Structures 5 Number Theory What is number theory? Study of the properties of number, focusing mainly on positive integers. Issues of divisibility, primes, etc.

Using the structure of numbers to form arguments odds and evens multiples divisibility

The sum of two odd numbers is even represents “odd number” (2n + 1) + (2m + 1) = 2n + 2m + 2 = 2(n + m + 1) various informal ways of thinking about this commonly: odd numbers can be expressed as pairs with one left over; two odd numbers have two left over which make a pair alternative algebraic proof (provides a method for addressing more complex cases not susceptible to diagrammatic proof to produce such proofs focus on: how to represent the givens (odd numbers) what form you want the result to be in (even number) 2( … ) is even

How can you represent “consecutive numbers”? The sum of 3 consecutive numbers is divisible by 3. What does “divisible by 3” look like? How can you represent “consecutive numbers”? Further example represent the givens: n, n+1, n+2 or n-1, n, n+1 result divisible by 3: 3( … ) Have a go together …

Is it true in general that the sum of k consecutive numbers is divisible by k? Form and prove conjectures. Investigate. Problem sheet

Primes prime factorisation how many prime numbers are there?

Primes and Factorisation 24 12 2 6 2 3 2 factor tree for 24

Primes and Factorisation 24 12 2 6 2 3 2

Primes and Factorisation 24 12 2 3 6 4 2 24 6 2 3 2 This wasn’t the only way to do it. But all trees end up with the same ‘leaves’ – a unique prime factorisation. (fundamental theorem of arithmetic)

How many different factor trees? How many different factors? Given a number expressed as a product of primes, how many different factors does it have? e.g. Consider the numbers 1 – 100. Which numbers have 1, 2, 3, 4, etc. different factors? For 24 – as a group generate all possible factor trees – how many factors? 24=2.2.2.3 no of factors = 4.2 = 8 1, 2, 4, 8, 3, 6, 12, 24 relate to combinations Investigate

Formulae for prime numbers? Mersenne numbers curiosities: x^2+x+41 generates prime numbers for x=0 to x=39. obviously x=41 is not prime (x=40 is also composite Mersenne numbers – some of these are prime p=2, M=3 p=3 M=7 p=5 M=31 p=7 M=127 used as a means of searching for large primes

How many prime numbers are there?

Approximations for the number of primes less than x 1 2 3 4 5 Tchebycheff, Gauss Legendre Riemann

Is there a largest prime number? Develop on white board Euclid’s proof suppose there is a finite number n of prime numbers, pn is the largest then the product p1p2p3…pn is divisible by all the primes but p1p2p3…pn+1 is not divisible by any of them so either there is another prime larger than pn that is a factor of this number or this number is itself prime and larger than pn hence pn is not the largest prime number and there is an infinite number of primes A proof by contradiction