 5 6

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The problem with cryptography The usual problem with sending a coded message is that you have to agree on a key for the code beforehand. A shift on a Caesar cipher The starting positions for an Enigma machine An actual key to unlock a box If anyone finds the key, they can decrypt all your messages.

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Public key cryptography In public key cryptography, there are two different keys: A public key which can be used to encrypt messages. A private key which can be used to decrypt messages.

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Public key cryptography Digitally, this system requires mathematical problems which are easy to do in one direction but very difficult to do backwards without the key. E.g. it is very easy to multiply two primes together, but very difficult to factorise the product without knowing one of the primes. 97 x 53 = 5141

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One-way problems Another example is that it is easy to compute powers modulo a prime, but difficult to find logarithms. 5 6 = 8 (mod 23)

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Diffie-Hellman key exchange We can use this idea to obtain a shared secret. Alice and Bob agree on a prime p and a number g. Alice chooses a secret number a and publishes A = g a (mod p). Bob chooses a secret number b and publishes B = g b (mod p). Now both can compute g ab (mod p).

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RSA cryptography RSA cryptography was originally invented by the Englishman Clifford Cocks in 1973. However, this remained secret until 1997 as the work was done for GCHQ. It is now named after Ron Rivest, Adi Shamir and Leonard Adleman who described the method in 1977.

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The RSA algorithm 1. Choose two nice big primes, p and q. 2. Compute n =pq. 3. Compute (n) = (p-1)(q-1) 4. Choose e so that e and (n) are coprime. 5. Publish e and n; this is your public key. 6. Find d = e -1 (mod (n)); this is your private key.

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Encryption Bob wants to send Alice a message M. Bob knows Alice’s public key: (e,n). He computes c = M e (mod n). Alice computes c d (mod n) and recovers M. (using the magic of modular arithmetic and some number theory!)

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Problems with RSA The two primes must be chosen randomly. If many people pick the same prime then the numbers are easy to factorise. The same plaintext message sent to many different people becomes easy to decrypt. Therefore M is randomised prior to sending. p and q must not be too close together, and d should be large.

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Breaking RSA Currently the largest number factorised by a general algorithm is 768 bits long. RSA keys are typically 1024 – 2048 bits long. Note: a proof of the Riemann Hypothesis is unlikely to break our security systems!

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Open prime problems Can every even number be written as the sum of two primes? (Goldbach conjecture) Are there infinitely many twin primes? Are there infinitely many Fermat primes or Mersenne primes? Is the Riemann Hypothesis true?

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Lessons to take home Prime numbers are the building blocks of arithmetic. There is no discernible pattern to the primes, although we understand how they are distributed. Primes underlie most of our cryptosystems.

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MathsPicnic!MathsPicnic!

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CSE 321 Discrete Structures Winter 2008 Lecture 8 Number Theory: Modular Arithmetic.

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