Planned intervention: On Wednesday April 3rd 05:30 UTC Zenodo will be unavailable for up to 2-10 minutes to perform a storage cluster upgrade.

There is a newer version of the record available.

Published February 16, 2020 | Version v16
Preprint Open

The Complexity of Number Theory

Creators

  • 1. Joysonic

Description

On the one hand, the Goldbach's conjecture has been described as the most difficult problem in the history of Mathematics. This conjecture states that every even integer greater than 2 can be written as the sum of two primes. This is known as the strong Goldbach's conjecture. The conjecture that all odd numbers greater than 7 are the sum of three odd primes is known today as the weak Goldbach conjecture. A major complexity class is NSPACE(S(n)) for some S(n). We show if the weak Goldbach's conjecture is true, then the problem PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n). However, if PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n), then the strong Goldbach's conjecture is true or this has an infinite number of counterexamples. Since Harald Helfgott proved that the weak Goldbach's conjecture is true, then the strong Goldbach's conjecture is true or this has an infinite number of counterexamples, where the case of infinite number of counterexamples statistically seems to be unlikely. On the other hand, we have the functional problem FACTORIZATION is not in NSPACE(S(n)) for all S(n) = o(log n), since this uses the decision problem PRIMES as subroutine. However, this implies that the Beal's conjecture is true. Since the Beal's conjecture is a generalization of Fermat's Last Theorem, then this is also a simple and short proof for that Theorem.

Files

manuscript.pdf

Files (365.6 kB)

Name Size Download all
md5:68bb85e0583b26bab3b49dfab148e69b
365.6 kB Preview Download