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Primary Problem (Posted on 2002-08-21) Difficulty: 4 of 5
Prove that there exists an infinitely large number of primes.

See The Solution Submitted by Dulanjana    
Rating: 3.5000 (8 votes)

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Solution Puzzle Solution | Comment 12 of 13 |

If possible let the total number of primes be finite and P be the largest prime.

Let  F(P) = 2*3*.........*P, so that F(P) corresponds to the product of all possible primes< = P

Now, (F(P), F(P) +1) = 1, so that each of the prime factors of  F(P) + 1 must be relatively prime to each of the prime factors of F(P).

Hence, it follows that each of the prime factors of F(P) + 1 must be different from all the known primes upto P.
This is a contradiction.

Consequently, it follows that there exists an infinity of prime numbers.

  Posted by K Sengupta on 2007-03-30 10:33:31
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