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For the case where is transcendental, we see that so no version of goldbach’s conjecture holds. Furthermore, in the case that , , so the only known version is Goldbach’s weak conjecture.
Now let for some natural number greater than 1. Then the atoms of are of the form where divides a power of and is a prime not dividing . Now take any . Then by goldbach's weak conjecture, can be written as the sum of at most 4 elements of the form for primes . For each term, if , the term is an atom. Otherwise, it can be written as for some . Let be a prime greater than . By Dirichlets theorem, there is some with prime. Because , . Take some where and . Then for some . So can be wrritten as a sum of atoms. So every element can be written as a sum of at most atoms.
Now let for some natural number greater than 1. Then the atoms of are of the form where divides a power of and is a prime not dividing . Now take any . Then by goldbach's weak conjecture, can be written as the sum of at most 4 elements of the form for primes . For each term, if , the term is an atom. Otherwise, it can be written as for some . Let be a prime greater than . By Dirichlets theorem, there is some with prime. Because , . Take some where and . Then for some . So can be wrritten as a sum of atoms. So every element can be written as a sum of at most atoms.
This post has been edited 2 times. Last edited by drhong, Nov 8, 2024, 3:27 PM