Consider the following statement.For every integer n, if n > 2 then there is a prime number p such that n < p < n!.The statement is true, but the following proposed proof is incorrect.Proposed proof:Let n be any integer that is greater than 2.By the infinitude of the primes theorem (Theorem 4.8.4), there is a prime number p such that n < p.Since n > 2, n! − 1 is an integer that is greater than 1.So by the divisibility by a prime theorem (Theorem 4.4.4), there is a prime number p such that p | (n! − 1).Thus, p ≤ n! − 1, and so p < n!.Therefore, by steps 2 and 5, there is a prime number p such that n < p < n!.Identify the error(s) in the proposed proof. (Select all that apply.)p | (n! − 1) does not imply that p is less than or equal to n! − 1.The statement can only be proved by contradiction.The divisibility by a prime theorem is incorrectly applied in step 4.The p in step 2 may not be the same as the p in step 5.The proof does not work if n is prime.The proof does not work if n! − 1 is prime.The infinitude of the primes theorem is incorrectly applied in step 2.

The infinitude of the primes theorem states that there are infinitely many prime numbers. However,…

Answered: Consider the following statement. For…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Consider the following statement.

For every integer n, if

n > 2

then there is a prime number p such that

n < p < n!.

The statement is true, but the following proposed proof is incorrect.

Proposed proof:

Let n be any integer that is greater than 2.
By the infinitude of the primes theorem (Theorem 4.8.4), there is a prime number p such that
n < p.
Since
n > 2, n! − 1
is an integer that is greater than 1.
So by the divisibility by a prime theorem (Theorem 4.4.4), there is a prime number p such that
p | (n! − 1).
Thus,
p ≤ n! − 1,
and so
p < n!.
Therefore, by steps 2 and 5, there is a prime number p such that
n < p < n!.

Identify the error(s) in the proposed proof. (Select all that apply.)

p | (n! − 1) does not imply that p is less than or equal to n! − 1.The statement can only be proved by contradiction.The divisibility by a prime theorem is incorrectly applied in step 4.The p in step 2 may not be the same as the p in step 5.The proof does not work if n is prime.The proof does not work if n! − 1 is prime.The infinitude of the primes theorem is incorrectly applied in step 2.

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