This question is about bogus proof. Please include detailed explanations in your

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This question is about bogus proof. Please include detailed explanations in your answer. Thumbs up if your answer helps. Thanks so much!!!

MCS 5.26: 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 repeated is wea decreasing. Here's a bogus proof of a very important true fact, that 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 hypoth- esis, P(n) be "n is a pusp So the lemma will follow if we prove that Pn) 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 S i r We must show that P n 1) holds, namely, that (n 1) is also a pusp. We argue by cases: If (n 1) is itself prime, then it is the product of a length one sequence consisting of itself This sequence is unique, since by definition of prime (n 1) has no other prime factors. So (n 1) is a pusp, that is P n 1) holds in this case Otherwise (n 1) is not prime, which by definition means n 1 km for some integers k and m such that 2 S k, m n 1. Now by the strong induction hypothesis, we know that k and m are pusps. It follows that by merging the unique prime sequences for k and m, in sorted order, we get a unique weakly decreasing sequence of primes whose product equals n 1. So n 1 is a pusp, in this case as well So P n 1) holds in any case, which completes the proof by strong induction that P(n) holds for all n

Explanation / Answer

Suppose that km and rs are two different non-trivial factorizations of n+1; how do you know that the merged prime sequence from km is the same as the merged prime sequence from rs? In other words, how do you know that you have uniqueness? Let us take an example that illustrates the breakdown. That example cannot be the ordinary integers, since Unique Factorization holds there.
Suppose that our world consists of the set E of even positive integers. Call such an integer eprime if it cannot be expressed as the product of elements of E. So for example 6 is eprime, while 12 is not.
One could use almost exactly the same argument to "prove" that every element of E is a pusep (the ep at the end is for eprime). However, note that 180=(18)(10)=(30)(6), and all four of 18, 10, 30, and 6 are eprime.

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