Question

Prove that there are infinitely many prime numbers

Answer 1

Answer:

BY EUCLUD'S THEOREM

Step-by-step explanation:

Assume there are a finite number, n , of primes , the largest being p n .

Consider the number that is the product of these, plus one: N = p 1 ... p n +1.

By construction, N is not divisible by any of the p i .

Hence it is either prime itself, or divisible by another prime greater than p n , contradicting the assumption.

q.e.d.

For example:

2 + 1 = 3, is prime

2*3 + 1 = 7, is prime

2*3*5 + 1 = 31, is prime

2*3*5*7 + 1 = 211, is prime

2*3*5*7*11 + 1 = 2311, is prime

2*3*5*7*11*13 + 1 = 30031 = 59*509

2*3*5*7*11*13*17 + 1 = 510511 = 19*97*277

2*3*5*7*11*13*17*19 + 1 = 9699691 = 347*27953

2*3*5*7*11*13*17*19*23 + 1 = 223092871 = 317*703763

2*3*5*7*11*13*17*19*23*29 + 1 = 6469693231 = 331*571*34231

2*3*5*7*11*13*17*19*23*29*31 + 1 = 200560490131, is prime

2*3*5*7*11*13*17*19*23*29*31*37 + 1 = 7420738134811 = 181*60611*676421

etc.

Answer 2

Answer:

I assume you know what a prime number is. There are infinitely many of them!

The following proof is one of the most famous, most often quoted, and most beautiful proofs in all of mathematics. Its origins date back more than 2000 years to Euclid of Alexandria who lived around 300 BC. Euclid's argument was different, but this is the proof that is most commonly given today:

See the attachment below to understand