1. © T Madas

2. 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

3. The Sieve of Eratosthenes can be used to find the prime numbers up to 100 very quickly 100 999897969594939291 90898887868584838281 80797877767574737271 70696867666564636261 60595857565554535251 50494847464544434241 40393837363534333231 30292827262524232221 20191817161514131211 10987654321

4. © T Madas The Sieve of Eratosthenes can be used to find the prime numbers up to 100 very quickly 100 999897969594939291 90898887868584838281 80797877767574737271 70696867666564636261 60595857565554535251 50494847464544434241 40393837363534333231 30292827262524232221 20191817161514131211 10987654321 Cross off the number 1

5. © T Madas 100 999897969594939291 90898887868584838281 80797877767574737271 70696867666564636261 60595857565554535251 50494847464544434241 40393837363534333231 30292827262524232221 20191817161514131211 1098765432 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

6. © T Madas 9997959391 8987858381 7977757371 6967656361 5957555351 4947454341 3937353331 2927252321 1917151311 9753 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

7. © T Madas 979591 898583 79777371 676561 595553 49474341 373531 292523 19171311 75 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

8. © T Madas 9791 8983 79777371 6761 5953 49474341 3731 2923 19171311 7 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

9. © T Madas 97 8983 797371 6761 5953 474341 3731 2923 19171311 7235 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

10. © T Madas The Prime Numbers up to 200

11. © T Madas Primes up to 200

12. © T Madas 200199198197196195194193192191 190189188187186185184183182181 180179178177176175174173172171 170169168167166165164163162161 160159158157156155154153152151 150149148147146145144143142141 140139138137136135134133132131 130129128127126125124123122121 120119118117116115114113112111 110109108107106105104103102101 100999897969594939291 90898887868584838281 80797877767574737271 70696867666564636261 60595857565554535251 50494847464544434241 40393837363534333231 30292827262524232221 20191817161514131211 10987654321 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

13. © T Madas 200199198197196195194193192191 190189188187186185184183182181 180179178177176175174173172171 170169168167166165164163162161 160159158157156155154153152151 150149148147146145144143142141 140139138137136135134133132131 130129128127126125124123122121 120119118117116115114113112111 110109108107106105104103102101 100999897969594939291 90898887868584838281 80797877767574737271 70696867666564636261 60595857565554535251 50494847464544434241 40393837363534333231 30292827262524232221 20191817161514131211 10987654321 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

14. © T Madas Interesting Facts Involving Primes

15. © T Madas Every even number other than 2, can be written as the sum of two primes 16= 3 + 13 = 5 + 11 22= 3 + 19 = 11 + 11 40= 3 + 37 = 11 + 29 52= 5 + 47 = 11 + 41 = 17 + 23 Write these even numbers as the sum of two primes, at least three different ways 50= 3 + 47 = 7 + 43 100= 3 + 97 = 11 + 89 150= 11 + 139 = 13 + 137 200= 3 + 197 = 7 + 193 = 19 + 131 = 13 + 37 = 17 + 83 = 13 + 187

16. © 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 1690 - 1764 L Euler 1707 - 1783

17. © T Madas Every odd number other than 1, can be written as the sum of a prime and a power of 2 3= 2 + 2 0 17= 13 + 2 2 35= 31 + 2 2 = 19 + 2 4 81= 79 + 2 1 = 17 + 2 6 = 3 + 2 5 Write these odd numbers as the sum of a prime and a power of 2 25= 23 + 2 1 = 17 + 2 3 75= 73 + 2 1 = 71 + 2 2 125= 109 + 2 4 = 64 + 2 6 175= 173 + 2 1 = 167 + 2 3 = 67 + 2 3 = 47 + 2 7

18. © 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 – 139 12= 211 – 199 14= 127 – 113 = 7 – 5 = 17 – 13

19. © 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 1 + 1 11= 6 x 2 – 1 13= 6 x 2 + 1 17= 6 x 3 – 1 19= 6 x 3 + 1 23= 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

20. © 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 1 + 1 13= 4 x 3 + 1 17= 4 x 4 + 1 29= 4 x 7 + 1 37= 4 x 9 + 1 41= 4 x 10 + 1 53= 4 x 13 + 1 61= 4 x 15 + 1 = 4 + 1 = 9 + 4 = 16 + 1 = 25 + 4 = 36 + 1 = 25 + 16 = 49 + 4 = 36 + 25

21. © 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 = 127 5 th Mersenne: 2 13 – 1 = 8191 6 th Mersenne: 2 17 – 1 = 131071 7 th Mersenne: 2 19 – 1 = 524287 8 th Mersenne: 2 31 – 1 = 2147483647 9 th Mersenne: 2 61 – 1 = 2305843009213693951 10 th Mersenne: 2 89 – 1 = 618970019642690137449562111 On May 15, 2004, Josh Findley discovered the 41 st known Mersenne Prime, 2 24,036,583 – 1. The number has 6 320 430 digits and is now the largest known prime number!

22. © T Madas Perfect Numbers

23. © 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: 1 + 2 + 3 = 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: 1 + 2 + 4 = 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: 1 + 2 + 3 + 4 + 6 = 16 Classify the numbers from 3 to 30 according to these categories

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

25. © 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

26. 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= 496 4 th Perfect: (2 7 – 1)2 7 – 1 = 127x 64= 8128 5 th Perfect: (2 13 – 1)2 13 – 1 = 33550336

27. © T Madas

28. Worksheets

29. © T Madas 100999897969594939291 90898887868584838281 80797877767574737271 70696867666564636261 60595857565554535251 50494847464544434241 40 393837363534333231 30292827262524232221 20 191817161514131211 10987654321 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 100999897969594939291 90898887868584838281 80797877767574737271 70696867666564636261 60595857565554535251 50494847464544434241 40393837363534333231 30292827262524232221 20191817161514131211 10987654321 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

30. © 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

31. © T Madas