CBSE BOARD X, asked by SRH1415, 1 year ago

show that there are infinitely many positive primes.

Answers

Answered by yalancy
1

Answer:

+ 1 is either prime or divisible by a prime larger than n. In either case, for every positive integer n, there is at least one prime bigger than n. The conclusion is that the number of primes is infinite.

Explanation:

Consider any finite list of prime numbers p1, p2, ..., pn. It will be shown that at least one additional prime number not in this list exists. Let P be the product of all the prime numbers in the list: P = p1p2...pn. Let q = P + 1. Then q is either prime or not:

If q is prime, then there is at least one more prime that is not in the list.

If q is not prime, then some prime factor p divides q. If this factor p were in our list, then it would divide P (since P is the product of every number in the list); but p also divides P + 1 = q, as just stated. If p divides P and also q, then p must also divide the difference[3] of the two numbers, which is (P + 1) − P or just 1. Since no prime number divides 1, p cannot be on the list. This means that at least one more prime number exists beyond those in the list.

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