In this study, I examined the Beals conjecture on the basis of formulas that produce composite numbers, and I developed some other conjectures that, if answered, would be an answer to the Beals conjecture. Apart from this, I have examined the Beals conjecture from another perspective and posed The Germany Problem. 

The Germany Problem for Beal's Conjecture

In this article we have constructed a basic methodology from which important perspectives for both the Collatz conjecture and Beal's conjecture can be derived. We falsified the conjecture we previously developed on Beal's conjecture and made the falsified conjecture usable for Collatz conjecture. In this regard, we have clearly expressed the numbers that can reach 1 in the Collatz conjecture. Then, using this, we created a limit for the numbers that should be considered for the Collatz conjecture. Thanks to the analysis of this limit, we have clearly expressed the recursive Collatz function in an equation. Using this structure, we wrote a function that indicates how many times the Collatz function repeats certain numbers to reach 1. 

Calculation of the Number of Transactions for Collatz Conjecture

In Chapter 2, we made extensive use of the fact that every positive integer can be written in one and only one way as a product of powers of distinct primes. This property of $$\mathbb Z$$ is basic to mathematics. It is so basic that many people don’t even think to question it. This is especially true in school, where students spend much of elementary school working with integers, using this unique factorizationUnique factorization property as if it were a law of nature. For example, young children build “factor trees” for whole numbers, and it is usually taken for granted that two different trees, like those in Figure 3.1, end up with the same set of prime factors. 

The Fundamental Theorem of Arithmetic

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The Fundamental Theorem of Arithmetic