primes????

Well, we know that a prime number can be expressed as 6k+1. So the product of first n primes (which contain 2 and 3), it will be a multiple of 6. Add one to it and you have a prime [1]

HOPE I'M NOT BLABBERING[4]

either evryone knows or they re too content with the 6k+1 thing
but hey you must have come across
Euclids proof that there are infinite no. of primes if you know that then this is too easy. in that case you can neglect this thread [1]

correct but i didnt understand what breaks down

surely you don mean euclids proof is wrong coz it is slightly different from the prob given above the prob is often used to test the understanding of the poof itself

what breaks down is the conjecture that p#+1 is a prime where p# is the product of all primes less than or equal to p.

Euclid's proof deals with proving that the set of primes is not finite. So, if there is a last prime number p_n consider p_n#+1.

If it is a prime we are done.

If its not, the fundamental theorem of algebra says that it has a prime factor. But none of the primes in the set (2,3,5,...,p_n) divides p_n#+1. Hence we have produced at least one prime distinct from the given set and since n is arbitrary, the number of primes is without bound.

according to the fundamental theorem of arithmetic a number X can be expressed as a product of primes less or = X. let product of n numbers be N
so N+1 is not divisible be any primes <N. thus we can say it itself is a prime

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