Why are Prime Numbers called prime & Sieve of Eratosthenes

Group Members – Umang ChandraSneh Lata GuptaShivam Rastogi

Rohan ChaudharyVivek Chaudhary

TOPIC

What is a Prime Number?

Natural Number >

1

no positive divisors

other than 1 and itself

Prime Number

A natural number greater than1 that is not a prime number is called a composite number. For example, 5 is prime because only 1 and 5 can divide it, whereas 6 is composite because it has the divisors 2,3, 1 and 6.

IMPORTANCE OF PRIME NUMBERS

Cryptography

Algorithm “RSA”

Alien Communications

To send out signals to

Aliens

Mathematical Discussion

Mathematical Conversations,

jokes etc

Why are they called Prime

– All other numbers (positive integers) are measured by primes, this makes primes first.

– We use the English word prime because the ancient Greeks saw them as multiplicatively first, so Billingsley translated Euclid's 'prôtos' as 'prime'.

– Other terms used for prime numbers – linear/ simple/ incomposite

Prôtos arithmos estin ho monadi monêi metroumenos - Euclid, (The Elements (book 7, definition 11)

Meaning- -is measured by a unit alone -are not multiples of other numbers

Also as we know that

?

Prime numbers are thus the first numbers, the numbers from which the

other numbers all arise

Thus they are primary numbers and hence are called as:

Method to find prime numbers in a given set of natural

numbers:

Eratosthenes’ Sieve or

Eratosthenes(ehr-uh-TAHS-thuh-neez)

Eratosthenes was the librarian at Alexandria, Egypt in 200 B.C.

Note every book was a scroll.

Eratosthenes(ehr-uh-TAHS-thuh-neez)

Eratosthenes was a Greek mathematician, astronomer, and geographer.He invented a method for finding prime numbers that is still used today.This method is called Eratosthenes’ Sieve.

Eratosthenes’ Sieve

A sieve has holes in it and is used to filter out the juice.Eratosthenes’s sieve filters out natural numbers to find the prime numbers.

Copyright © 2000 by Monica Yuskaitis

Now lets see the main steps how to find the prime numbers using

the sieve of Eratosthenes’

1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100

1 – Cross out 1; it is not prime.

Hint For Next Step

Remember all numbers divisible by 2 are even numbers.

Like 2,4,6,8,10,12,14………..

1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100

2 – Leave 2; cross out multiples of 2

Hint For Next StepTo find multiples of 3, add the digits of a number; see if you can divide this number evenly by 3; then the number is a multiple of 3.

2 6 7Total of digits = 15

3 divides evenly into 15267 is a multiple of 3

1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100

3– Leave 3; cross out multiples of 3

To find the multiples of 5 look for numbers that end with the digit 0 and 5.

Hint For the Next Step

385 is a multiple of 5& 890 is a multiple of 5

because the last digitends with 0 or 5.

1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100

4– Leave 5; cross out multiples of 5

1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100

5– Leave 7; cross out multiples of 7

1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100

6–Leave 11; cross out multiples of 11

1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100

All the numbers left are prime

The Prime Numbers from 1 to 100 are as follows:

2,3,5,7,11,13,17,19,23,31,37,41,43,47,53,59,61,67,71,73,

79,83,89,97

Similarly if we want to find the years of our century

i.e.2000-2099 which are prime numbers we follow the same step

by first making the grid of numbers and then crossing the

years which are not prime using the above stated method.

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

2010 2011 2012 2013 2014 2015 2016 2017 2018 2019

2020 2021 2022 2023 2024 2025 2026 2027 2028 2029

2030 2031 2032 2033 2034 2035 2036 2037 2038 2039

2040 2041 2042 2043 2044 2045 2046 2047 2048 2049

2050 2051 2052 2053 2054 2055 2056 2057 2058 2059

2060 2061 2062 2063 2064 2065 2066 2067 2068 2069

2070 2071 2072 2073 2074 2075 2076 2077 2078 2079

2080 2081 2082 2083 2084 2085 2086 2087 2088 2089

2090 2091 2092 2093 2094 2095 2096 2097 2098 2099

21st Century 2000-2099

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

2010 2011 2012 2013 2014 2015 2016 2017 2018 2019

2020 2021 2022 2023 2024 2025 2026 2027 2028 2029

2030 2031 2032 2033 2034 2035 2036 2037 2038 2039

2040 2041 2042 2043 2044 2045 2046 2047 2048 2049

2050 2051 2052 2053 2054 2055 2056 2057 2058 2059

2060 2061 2062 2063 2064 2065 2066 2067 2068 2069

2070 2071 2072 2073 2074 2075 2076 2077 2078 2079

2080 2081 2082 2083 2084 2085 2086 2087 2088 2089

2090 2091 2092 2093 2094 2095 2096 2097 2098 2099

Thus the prime numbers in this century are:

2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099TOTAL 14 SUCH

YEARS WHICH ARE PRIME

It’s one of the educational advantage is that it helps to develop our ability to see and extend pattern.

It is a good method to quickly make a short list of prime no.s.

It is the best intuitive method of finding a list of prime no.s.

ADVANTAGES OF THE SIEVE OF ERATOSTHENES

It’s disadvantage is that in this method we have to allocate the array at the start and that uses a bunch of memory.

It is a time consuming method because if we want to make a long list of prime no.s then it can take a lot of time.

DISADVANTAGES OF THE SIEVE OF ERATOSTHENES

THANK YOU