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A prime number or simply a prime, is a number with exactly two factors. These two factors are always the number 1 and the prime number itself All prime numbers are odd except number 2 1 is not a prime number. 2 is the smallest prime There is no largest prime. There are infinite prime numbers

The Sieve of Eratosthenes can be used to find the prime numbers up to 100 very quickly

© T Madas The Sieve of Eratosthenes can be used to find the prime numbers up to 100 very quickly Cross off the number 1

© T Madas The Sieve of Eratosthenes can be used to find the prime numbers up to 100 very quickly Cross off all the multiples of 2 except 2

© T Madas The Sieve of Eratosthenes can be used to find the prime numbers up to 100 very quickly Cross off all the multiples of 3 except 3 2

© T Madas The Sieve of Eratosthenes can be used to find the prime numbers up to 100 very quickly Cross off all the multiples of 5 except 5 23

© T Madas The Sieve of Eratosthenes can be used to find the prime numbers up to 100 very quickly Cross off all the multiples of 7 except 7 235

© T Madas The Sieve of Eratosthenes can be used to find the prime numbers up to 100 very quickly These are the prime numbers up to 100

© T Madas The Prime Numbers up to 200

© T Madas Primes up to 200

© T Madas Cross off: number 1 multiples of 2 except 2 multiples of 3 except 3 multiples of 5 except 5 multiples of 7 except 7 multiples of 11 except 11 multiples of 13 except 13 Primes up to 200

© T Madas Primes up to 200 Cross off: number 1 multiples of 2 except 2 multiples of 3 except 3 multiples of 5 except 5 multiples of 7 except 7 multiples of 11 except 11 multiples of 13 except 13

© T Madas Interesting Facts Involving Primes

© T Madas Every even number other than 2, can be written as the sum of two primes 16= = = = = = = = = Write these even numbers as the sum of two primes, at least three different ways 50= = = = = = = = = = = =

© T Madas Every even number other than 2, can be written as the sum of two primes This statement is known as the Goldbach conjecture. In 1742 Christian Goldbach requested from Leonhard Euler, the most prolific mathematician of all times, for a proof for his conjecture. Euler could not prove this statement, nor has anyone else to this day, although no counter example can be found. C Goldbach L Euler

© T Madas Every odd number other than 1, can be written as the sum of a prime and a power of 2 3= = = = = = = Write these odd numbers as the sum of a prime and a power of 2 25= = = = = = = = = =

© T Madas Every even number can be written as the difference of 2 consecutive primes 2= 5 – 3 4= 11 – 7 6= 29 – 23 = 37 – 31 8= 97 – 89 = 59 – 53 Write these even numbers as the difference of 2 consecutive primes 10= 149 – = 211 – = 127 – 113 = 7 – 5 = 17 – 13

© T Madas Every prime number greater than 3 is of the form 6n ± 1, where n is a natural number 5= 6 x 1 – 1 7= 6 x = 6 x 2 – 1 13= 6 x = 6 x 3 – 1 19= 6 x = 6 x 4 – 1 29= 6 x 5 – 1 Careful because the converse statement is not true: Every number of the form 6n ± 1 is not a prime number

© T Madas Every prime number of the form 4n + 1, where n is a natural number, can be written as the sum of 2 square numbers 5= 4 x = 4 x = 4 x = 4 x = 4 x = 4 x = 4 x = 4 x = = = = = = = =

© T Madas Prime numbers which are of the form 2 n – 1, where n is a natural number, are called Mersenne Primes 1 st Mersenne: 2 2 – 1 = 3 2 nd Mersenne: 2 3 – 1 = 7 3 rd Mersenne: 2 5 – 1 = 31 4 th Mersenne: 2 7 – 1 = th Mersenne: 2 13 – 1 = th Mersenne: 2 17 – 1 = th Mersenne: 2 19 – 1 = th Mersenne: 2 31 – 1 = th Mersenne: 2 61 – 1 = th Mersenne: 2 89 – 1 = On May 15, 2004, Josh Findley discovered the 41 st known Mersenne Prime, 2 24,036,583 – 1. The number has digits and is now the largest known prime number!

© T Madas Perfect Numbers

© T Madas A perfect number is a number which is equal to the sum of its factors, other than the number itself. 6 is perfect because: = 6 A deficient number is a number which is more than the sum of its factors, other than the number itself 8 is deficient because: = 7 An abundant, or excessive number is a number which is less than the sum of its factors, other than the number itself 12 is abundant because: = 16 Classify the numbers from 3 to 30 according to these categories

A =42 30D =15 16 D 1 29D 1+3+5=9 15 P =28 28D 1+2+7=10 14 D 1+3+9=13 27D 1 13 D =16 26A =16 12 D 1+5=6 25D 1 11 A =28 24D 1+2+5=8 10 D 1 23D 1+3=4 9 D =14 22D 1+2+4=7 8 D 1+3+7=11 21D 1 7 A =22 20P 1+2+3=6 6 D 1 19D 1 5 A =21 18D 1+2=3 4 D 1 17D 1 3

© T Madas The definition of a perfect number dates back to the ancient Greeks. It was in fact Euclid that proved that a number of the form (2 n – 1)2 n – 1 will be a perfect number provided that: 2 n – 1 is a prime, which is known as Mersenne Prime Since the perfect numbers are connected to the Mersenne Primes, there are very few perfect numbers that we are aware of, given we only know 41 Mersenne Primes

The definition of a perfect number dates back to the ancient Greeks. It was in fact Euclid that proved that a number of the form (2 n – 1)2 n – 1 will be a perfect number provided that: 2 n – 1 is a prime, which is known as Mersenne Prime 1 st Perfect: (2 2 – 1)2 2 – 1 = 3= 3x 2x 2= 6= 6 1 st Mersenne: 2 2 – 1, 2 nd Mersenne: 2 3 – 1, 3 rd Mersenne: 2 5 – 1, 4 th Mersenne: 2 7 – 1, 5 th Mersenne: 2 13 – 1, 2 nd Perfect: (2 3 – 1)2 3 – 1 = 7= 7x 4x 4= 28 3 rd Perfect: (2 5 – 1)2 5 – 1 = 31x 16= th Perfect: (2 7 – 1)2 7 – 1 = 127x 64= th Perfect: (2 13 – 1)2 13 – 1 =

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Worksheets

© T Madas The Sieve of Eratosthenes can be used to find the prime numbers up to 100 very quickly Cross off: number 1 multiples of 2 except 2 multiples of 3 except 3 multiples of 5 except 5 multiples of 7 except 7 Primes up to The Sieve of Eratosthenes can be used to find the prime numbers up to 100 very quickly Cross off: number 1 multiples of 2 except 2 multiples of 3 except 3 multiples of 5 except 5 multiples of 7 except 7 Primes up to 100

© T Madas Cross off: number 1 multiples of 2 except 2 multiples of 3 except 3 multiples of 5 except 5 multiples of 7 except 7 multiples of 11 except 11 multiples of 13 except 13 Primes up to 200 The Sieve of Eratosthenes can be used to find the prime numbers up to 100 very quickly

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