Transcript Why are Prime Numbers called Prime & sieve of

Why are Prime Numbers called
prime & Sieve of Eratosthenes
Group Members –
Umang Chandra
Sneh Lata Gupta
Shivam Rastogi
Rohan Chaudhary
Vivek Chaudhary
What is a Prime Number?
Natural
Number >
1
Prime
Number
no positive
divisors
other than
1 and itself
IMPORTANCE OF PRIME NUMBERS
Why are they called Prime ?
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
– 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
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 – Cross out 1; it is not prime.
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
Hint For Next Step
Remember all numbers
divisible by 2 are even numbers.
Like 2,4,6,8,10,12,14………..
2 – Leave 2; cross out multiples of 2
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
Hint For Next Step
To 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.
267
Total of digits = 15
3 divides evenly into 15
267 is a multiple of 3
3– Leave 3; cross out multiples of 3
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
Hint For the Next Step
To find the multiples of 5 look
for numbers that end with the
digit 0 and 5.
385 is a multiple of 5
& 890 is a multiple of 5
because the last digit
ends with 0 or 5.
4– Leave 5; cross out multiples of 5
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
5– Leave 7; cross out multiples of 7
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
6–Leave 11; cross out multiples of 11
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
All the numbers left are prime
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
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.
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
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, 2099
TOTAL 14 SUCH YEARS
WHICH ARE PRIME
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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.


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.