is ‘n’ a prime number ?
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IS ‘N’ A PRIME NUMBER ?
DONE BY:ISHTIAQUE KHAN
1757, CSEDU
THEORY ONE
• WHAT IS A PRIME NUMBER ?
A Prime Number is any integral number that has exactly 2 divisors.
THEORY TWO
• IS 0 A PRIME NUMBER?
No, 0 is not a Prime Number.
• IS 1 A PRIME NUMBER?
No, 1 is not a Prime Number.
• IS 2 A PRIME NUMBER?
Yes, 2 is a Prime Number.
THEORY THREE
• IF A POSITIVE INTEGER N IS NOT DIVISIBLE BY ANY NUMBER, X, IN THE RANGE:
2 <= X <= ,
THEN N IS NOT DIVISIBLE BY ANY NUMBER, X, IN THE RANGE:
< X < N
N
N
LET N BE 37
37
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LET N BE 37
SO,2<=X<=SQUARE ROOT
(N)IS
2 <= X <= 6.08SINCE X IS INTEGER
2 <= X <= 6
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LET N BE 37
37 2
18.5
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LET X BE 2
THEREFORE N IS NOT DIVISIBLE BY 2 OR ANY OTHER MULTIPLE OF 2
LET N BE 37
2 1 2
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LET N BE 37
2 2 4
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LET N BE 37
2 3 6
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LET N BE 37
2 4 8
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LET N BE 37
2 5 10
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LET N BE 37
2 6 12
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LET N BE 37
2 7 14
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LET N BE 37
2 8 16
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LET N BE 37
2 9 18
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LET N BE 37
2 10
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LET N BE 37
2 11
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LET N BE 37
2 12
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LET N BE 37
2 13
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LET N BE 37
2 14
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LET N BE 37
2 15
30
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LET N BE 37
2 16
32
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LET N BE 37
2 17
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LET N BE 37
2 18
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LET N BE 37
37 3
12.3
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LET X BE 3
THEREFORE N IS NOT DIVISIBLE BY 3 OR ANY OTHER MULTIPLE OF 3
LET N BE 37
3 1 3
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LET N BE 37
3 2 6
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LET N BE 37
3 3 9
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LET N BE 37
3 4 12
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LET N BE 37
3 5 15
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LET N BE 37
3 6 18
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LET N BE 37
3 7 21
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LET N BE 37
3 8 24
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LET N BE 37
3 9 27
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LET N BE 37
3 10
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LET N BE 37
3 11
33
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LET N BE 37
3 12
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LET N BE 37
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LET X BE 4
BUT 4 IS ALREADY CHECKED
LET N BE 37
37 5
7.4
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LET X BE 5
THEREFORE N IS NOT DIVISIBLE BY 5 OR ANY OTHER MULTIPLE OF 5
LET N BE 37
5 1 5
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LET N BE 37
5 2 10
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LET N BE 37
5 3 15
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LET N BE 37
5 4 20
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LET N BE 37
5 5 25
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LET N BE 37
5 6 30
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LET N BE 37
5 7 35
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LET N BE 37
11
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LET X BE 6
BUT 6 IS ALREADY CHECKED
LET N BE 37
NEXT WE HAVE ONLY PRIME NUMBERS
REMAINING. SO NON OF THEM IS A MULTIPLE OF ANY
OF THE OTHER.
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LET N BE 37
THE LEAST OF THE
REMAINING NUMBERS
IS 7
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LET N BE 37
SO THE LEAST OF THE PRODUCT OF
ANY TWO REMAINING NUMBERS IS
7 x 7 = 49WHICH IS MORE
THAN 37
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LET N BE 37
THAT MEANS ALL OTHER COMBINATION
LIKE 7 X 11 = 77
OR 29 X 29 = 841
OR 13 X 31 = 403
HAVE TO BE MORE THAN 37.
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LET N BE 37
SO, NON OF THE OTHER REMAINING NUMBERS
CAN DIVIDE 37.
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Therefore it is proof enough that 37 is not divisible by any number, X, in the range: < X < 37
If it is possible to show that 37 is not divisible by any number, X, in the range:2 <= X <= ,
That is 2 <= X <= 6
37
37
So it is possible to prove ‘if an integer N is a Prime number’ by dividing it with the integers, X, in the range:
2 <= X <=
N
THEORY FOUR• IF N AND X ARE TWO POSITIVE INTEGERS, THEN IN THE RANGE R1:
2 <= X <= ,
IF WE KEEP CHECKING EACH X IN THE SEQUENCE
2, 3, 4, 5, ………. ,
ALONG WITH ALL THE MULTIPLES OF X IN THE RANGE R1, THEN WHEN
X > ,
THE ONLY REMAINING UNCHECKED ELEMENTS IN R1 WILL ALL BE PRIME NUMBERS
N
N
N
LET N BE 37
37
11
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LET N BE 37
11
12
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1
2
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10
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THEREFORE THE RANGE R1 IS:2 <= X <= SQUARE ROOT (37)
WHICH IS
2 <= X <= 6.08
SINCE X IS INTEGRAL THEREFORE R1 IS
2 <= X <= 6
LET N BE 37
11
12
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2
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THEREFORE THE RANGE R1 IS:2 <= X <= SQUARE ROOT (37)
WHICH IS
2 <= X <= 6.08
SINCE X IS INTEGRAL THEREFORE R1 IS
2 <= X <= 6
LET N BE 37
37 2
18.5
11
12
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1
2
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LET X BE 2
THEREFORE N IS NOT DIVISIBLE BY 2 OR ANY OTHER MULTIPLE OF 2
LET N BE 37
2 1 2
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LET N BE 37
2 2 4
11
12
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20
1
2
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LET N BE 37
2 3 6
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
LET N BE 37
NOW, X = 3THEREFORE,
SQUARE ROOT (6) < XTHAT IS2.44 < 3
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
LET N BE 37
NOW THE ONLY NUMBERS REMAINING
IN THE RANGE R1:
2 <= X <= 6
ARE PRIME NUMBERS:3 AND 5
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
LET N BE 401
401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
THEREFORE THE RANGE R1 IS:2 <= X <= SQUARE ROOT (401)
WHICH IS
2 <= X <= 20.02
SINCE X IS INTEGRAL THEREFORE R1 IS
2 <= X <= 20
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
THEREFORE THE RANGE R1 IS:2 <= X <= SQUARE ROOT (401)
WHICH IS
2 <= X <= 20.02
SINCE X IS INTEGRAL THEREFORE R1 IS
2 <= X <= 20
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
401
2200.5
LET X BE 2
THEREFORE N IS NOT DIVISIBLE BY 2 OR ANY OTHER MULTIPLE OF 2
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
2 1 2
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
2 2 4
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
2 3 6
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
2 4 8
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
2 5 10
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
2 6 12
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
2 7 14
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
2 8 16
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
2 9 18
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
2 10
20
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
LET X BE 3
401
3133.7
THEREFORE N IS NOT DIVISIBLE BY 3 OR ANY OTHER MULTIPLE OF 3
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
3 1 3
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
3 2 6
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
3 3 9
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
3 4 12
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
3 5 15
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
3 6 18
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
NOW, X = 5THEREFORE,
20 < XTHAT IS4.47 < 5
LET N BE 401
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
..
..
..
..
..
..
..
..
397
398
399
400
401
402
403
NOW THE ONLY NUMBERS REMAINING
IN THE RANGE R1:
2 <= X <= 20
ARE PRIME NUMBERS:5, 7, 11, 13, 17 AND 19
Thank you!
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