Number Theory


An Introduction to the $n$-Irreducible Sequents and the $n$-Irreductible Number

Authors: A. Zaganidis

In this work, we introduce the $n$-irreductible sequents and the $n$-irreductible numbers defined with the help of the second order logic. We give many concrete examples of $n$-irreductible numbers and $n$-irreductible sequents with the Peano's axioms and the axioms of the real numbers. Shortly, a sequent is $n$-irreductible iff the sequent is composed by some closed hypotheses and a $n$-irreductible formula (a close formula with one internal variable such that the formula is only true when we set that variable to the unique natural number $n$), and it does not exist some strict sub-sequent which are composed by some closed sub-hypotheses and some sub-$m$-irreductible formula with $m>1$. The definition is motivated by the intuition that \Nathypo do not carry natural numbers or "hidden natural numbers" except for the numbers $0$ and $1$, i.e., they can be used in a $n$-irreductible sequent. Moreover, we postulate at second order of logic that \Nathypo are not chosen randomly: \Nathypo are the only hypotheses which give the largest $n$-irreductible number $N_Z \NZ$. The Goldbach's conjecture, \poubelle{the Dubner's conjecture,}the Polignac's conjecture, the Firoozbakht's conjecture, the Oppermann's conjecture, the Agoh-Giuga conjecture, the generalized Fermat's conjecture and the Schinzel's hypothesis H are reviewed with this new (second order logic) $n$-irreductible axiom. Finally, two open questions remain: Can we prove that a natural number is not $n$-irreductible? If a $n$-irreductible number $n$ is found with a function symbol $f$ where its outputs values are only $0$ and $1$, can we always replace the function symbol $f$ by a another function symbol $\tilde{f}$ such that $\tilde{f}=1-f$ and the new sequent is still $n$-irreductible?

Comments: 19 Pages. I have chosen the category Number Theory since most of the consequences of the present article are inside the number theory.

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Submission history

[v1] 2017-04-03 11:20:53
[v2] 2017-06-02 04:16:44
[v3] 2018-03-19 23:59:11

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