Structures 5

Number Theory

Using the structure of numbers to form arguments

• odds and evens• multiples• divisibility

The sum of two odd numbers is even

(2n + 1) + (2m + 1)

= 2n + 2m + 2

= 2(n + m + 1)

2n + 1represents

“odd number”

2( … )is even

The sum of 3 consecutive numbers is divisible by 3.

How can you represent “consecutive numbers”?

What does “divisible by 3”

look like?

Is it true in general that the sum of k consecutive numbers is divisible by k?

Form and prove conjectures.

Primes

• prime factorisation• how many prime numbers are there?

Primes and Factorisation24

12 2

6 2

3 2

factor tree for 24

Primes and Factorisation24

12 2

6 2

3 2

24 =3×2×2×2

24 =3⋅23

Primes and Factorisation24

12 2

6 2

3 2

24 =3×2×2×2

24 =3⋅23

3

6 4

2 2

24

2

• 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?

23 ⋅35 ⋅7

Formulae for prime numbers?

x2 + x+ 41

2 p −1 (p prime) Mersenne numbers

How many prime numbers are there?

Approximations for the number of primes less than x

1

2

3

4

5

Tchebycheff, Gauss

Legendre

Riemann

Li(x)= dt / lnt2

x

∫x/ (lnx−1.08366)

Li(x)−12 Li(x

12 )

x/ (lnx−1−1/ lnx)x/ (lnx−1)

Is there a largest prime number?