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Prime Numbers from the pre-classical to the avant-garde Paul Erdős (1913-1996) The Book ‘You don’t have to believe in God, but you should believe in The Book’ -Paul Erdős G.H. Hardy “There is a very high degree of unexpectedness, combined with inevitability and economy.” A Good Problem (definition) Easy to state Hard to solve Of fundamental importance in mathematics Euler 2016 = 16 x 126 = 32 x 63 = 32 x 9 x 7 = 25 x 32 x 7 Euclid 325-265 BC, Alexandria The guy explaining the maths ‘It’s all Greek to me’ ab + (a-b)2 = (a+b)2 4 4 If a straight line be cut into equal and unequal segments, the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section is equal to the square on the half. Reductio ad absurdum To prove SOMETHING. Suppose the OPPOSITE of the something is true. Construct a sequence of implications which lead to a FALSEHOOD. Then the OPPOSITE cannot have been true to start off with. Hence, SOMETHING is true. Thomas Hardy: "Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game. Euclid’s Proof Suppose the opposite: « The primes are not infinite. In fact, I can make a list of them 2,3,5,7,11,…,P, where P is the biggest prime. » Now to find the contradiction… Paul Erdős, 1913-1996 Child prodigy, he could calculate at the age of three how many seconds his family’s friends had lived. “A mathematician is a machine for turning coffee into theorems.” 1949 Elementary Proof of the Prime Number Theorem Key Idea 1: Square and no square 2016 = 25 × 32 × 7 = 25 × 32 × 50 × 71 = (21 × 30 × 50 × 71) × (24 × 32) = (21 × 30 × 50 × 71) × (22 × 31)2 SQUARE-FREE PART All powers are 0 or 1 SQUARE PART Key Idea 2: Counting divisors How many multiples of 7 less than 24? 7 14 21 28 35 24 7 = 3.4285714… 24 7 = 3 A stronger Result 1 1 1 1 ── + ── + ── + ── + … = ∞ 2 3 5 7 ? Not all infinite sequences add up to ∞ There are infinitely many primes Suppose the opposite « 1 1 1 1 ── + ── + ── + ── + … adds up to less than ∞. 2 3 5 7 Then there will be a TAIL 1 1 1 1 1 1 ── + ── + … + ── + ── + ── + ── + … 2 3 P q r s We need to go n primes along the sequence for the tail to add to less than a half SMALL PRIMES < 1/2 BIG PRIMES » The Proof Produce statement through logical steps: X M = any number at all = counts the numbers less than X divisible only by ‘small primes’ Make the contradiction: Since X can be anything, pick a value which makes the statement false Notation notation notation p = biggest of small primes X = any number at all M = counts the numbers less than X divisible only by ‘small primes’ k = any number less than X In how many ways can numbers less than X be constructed? Certainly, X cannot be more than this.