# I saw this question, in a sample paper for Maths Olympiad:Prove that there are infinitely many prime numbers.Ofcourse this one is obvious, but how to prove it mathematically. Pls Hlp.

290 Points
10 years ago

Hi Speed Racer,

Firstly let's assume there are "n" prime numbers P1,P2.....Pn {where n is finite}

Now consider X = P1*P2*.....*Pn + 1.

For any integer number "a" (where a>1), aT is a multiple of "a" (T is any natural number). And hence aT+1, will not be divisible by "a" ------------------Statement (1)

Now consider the above X (it can be either a prime number or a composite number)

In case it is a prime number, then we have one more prime number other than P1,P2,P3,...Pn

In case it is a composite number, then X should be divisible by some prime number (But claearly as mentioned in statement (1), X cannot be divisible by any of the prime number P1,P2,....,Pn) So X must be divisible by some prime number that is different from P1,P2,...Pn.

So in any case, we have one more prime number (say P0), which is different from P1,P2,....Pn.

So now take the set of numbers.... P0,P1,P2,....,Pn and repeat the same, you will have more prime number P(n+1).

Keep on repeating this, and you will keep on generating prime numbers (And hence there are infinitely many prime numbers).

Hope that helps.

All the best.

Regards,

Chetan Mandayam Nayakar
312 Points
10 years ago

Dear Speed,

Let tus assume that here are finite number of prime numbers: p1,p2,...pn

Then,p1*p2*p3...pn+1 is prime

therefore, by self-contradiction the assumption that there are finitely many prime numbers is wrong.

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All the best Speed !!!

Regards,

CHETAN MANDAYAM NAYAKAR

Speed Racer
15 Points
10 years ago

Chetan Sir,

Please tell me how is p1p2p3...pn+1 is prime??

Say p1=3, p2=5.

How is p1p2+1 = 16 a prime?