The Prime Number Theory
The Prime Number Theory
- Definition of Prime number
- Difference between prime numbers and odd numbers
- Euclid's proof,
- nth term of prime number series
- The value of x
- Square Subtraction Theorem (S.S.T.)
- Solution for Goldbach's conjecture
Definition of Prime number :
➢ The natural numbers, which are only divisible by 1 and itself, are called prime numbers.
➢ The prime number – series :
2,3,5,7,11,13,17,19,23,29,31,37,41,43…
Difference between Prime numbers and Odd numbers :
➢ The prime numbers are not square numbers nor cube numbers. A prime is not divisible by any other prime.
➢ ex. 9 is not prime (32 = 9)
➢ If we divide prime by any other prime we get a fraction instead of integer.
➢ i.e. prime/prime integer (Z)
Euclide's Proof
Euclid's Elements
Book IX
Proposition 20
n-th term of prime number series :
➢ The series of odd number set (starting from 3) have all the members of prime number set (except 2).
➢ Odd numbers :
3,5,7,9,11,13,15,17,19,21,23,25...
➢ Prime numbers :
2,3,5,7,11,13,17,19,23,29,31,37…
➢ If we find the formula of Tn of odd numbers series, may we can find Tn of prime number series.
➢ Tn = a + (n - 1)d
Here, a = 3 ( we will consider 3 as the first term)
d= T2 - T1 = 5 - 3 = 2
➢ Thus, Tn = 3 + (n-1)2
= 3 + 2n -2
∴ Tn = 2n +1 n ∈ N
➢ We can observe that this formula can be used for prime numbers too.
➢ Here, we can see that the 4th term of the prime number series is 11, But by taking n = 4 in formula, we get 9 instead of 11. Here 9 is an odd composite number.
➢ When we take n=6 in formula, we get 13 instead of 17.
➢ When we take n=10 in formula, we get 21 instead of 31.
➢ That’s how, we knew that this formula can be used for both- prime numbers and odd numbers.
➢ We should make the formula that it can be used only for the prime numbers.
➢ In the explanation given above, we saw that for some value of n, we have to add some number, to get the right value of Tn.
➢ By observing carefully, we see that - after every odd composite number, there is addition of 1 in the value of n.
➢ Suppose, no. of odd composite number = x.
➢ ∴ Tn = 2(n + x) + 1
= 2n + 2x +1 n ∈ N & x ∈ W
➢ But, here n is no. of term ( in prime number series) starting from 3 instead of 2.
➢ So, we have to take (n-1) in place of n, to get the right value of Tn.
➢ ∴ Tn = 2(n – 1) + 2x + 1
= 2n - 2 + 2x + 1
∴ Tn = 2n + 2x -1 n ∈ N & x ∈ W
▪ nth term of prime number series :
➢Tn = 2n + 2x - 1 n ∈ N & x ∈ W
Where, n = no. of term (in prime number series), n 1
(use n = 1.5 instead of 1 for 1st term)
x = no. of odd composite numbers < Tn.
The value of x :
➢ The value of x can not be expressed by any formula or mathematical calculation.
➢ We don’t know how many odd composite numbers are there, less than Tn.
➢ The value of x for few n :
➢ ex. n = 39, x = 45
➢ Thus, Tn = 2(39) + 2(45) - 1
= 78 + 90 – 1 = 167
Hence, 39th prime number is 167.
Square Subtraction Theorem :
Statement : Every prime number ( greater than 2 ) can be expressed as a Subtraction of two square numbers.
➢ P = ( k + 1 )2 - k2 k ∈ N
➢ Algebraic Proof : Here, This Theorem is for prime number- greater than 2, so we can use the formula p = 2n + 2x + 1 instead of p = 2n + 2x -1.
➢ ∴ P = 2n + 2x + 1
= (n+x)2 + (2n+2x+1) - (n+x)2
= (n2+x2+2nx+2n+2x+1) - (n+x)2
= (n2+x2+1+2nx+2n+2x) - (n+x)2
= (n+x+1)2 - (n+x)2
Suppose, (n+x) = k
➢ ∴ P = (k+1)2 – (k)2
Here, n ∈ N & x ∈ W
∴ (n+x) ∈ N
∴ k ∈ N
➢ Geometrical Proof : Suppose, a right angle triangle ∆ABC, with the sides a, b and c.
➢ a is hypotenuse,
➢ ∴ m∠B = 90°
m∠C = 60°
m∠A = 30°
➢ ∴ SinA = Sin(30) = ½ = b/a
➢ ∴ SinC = Sin(60) = √3/2 = c/a
➢ ∴ SinB = Sin(90) = 1 = a/a
➢ ∴ a = 2, b = 1, c = √3
➢ Here, using Pythagorean Theorem,
➢ a2 = b2 + c2
➢ ∴ (2)2 = (1)2 + (√3)2
➢ ∴ (2)2 - (1)2 = (√3)2
➢ ∴ 3 = (2)2 - (1)2
➢ The few examples of S.S.T. :
➢ 3 = (2)2 - (1)2
5 = (3)2 - (2)2
7 = (4)2 - (3)2
11 = (6)2 - (5)2
Solution for Goldbach's Conjecture :
Goldbach's conjecture is one of the oldest and best known unsolved problems in number theory and all of mathematics. It states :
Every even integer greater than 2 can be expressed as the sum of two primes.
Proof 1 :
Suppose, a and b are two prime numbers.
According to Square Subtraction theorem,
a and b can be expressed as ...
➢ a = (k+1)2 - k2 a,k ∈ N
b = (v+1)2 - v2 b, v ∈ N
➢ ∴ a + b = (k+1)2 + (v+1)2 - k2 - v2
= (k2 + 2k + 12 - k2) + (v2 + 2v + 12 - v2)
= (2k + 2v +2)
= (2k + 2v +2)
= 2(k + v + 1)
➢ here, k, v ∈ N
∴ (k + v + 1) ∈ N
∴ [(a+b)/2] ∈ N
➢ ∴ (a+b) is an even number.
Proof 2 :
Suppose, a and b are two prime numbers.
According to formula of Tn (for prime number series) a and b can be expressed as ...
➢ a = 2n + 2x - 1 a, n ∈ N x ∈ W
b = 2m + 2y - 1 b, m ∈ N y ∈ W
➢ ∴ a + b = (2n + 2x - 1) + (2m + 2y - 1)
= 2n + 2x - 1 + 2m + 2y - 1
= 2n + 2m + 2x + 2y – 2
= 2(n + m + x + y - 1)
➢ here, n, m ∈ N & x, y ∈ W
∴ (n + m + x + y - 1) ∈ N
∴ [(a+b)/2] ∈ N
➢ ∴ (a+b) is an even number.
➢ Thus, Every even integer greater than 2 can be expressed as the sum of two primes.
References
- ↑ "Euclid's Elements, Book IX, Proposition 20". mathcs.clarku.edu. Retrieved 2018-10-10.
This article incorporates text from a publication now in the public domain: Sir Thomas Little Heath (1908). The thirteen books of Euclid's Elements. Cambridge: Cambridge University Press.
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