Exercises outline two additional proofs that there are infinitely many primes.This exercis...

Exercises outline two additional proofs that there are infinitely many primes.

This exercise develops a proof that there are infinitely many primes based on the fundamental theorem of arithmetic published by A. Auric in 1915. Assume that there are exactly r primes, P1 < P2 < ⋯r. Suppose that n is a positive integer and let .

a) Show that an integer m with 1 ≤ m ≤Q can be written uniquely as , where ei ≥ 0 for i = 1, 2, … , r Furthermore, show that for the integer m with this factorization, .

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b) Let C = (log pr)/(log p1). Show that ei ≤ n C for i = 1, 2, … , r and that Q does not exceed the number of r-tuples (e1, e2, … , er) of exponents in the prime-power factorizations of integers m with 1 ≤ m ≤ Q.

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c) Conclude from part (b) that .

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d) Show that the inequality in part (c) cannot hold for sufficiently large values of n. Conclude that there must be intinitely many primes.