I have found some beautiful techniques to solve complete or partially problems such as:
1- Goldbach’s conjecture
2- Riemann hypothesis
See more in my notions:
https://www.notion.so/ce1d7a1822844af7bc6df4e92cb3aca6?v=615f234f5cdd4da1b62ccb0c73ab34ad
Dear Frank Vega
I read your notes, but I got nothing! You described some open problems. Complexity theory is useful in the presence of an algorithm to tackle the problem.
First, you need to show a concrete theory and then build your algorithm with a suitable complexity time to support your proofs.
We have nothing to do with complexity in the absence of the theory.
Best regards
Dear Issam Kaddoura,
There are cases where there is no an algorithm behind. For example, if we try to show some language is non-regular, then we do not need explicitly an algorithm for that. Actually, this is one of the techniques that we used in the paper "The Complexity of Number Theory". See more about that technique in:
https://en.wikipedia.org/wiki/Pumping_lemma_for_regular_languages
Best regards
Dear Frank Vega
I read the mentioned technique. It is useful to show (a particular language is irregular) by contradiction. But it is not clear how you will employ this approach to show any of the mentioned open problems.
Best wishes
Dear Issam Kaddoura
We prove there are non-regular languages that define mathematical problems. Indeed, if those math problems are not true, then they have a finite or infinite number of counterexamples (the complement languages contain the counterexample elements). However, we know every finite language is regular. Moreover, we know the complexity class REG (the class of regular languages) is closed under complement. Therefore, those languages are true or they have an infinite number of counterexamples. Indeed, that tiny result shows problems like the Beal's conjecture are true. Certainly, there is a well-know theorem called "Darmon-Granville theorem" which proves there are not an infinite number of counterexamples for a specific choice of exponents. Consequently, our technique showed the Beal's conjecture is true and found a partial or complete proof for other mathematical problems.
Best Regard
Dear Frank Vega
Wish you good luck. I hope to see the results of the work when they appear.
Best regards
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