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

[question]

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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

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

Transdenumerability of the reals

[white paper]

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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

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

A unidimensional windmill in the plane with varying pivot points

[white paper]

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We show that we can choose a point in the plane such that the resulting windmill process with varying pivot uses each point of the plane infinitely many times.

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

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 148, 2020 [FT]

Generating finitely many circles from one circle using Banach-Tarski decomposition paradox

[microreview]

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We prove the Banach-Tarski decomposition paradox applied to a circle.

https://doi.org/10.31219/osf.io/mh5gx

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 142, 2020 [FN]

A subgroup has equally many left and right cosets

[microreview]

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We prove that a subgroup has the same number of left and right cosets.

https://doi.org/10.31219/osf.io/t4snh

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 137, 2020 [FH]

On the arithmetic of automated theorem proving

[white paper]

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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

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

Test functions and Distributions (Open Mathematics Knowledge Base)

[knowledge base]

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In this work, we present a list of mathematical results about test functions and distributions theory for future implementation in a digital Open Mathematics Knowledge Base.

https://doi.org/10.31219/osf.io/xne52

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 124, 2020 [EU]

Logical Loop

[microresearch]

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We discuss the dubbed term “logical loop” and it’s implication regarding provability in undecidable theorems.

https://doi.org/10.31219/osf.io/c5ezm

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 123, 2020 [ET]

Differential Forms (Open Mathematics Knowledge Base)

[knowledge base]

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We present a list of mathematical results about differential forms for future implementation in a digital Open Mathematics Knowledge Base.

https://doi.org/10.31219/osf.io/g7uqb

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 122, 2020 [ES]

A semigroup is a rectangular band if and only if it is nowhere commutative

[microreview]

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We prove the proposition addressed in the title of this paper.

https://doi.org/10.31219/osf.io/98khs

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 120, 2020 [EQ]

Using prime numbers for automatic theorem proving

[original idea]

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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

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

An element of a finite monoid is right invertible if and only if it is left invertible

[microreview]

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We prove the proposition addressed in the title of this paper.

https://doi.org/10.31219/osf.io/7vm92

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 118, 2020 [EO]

A finite cancellative semigroup is a group

[microreview]

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We prove the proposition addressed in the title of this paper.

https://doi.org/10.31219/osf.io/34vbp

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 117, 2020 [EN]

A right zero semigroup is left-cancellative

[microreview]

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We prove the proposition addressed in the title of this paper.

https://doi.org/10.31219/osf.io/g6snd

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 116, 2020 [EM]

A semigroup with a right zero equal to a right identity is trivial

[microreview]

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We prove the proposition addressed in the title of this paper.

https://doi.org/10.31219/osf.io/zpwb5

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 115, 2020 [EL]

A simple group of order 168 is isomorphic to PGL(2,7)

[question]

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We present a sketch on a problem related to the isomorphism between the simple group of order 168 and the projective general linear group.

https://doi.org/10.31219/osf.io/ydex4

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 114, 2020 [EK]

A simple group of order 60 is isomorphic to a subgroup of S6

[microreview]

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We show that the permutation of six Sylow 5-subgroups by conjugation is a faithful action, so that G is isomorphic to a subgroup of S6.

https://doi.org/10.31219/osf.io/g43yj

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 113, 2020 [EJ]

Non-classical logic and undecidability

[conjecture]

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We elaborate a conjecture by applying the definition of Non-Axiomatic System [1] in Non-Classical Logic.

https://doi.org/10.31219/osf.io/h4f7p

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 112, 2020 [EI]

3-cycle commutator

[microreview]

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We show within the maximum number of steps that if only one point is moved by two permutations, the commutator is a 3-cycle.

https://doi.org/10.31219/osf.io/kxnt8

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 106, 2020 [EC]

Complex complex numbers

[original idea]

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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

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

Open Mathematics Knowledge Base

[knowledge base]

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We present a list of mathematical results for future implementation in a digital Open Mathematics Knowledge Base.

https://doi.org/10.31219/osf.io/evq8a

Open Mathematics Collaboration

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Op. J. Math. Phys.
Volume 2, Article 103, 2020 [DZ]