(a) What does it mean to say that an integer is prime? (b) Show that every integ

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(a) What does it mean to say that an integer is prime?

(b) Show that every integer n  2 has a prime factor.

(c) Show that there are in nitely many prime numbers.

Apr 18th, 2015

(a) An integer n ≥ 2 is a prime if its only divisors are 1 and n itself (that is, if k | n, then either k = 1 or

k = n).

(b) Take any integer n ≥ 2. Prove by contradiction. Assume that n has no prime factors. Then n is not a prime because n = n × 1 and n is a factor of itself. Since n is not a prime, it has a factor m1 such that

1 < m1 < n, n = m1 × k1. By applying the same reasoning to the number m1 we get its factor m2 , 1 < m2 < m1, m1 = m2 × k2  etc. However, the sequence m1 > m2 > m3 > … cannot be continues indefinitely because there is only a finite number of integers between n and 1. We arrived at a contradiction, thus, n must have a prime factor.

(c) Prove by contradiction. Suppose that there is a finite set of prime numbers: p1, p2, p3, … , pn. Construct a number N = p1p2p3…pn + 1. Since (b) is true, it must have a prime factor p, however, no prime number can be a factor of N. Q.E.D.


Apr 18th, 2015

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