Explain in detail what is incorrect in the proof. A sequence of numbers is weakl

Question

Explain in detail what is incorrect in the proof.

A sequence of numbers is weakly decreasing when each number in the sequence is the numbers after it. (This implies that a sequence of just one number is weakly decreasing.) Here's a bogus proof of a very important true fact, every integer greater than 1 is a product of a unique weakly decreasing sequence of primes -a pusp. for short. Explain what's bogus about the proof. Lemma. Every integer greater than 1 is a pusp. For example. 252 = 7 . 3 . 3 . 2 . 2, and no other weakly decreasing sequence of primes will have a product equal to 252. Bogus proof. We will prove the lemma by strong induction, letting the induction hypothesis, P(n), be n is a pusp. So the lemma will follow if we prove that P(n) holds for all n 2. Base Case (n = 2): P(2) is true because 2 is prime, and so it is a length one product of primes, and this is obviously the only sequence of primes whose product can equal 2. Inductive step: Suppose that n 2 and that i is a pusp for every integer i where 2 i

Explanation / Answer

Suppose you specify another composite number n+1 = r*s, such that k*m and r*s both have different factorizations of n+1. There is no additional information in the proof that shows that the prime factorizations of k and m are unique compared to the prime factorizations of r and s. Obviously the fundamental theorem of arithmetic holds (i.e. each positive integer has a unique factorization) so we know that individually, k, m, r, and s have unique factorizations. The question now is: when you merge k and m and when you merge r and s, how can you be sure that the resulting sequences for each pair is the SAME?

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