# Goldbach Conjecture, Twin Primes Conjecture, and Bounded Gap Theorem in the language of number theory

[white paper: pedagogical]

We write the formulas of the theorem and the conjectures highlighted in the title of this white paper in the language of number theory for pedagogical purpose in first-order logic.

https://doi.org/10.31219/osf.io/u45y3

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 3, Article 229, 2021 [IZ]

# Is there a proof for the (non)existence of a formula for prime numbers?

[question]

We discuss about whether it is possible or not to prove or disprove the existence of a formula for the prime numbers.

https://doi.org/10.31219/osf.io/sb7td

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 2, Article 167, 2020 [GO]

# Transdenumerability of the reals

We show that the real numbers are transdenumerable by setting a one to one map with the set of the transfinite ordinals introduced by Cantor.

https://doi.org/10.31219/osf.io/fu7bx

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 2, Article 156, 2020 [GD]

# On the arithmetic of automated theorem proving

We propose a model to assign prime numbers to axioms and theorems, then by comparing equivalent numbers, it results in new equivalent theorems.

https://doi.org/10.31219/osf.io/wbf85

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 2, Article 132, 2020 [FC]

# Using prime numbers for automatic theorem proving

[original idea]

We apply an analogous setting from Gödel’s numbering system to automatic theorem proving.

https://doi.org/10.31219/osf.io/g7usc

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 2, Article 119, 2020 [EP]

# Complex complex numbers

[original idea]

We apply in the complex numbers the same line of thought that led to the very creation of the complex themselves. In addition, we consider multiple imaginary numbers and generalize both ideas altogether.

https://doi.org/10.31219/osf.io/485kj

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 2, Article 105, 2020 [EB]

# Cycle decomposition for the permutations of an infinite set

[original insight]

We present two cycle decompositions for the permutations of an infinite set.

https://doi.org/10.31219/osf.io/u6zwt

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 2, Article 92, 2020 [DO]

# Subtraction of Transfinite Ordinals

[microresearch]

We present the subtraction of transfinite ordinal numbers.

https://doi.org/10.31219/osf.io/yvrf3

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 2, Article 91, 2020 [DN]

# Sum of all natural numbers

[microresearch]

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + … = ?

https://doi.org/10.31219/osf.io/yx28b

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 2, Article 83, 2020 [DF]

# Arbitrarily Large Sequences with no Prime Numbers

[mathematical insight]

We present a discussion on the ingenious mathematical insight, x=(n+1)!+2, regarding the proof of the theorem “For every positive integer n, there is a sequence of n consecutive positive integers containing no primes.”

https://doi.org/10.31219/osf.io/3cr7b

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 1, Article 64, 2019 [CL]

# Greatest Common Divisors of the Fibonacci numbers and their indices

[microreview]

A very interesting property of the Fibonacci numbers alongside with the steps involved in its proof are presented.

https://doi.org/10.31219/osf.io/3tqeg

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 1, Article 63, 2019 [CK]

# Periodicity of the rational numbers

[microresearch]

In this article, we prove that alpha is a rational number if, and only if, its decimal representation possesses a period.

https://doi.org/10.31219/osf.io/gwzhv

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 1, Article 42, 2019 [BP]

[microresearch]