Question

Is there a finite number of prime numbers?

Answer 1

★ THEORY OF NUMBERS ★

Utilizing mathematical generality at it's best :

If possible , suppose that the numbers of prime is finite

∃ the greatest prime say q

Let b denote the product of these primes , 2 , 3 , 5... q

i.e., let

b = 2 × 3 × 5 × ... q ... (i)

let , a = b + 1 ... ( ii )

Surely , a ≠ 1

aslike : a = b + 1 > 1

The Number " a " must have a prime say factor p

i.e. , p | a

Now , p is one of the primes 2 , 3 , 5 , 7 , ... q

Accordingly to our assumptions , it's the only primes series , that's why putting up in consideration

b = 2 ( 3 ) ( 5 ) ... q

p | b

Again , p | a and p | b

p | a - b OR p | 1

From eqn ( ii ) ... a - b = 1

p = 1 [ which is impossible ]

Because , 1 is not a prime number

So , our supposition is false ,

HENCE ,

THE NUMBER OF PRIMES IS INFINITE

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