© T Madas. A prime number or simply a prime, is a number with exactly two factors. These two...

© T Madas

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

© T Madas

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

100999897969594939291

90898887868584838281

80797877767574737271

70696867666564636261

60595857565554535251

50494847464544434241

40393837363534333231

30292827262524232221

20191817161514131211

10987654321

© T Madas


100999897969594939291

90898887868584838281

80797877767574737271

70696867666564636261

60595857565554535251

50494847464544434241

40393837363534333231

30292827262524232221

20191817161514131211

10987654321

Cross off the number 1

© T Madas

100999897969594939291

90898887868584838281

80797877767574737271

70696867666564636261

60595857565554535251

50494847464544434241

40393837363534333231

30292827262524232221

20191817161514131211

1098765432


Cross off all the multiples of 2 except 2

© T Madas

9997959391

8987858381

7977757371

6967656361

5957555351

4947454341

3937353331

2927252321

1917151311

9753



2

© T Madas

979591

898583

79777371

676561

595553

49474341

373531

292523

19171311

75



2 3

© T Madas

9791

8983

79777371

6761

5953

49474341

3731

2923

19171311

7



2 3 5

© T Madas

97

8983

797371

6761

5953

474341

3731

2923

19171311

72 3 5


These are the prime numbers up to 100

© T Madas

ThePrime Numbersup to 200

© T Madas200199198197196195194193192191

190189188187186185184183182181

180179178177176175174173172171

170169168167166165164163162161

160159158157156155154153152151

150149148147146145144143142141

140139138137136135134133132131

130129128127126125124123122121

120119118117116115114113112111

110109108107106105104103102101

100999897969594939291

90898887868584838281

80797877767574737271

70696867666564636261

60595857565554535251

50494847464544434241

40393837363534333231

30292827262524232221

20191817161514131211

10987654321

Primes up to 200

© T Madas200199198197196195194193192191

190189188187186185184183182181

180179178177176175174173172171

170169168167166165164163162161

160159158157156155154153152151

150149148147146145144143142141

140139138137136135134133132131

130129128127126125124123122121

120119118117116115114113112111

110109108107106105104103102101

100999897969594939291

90898887868584838281

80797877767574737271

70696867666564636261

60595857565554535251

50494847464544434241

40393837363534333231

30292827262524232221

20191817161514131211

10987654321

Cross off:number 1multiples of 2 except 2multiples of 3 except 3multiples of 5 except 5multiples of 7 except 7multiples of 11 except 11multiples of 13 except 13

Primes up to 200

© T Madas200199198197196195194193192191

190189188187186185184183182181

180179178177176175174173172171

170169168167166165164163162161

160159158157156155154153152151

150149148147146145144143142141

140139138137136135134133132131

130129128127126125124123122121

120119118117116115114113112111

110109108107106105104103102101

100999897969594939291

90898887868584838281

80797877767574737271

70696867666564636261

60595857565554535251

50494847464544434241

40393837363534333231

30292827262524232221

20191817161514131211

10987654321

Primes up to 200Cross off:number 1multiples of 2 except 2multiples of 3 except 3multiples of 5 except 5multiples of 7 except 7multiples of 11 except 11multiples 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 = 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

© 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 Goldbach1690 - 1764

L Euler1707 - 1783

© T Madas

Every odd number other than 1, can be written as the sum of a prime and a power of 23 = 2 + 20

17 = 13 + 22

35 = 31 + 22= 19 + 24

81 = 79 + 21= 17 + 26

= 3 + 25

Write these odd numbers as the sum of a prime and a power of 2

25 = 23 + 21= 17 + 23

75 = 73 + 21= 71 + 22

125

= 109 + 24= 64 + 26

175

= 173 + 21= 167 + 23

= 67 + 23

= 47 + 27

© 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

© T Madas

Every prime number greater than 3 is of the form 6n ± 1, where n is a natural number5 = 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

© 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

© T Madas

Prime numbers which are of the form 2 n – 1,

where n is a natural number, are called Mersenne Primes1st Mersenne: 22 – 1 = 32nd Mersenne: 23 – 1 = 73rd Mersenne: 25 – 1 = 314th Mersenne: 27 – 1 = 1275th Mersenne: 213 – 1 = 81916th Mersenne: 217 – 1 = 1310717th Mersenne: 219 – 1 = 5242878th Mersenne: 231 – 1 = 21474836479th Mersenne: 261 – 1 = 230584300921369395110th Mersenne: 289 – 1 = 618970019642690137449562111On May 15, 2004, Josh Findley discovered the 41st known Mersenne Prime, 224,036,583 – 1. The number has 6 320 430 digits and is now the largest known prime number!

© 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

© T Madas

A1+2+3+5+6+10+15=4230D1+2+4+8=1516

D129D1+3+5=915

P1+2+4+7+14=2828D1+2+7=1014

D1+3+9=1327D113

D1+2+13=1626A1+2+3+4+6=1612

D1+5=625D111

A1+2+3+4+6+12=2824D1+2+5=810

D123D1+3=49

D1+2+11=1422D1+2+4=78

D1+3+7=1121D17

A1+2+4+5+10=2220P1+2+3=66

D119D15

A1+2+3+6+9=2118D1+2=34

D117D13

© 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 (2n – 1)2n – 1 will be a perfect number provided that:

2n – 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

© 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 (2n – 1)2n – 1 will be a perfect number provided that:

2n – 1 is a prime, which is known as Mersenne Prime

1st Perfect: (22 – 1)22 – 1 = 3 x 2= 61st Mersenne: 22 – 1,

2nd Mersenne: 23 – 1,

3rd Mersenne: 25 – 1,

4th Mersenne: 27 – 1,

5th Mersenne: 213 – 1,

2nd Perfect: (23 – 1)23 – 1= 7 x 4= 28

3rd Perfect: (25 – 1)25 – 1 = 31 x 16 = 496

4th Perfect: (27 – 1)27 – 1 = 127 x 64 = 8128

5th Perfect: (213 – 1)213 – 1 = 33550336

© T Madas

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© T Madas. A prime number or simply a prime, is a number with exactly two factors. These two...

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Transcript of © T Madas. A prime number or simply a prime, is a number with exactly two factors. These two...