How many prime numbers are there? The following problem anticipates some of the

Question

How many prime numbers are there? The following problem anticipates some of the topics we will go through later on in the course. However, all you need to know to be able to solve it is some look at definition of prime number, divisibility and factorization (reference material here). Focus first on the questions marked with (*). Read the two proofs at this link and understand them, focusing on whichever one you want (first is simpler in a way, but the second may help you answer (a) ). How do we formalize in logic the statement of the theorem? (*) What are some logical inference rules we are using in the proof? Would the proof work anyway if instead of doing “+1” we did “+3” at the crucial step? (*) How would the proof (and the theorem statement) change if we only wanted to prove that there is not an even number of primes? What is the logical connection between the original statement of the theorem and the one at (d)? For example, does one imply the other? (*)

Explanation / Answer

Prime Numbers are those numbers which have just two factors(including the number itself).

Ex.2,3,5,7,11,13 etc.

But to effectively calculate whether a number is prime or not , we need not check every factor but only need to count the factors till the square root of the number----Sieve of Erastothenes.

a) the logic is formulizes as square root theorem named above.

Note. Your question does not specify the proof it is talking about so the middle part is unanswerable.Also part (d) cannot be answered on the basis of the information provided by you.

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