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Preprint ARTICLE | doi:10.20944/preprints202306.0163.v1

Posner’s Theorem and *-Centralizing Derivations on Prime Ideals with Applications

Shakir Ali, Turki M. Alsuraiheed, Mohammad Salahuddin Khan, Cihat Abdioglu, Mohammed Ayedh, Naira N. Rafique

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Derivation; *-centralizing derivation; *-commuting derivation; involution; prime ideal; prime ring; semiprime ring

Online: 2 June 2023 (08:59:17 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202304.0931.v2

Proof the Collatz Conjecture in Binary Strings

Jishe Feng

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Collatz conjecture; binary number; ultimately periodic; chaos

Online: 19 May 2023 (16:16:49 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202303.0420.v7

A Common Approach to Three Open Problems in Number Theory

Apoloniusz Tyszka

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Brocard's problem; Brocard-Ramanujan equation x!+1=y^2; composite Fermat numbers; composite numbers of the form 2^(2^n)+1; Erd\"os' equation x(x+1)=y!

Online: 18 May 2023 (08:25:27 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202305.1118.v1

The Recursive Structures of Manin Symbols over $\mathbb{Q}$, Cusps and Elliptic Points on $X_0(N)$

SanMin Wang

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: modular curve; elliptic curve; recursive structure; Manin's symbols over $\mathbb{Q}$; cusps; elliptical points; algorithmic number theory

Online: 16 May 2023 (07:23:25 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202305.0905.v1

Transposed Poisson Superalgebra

Viktor Abramov, Olga Liivapuu

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Commutative superalgebra; Lie superalgebra, Poisson algebra; transposed Poisson algebra; graded differential algebra; quantum Yang-Mills theory

Online: 12 May 2023 (08:38:03 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202303.0420.v6

A Common Approach to Three Open Problems in Number Theory

Apoloniusz Tyszka

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Brocard's problem; Brocard-Ramanujan equation x!+1=y^2; composite Fermat numbers; composite numbers of the form 2^(2^n)+1; Erd\"os' equation x(x+1)=y!

Online: 11 May 2023 (04:42:23 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202304.1241.v3

Closed Form Solution to the Roots of a Quadratic Equation over the Quaternions

Edward King Solomon

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Quaternion; Quadratic Equation; Roots

Online: 9 May 2023 (04:36:31 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202304.0093.v6

Resolution of the 3n+1 Problem Using Inequality Relation Between Indices of 2 and 3

Gaurav Goyal

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Collatz conjecture; 3n+1; inequality relations

Online: 9 May 2023 (04:15:37 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202304.1241.v2

Closed Form Solution to the Roots of a Quadratic Equation over the Quaternions

Edward King Solomon

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Quaternion; Quadratic Equation; Roots

Online: 6 May 2023 (04:13:34 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202304.0093.v5

Resolution of the 3n+1 Problem Using Inequality Relation Between Indices of 2 and 3

Gaurav Goyal

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Collatz conjecture; 3n+1; inequality relations

Online: 5 May 2023 (10:18:23 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202305.0071.v1

Does the Collatz Sequence Eventually Reach 1 for All Positive Integer Initial Values?

Masoud Allame, Abdollah Hadi-Vencheh

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Collatz Conjecture; 3x + 1 Conjecture; base-2 numeral system

Online: 2 May 2023 (08:10:49 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202304.1241.v1

Closed Form Solution to the Roots of a Quadratic Equation over the Quaternions

Edward King Solomon

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Quaternion; Quadratic Equation; Roots

Online: 30 April 2023 (02:43:13 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202303.0420.v5

A Common Approach to Three Open Problems in Number Theory

Apoloniusz Tyszka

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Brocard's problem; Brocard-Ramanujan equation x!+1=y^2; composite Fermat numbers; composite numbers of the form 2^(2^n)+1; Erd\"os' equation x(x+1)=y!

Online: 27 April 2023 (04:44:46 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202304.0931.v1

Proof the Collatz Conjecture in Binary Strings

Jishe Feng

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Collatz conjecture; binary number; ultimately periodic; chaos

Online: 26 April 2023 (03:01:31 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202304.0093.v4

Resolution of the 3n+1 Problem Using Inequality Relation Between Indices of 2 and 3

Gaurav Goyal

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Collatz conjecture; 3n+1; inequality relations

Online: 21 April 2023 (09:25:33 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202303.0420.v4

A Common Approach to Three Open Problems in Number Theory

Apoloniusz Tyszka

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Brocard's problem; Brocard-Ramanujan equation x!+1=y^2; composite Fermat numbers; composite numbers of the form 2^(2^n)+1; Erd\"os' equation x(x+1)=y!

Online: 19 April 2023 (05:34:48 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202108.0146.v25

A Proof Of The Riemann Hypothesis Based On A New Expression Of The Completed Zeta Function

Weicun Zhang

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Riemann Hypothesis (RH); Proof; Completed zeta function

Online: 17 April 2023 (04:41:40 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202303.0420.v3

A Common Approach to Three Open Problems in Number Theory

Apoloniusz Tyszka

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Brocard's problem; Brocard-Ramanujan equation x!+1=y^2; composite Fermat numbers; composite numbers of the form 2^(2^n)+1; Erd\"os' equation x(x+1)=y!

Online: 17 April 2023 (04:05:43 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202304.0093.v3

Resolution of the 3n+1 Problem Using Inequality Relation Between Indices of 2 and 3

Gaurav Goyal

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Collatz conjecture; 3n+1; inequality relations

Online: 11 April 2023 (10:05:24 CEST)

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Collatz conjecture states that an integer n reduces to 1 when certain simple operations are applied to it. Mathematically, the Collatz function is written as .

f^{k} (n) = \frac{3^{k} n + C}{2^{z}}

, where z, k, C

\geq

1.Suppose the integer n violates Collatz conjecture by reappearing, then the equation modifies to

n = \frac{3^{k}}{2^{z}} n + \frac{C}{2^{z}}

. The article takes an elementary approach to this problem by calculating the maximum value of

\frac{C}{2^{Z}}

. Correspondingly, an upper limit on the integer n is placed that re-appears in the sequence. The limit is found to be

n < 3.5

. Next, it shown that the integer n repeats in the sequence if

n < 0

. Finally, it is shown that no integer chain exists that does not lead to 1.

Preprint ARTICLE | doi:10.20944/preprints202304.0093.v2

Resolution of the $3n+1$ Problem Using Inequality Relation Between Indices of 2 and 3

Gaurav Goyal

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Collatz conjecture; 3n+1; inequality relations

Online: 10 April 2023 (08:40:06 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202304.0093.v1

Resolution of the $3n+1$ Problem Using Inequality Relation Between Indices of 2 and 3

Gaurav Goyal

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Collatz conjecture; 3n+1; inequality relations

Online: 6 April 2023 (11:21:24 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202304.0043.v1

Towards Finding Equalities Involving Mixed Products of the Moore–Penrose and Group Inverses of a Matrix

Yongge Tian

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: block matrix; group inverse; Moore--Penrose inverse; range; rank; reverse-order law

Online: 4 April 2023 (12:16:28 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202303.0420.v2

A Common Approach to Three Open Problems in Number Theory

Apoloniusz Tyszka

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Brocard's problem; Brocard-Ramanujan equation x!+1=y^2; composite Fermat numbers; composite numbers of the form 2^(2^n)+1; Erd\"os' equation x(x+1)=y!

Online: 29 March 2023 (03:34:34 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202303.0420.v1

A Common Approach to Three Open Problems in Number Theory

Apoloniusz Tyszka

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Brocard's problem; Brocard-Ramanujan equation x!+1=y^2; composite Fermat numbers; composite numbers of the form 2^(2^n)+1; Erd\"os' equation x(x+1)=y!

Online: 24 March 2023 (02:00:47 CET)

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Preprint ARTICLE | doi:10.20944/preprints202301.0541.v4

A Solution of The Collatz Conjecture Problem

baoyuan duan

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: collatz conjecture; (*3+2^m-1)/2^k odd sequence,weight function

Online: 13 March 2023 (03:05:38 CET)

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Preprint ARTICLE | doi:10.20944/preprints202303.0146.v1

A Series of New Formulas To Approximate the Sine and Cosine Functions

Mashrur Azim, Asma Akter Akhi, Md Kamrujjaman

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: sine; cosine; Bhaskara I’s formula

Online: 8 March 2023 (06:25:57 CET)

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Preprint ARTICLE | doi:10.20944/preprints202301.0541.v3

A Solution of The Collatz Conjecture Problem

baoyuan duan

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: collatz conjecture; (*3+2^m-1)/2^k odd sequence,weight function

Online: 6 March 2023 (04:14:16 CET)

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Preprint ARTICLE | doi:10.20944/preprints202303.0065.v1

A Fundamental Study of Composite Numbers as a Different Perspective on Problems Related to Prime Numbers

Ahmet Furkan Gocgen, Ejder Aysun

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: prime numbers; composite numbers; prime-composite relationship; formula of composite; formula of all composite; formula of all odd composite

Online: 3 March 2023 (09:13:27 CET)

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Preprint ARTICLE | doi:10.20944/preprints202003.0145.v4

Gilbreath Equation, Gilbreath Polynomials, Upper and Lower Bound for Gilbreath Conjecture

Riccardo Gatti

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Gilbreath conjecture; Gilbreath sequence; Gilbreath equation; integer sequence; prime numbers; number theory

Online: 21 February 2023 (14:30:09 CET)

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Preprint ARTICLE | doi:10.20944/preprints202301.0163.v5

Solve the 3x+1 Problem by the Multiplication and Division of Binary Numbers

Jishe Feng

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: The Collatz conjecture

Online: 20 February 2023 (10:26:07 CET)

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Preprint ARTICLE | doi:10.20944/preprints202302.0329.v1

Constructions of Mixed Reverse-Order Laws for the Products of Matrices Involving Moore–Penrose Inverses and Group Inverses

Yongge Tian

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: block matrix; group inverse; Moore--Penrose inverse; range; rank; reverse-order law

Online: 20 February 2023 (07:48:05 CET)

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Preprint ARTICLE | doi:10.20944/preprints202301.0541.v2

A Solution of The Collatz Conjecture Problem

baoyuan duan

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: collatz conjecture

Online: 13 February 2023 (02:53:06 CET)

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Preprint ARTICLE | doi:10.20944/preprints202301.0163.v4

Solve the 3x+1 Problem by the Multiplication and Division of Binary Numbers

Jishe Feng

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: The Collatz conjecture

Online: 30 January 2023 (09:23:25 CET)

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Preprint ARTICLE | doi:10.20944/preprints202301.0541.v1

A Solution Of The Collatz Conjecture Problem

baoyuan duan

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: collatz conjecture

Online: 30 January 2023 (06:21:02 CET)

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Preprint ARTICLE | doi:10.20944/preprints202301.0163.v3

Solve the 3x+1 Problem by the Multiplication and Division of Binary Numbers

Jishe Feng

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: The Collatz conjecture

Online: 27 January 2023 (10:43:34 CET)

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Preprint ARTICLE | doi:10.20944/preprints202301.0163.v2

Solve the 3x+1 Problem

Jishe Feng

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: The Collatz conjecture

Online: 23 January 2023 (09:33:04 CET)

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Preprint ARTICLE | doi:10.20944/preprints202108.0146.v24

A Proof Of The Riemann Hypothesis Based On A New Expression Of The Completed Zeta Function

Weicun Zhang

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Riemann Hypothesis (RH); Proof; Completed zeta function

Online: 20 January 2023 (15:09:37 CET)

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Preprint ARTICLE | doi:10.20944/preprints202301.0225.v1

On the Solution to Riemann Hypothesis

Michael Mark Anthony

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Gamma function; Zeta function; Bernoulli numbers

Online: 12 January 2023 (10:38:15 CET)

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Preprint ARTICLE | doi:10.20944/preprints202301.0163.v1

Solve the 3x+1 Problem

Jishe Feng

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: The Collatz conjecture

Online: 9 January 2023 (11:00:24 CET)

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Preprint ARTICLE | doi:10.20944/preprints202110.0087.v17

On the Additive and Subtractive Representation of Even Numbers from Primes

Ali Shehu, Jetmira Uka

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Sieve of Eratosthenes; Goldbach conjecture; Polignac conjecture; Twin Prime conjecture

Online: 30 December 2022 (03:45:58 CET)

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Preprint ARTICLE | doi:10.20944/preprints202110.0087.v16

On the Additive and Subtractive Representation of Even Numbers from Primes

Ali Shehu, Jetmira Uka

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Sieve of Eratosthenes; Goldbach conjecture; Polignac conjecture; Twin Prime conjecture

Online: 13 December 2022 (02:17:30 CET)

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Preprint ARTICLE | doi:10.20944/preprints202212.0171.v1

Proof of Two-dimensional Jacobian Conjecture

Yucai Su

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Jacobian conjecture; Keller maps; injectivity of Keller maps

Online: 9 December 2022 (07:12:06 CET)

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Preprint ARTICLE | doi:10.20944/preprints202010.0374.v2

On Soft Lebesgue Measure

Sujay Goldar, Subhasis Ray

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Soft Set, Soft Element, Soft Real Number, Soft Interval, Soft Sequence, Soft Lebesgue Measure.

Online: 8 December 2022 (07:52:16 CET)

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Preprint ARTICLE | doi:10.20944/preprints202211.0566.v1

Determine the Exact Value of the Square Root of 2

Qing Li

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: the square root of 2 ,the decimal point; jump; change in direction; the infinitely great

Online: 30 November 2022 (06:18:32 CET)

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Preprint ARTICLE | doi:10.20944/preprints202211.0182.v2

Polyadic Rings of p-adic Integers

Steven Duplij

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: polyadic semigroup; polyadic ring; arity; querelement; residue class; representative; p-adic integer

Online: 29 November 2022 (03:39:52 CET)

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Preprint ARTICLE | doi:10.20944/preprints202110.0087.v15

On the Additive and Subtractive Representation of Even Numbers from Primes

Ali Shehu, Jetmira Uka

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Sieve of Eratosthenes; Goldbach conjecture; Polignac conjecture; Twin Prime conjecture

Online: 22 November 2022 (03:01:41 CET)

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Preprint ARTICLE | doi:10.20944/preprints202210.0432.v3

Proof of the Collatz Conjecture

Bülent Sukuşu

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Collatz operation, Collatz inverse operation and Collatz numbers

Online: 15 November 2022 (10:59:46 CET)

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Preprint ARTICLE | doi:10.20944/preprints202110.0087.v14

On the Additive and Subtractive Representation of Even Numbers from Primes

Ali Shehu, Jetmira Uka

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Sieve of Eratosthenes; Goldbach conjecture; Polignac conjecture; Twin Prime conjecture

Online: 15 November 2022 (06:35:57 CET)

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Preprint ARTICLE | doi:10.20944/preprints202211.0182.v1

Polyadic rings of p-adic integers

Steven Duplij

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: polyadic semigroup; polyadic ring; arity; querelement; residue class; representative; p-adic integer

Online: 10 November 2022 (02:00:13 CET)

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Preprint ARTICLE | doi:10.20944/preprints202211.0045.v1

Special Cases of Generalized Leonardo Numbers: Modified p-Leonardo, p-Leonardo-Lucas and p-Leonardo Numbers

Yüksel Soykan

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Leonardo numbers, Leonardo-Lucas numbers, p-Leonardo numbers, p-Leonardo-Lucas numbers, Tribonacci numbers, Fibonacci numbers, Lucas numbers.

Online: 2 November 2022 (04:43:22 CET)

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Preprint ARTICLE | doi:10.20944/preprints202210.0432.v2

Proof of the Collatz Conjecture

Bülent Sukuşu

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Collatz operation; Collatz inverse operation and Collatz numbers

Online: 31 October 2022 (09:24:48 CET)

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Preprint ARTICLE | doi:10.20944/preprints202210.0432.v1

Proof of The Collatz Conjecture

Bülent Sukuşu

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Collatz operation; Collatz inverse operation and Collatz numbers

Online: 27 October 2022 (10:58:37 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202210.0420.v1

A Concise Proof of Goldbach Conjecture

Xin Wang

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: prime number; periodicity; Goldbach Conjecture; proof

Online: 27 October 2022 (03:49:54 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202108.0146.v23

A Proof Of The Riemann Hypothesis Based On A New Expression Of The Completed Zeta Function

Weicun Zhang

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Riemann Hypothesis (RH); Proof; Completed zeta function

Online: 25 October 2022 (12:29:03 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202108.0146.v22

A Proof Of The Riemann Hypothesis Based On A New Expression Of The Completed Zeta Function

Weicun Zhang

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Riemann Hypothesis (RH); Proof; Completed zeta function

Online: 12 October 2022 (10:19:00 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202110.0087.v13

On the Additive and Subtractive Representation of Even Numbers from Primes

Ali Shehu, Jetmira Uka

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Sieve of Eratosthenes; Goldbach conjecture; Polignac conjecture; Twin Prime conjecture

Online: 11 October 2022 (03:22:25 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202210.0077.v1

Goldbach Conjecture and the Trichotomy Law

Aribam Uttam Sharma

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Goldbach Conjecture; Trichotomy Law

Online: 7 October 2022 (08:34:21 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202209.0438.v1

Indefinite Halmos, Egervary and Sz.-Nagy Dilations

K. MAHESH KRISHNA

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Dilation; Indefinite inner product space; Module

Online: 29 September 2022 (02:48:49 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202110.0087.v12

On the Additive and Subtractive Representation of Even Numbers from Primes

Ali Shehu, Jetmira Uka

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Sieve of Eratosthenes; Goldbach conjecture; Polignac conjecture; Twin Prime conjecture

Online: 27 September 2022 (03:49:33 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202104.0207.v5

Exponentials and Logarithms Properties in an Extended Complex Number Field

Daniel Tischhauser

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Complex number field; Complex exponentiation; Complex logarithm; Exponential and logarithm identities

Online: 14 September 2022 (03:53:54 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202110.0087.v11

On the Additive and Subtractive Representation of Even Numbers from Primes

Ali Shehu, Jetmira Uka

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Sieve of Eratosthenes; Goldbach conjecture; Polignac conjecture; Twin Prime conjecture

Online: 13 September 2022 (10:15:57 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202209.0155.v1

Collatz Dynamics is Partitioned by Residue Class Regularly

Wei Ren

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Collatz conjecture; 3x+1 problem; Dynamics System; Residue Class

Online: 13 September 2022 (03:21:20 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202108.0146.v21

A Complete Proof Of The Riemann Hypothesis Based On A New Expression Of $\xi(s)$

Weicun Zhang

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Riemann Hypothesis (RH); Proof; Completed zeta function

Online: 5 September 2022 (13:15:47 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202209.0044.v1

Multiple Sums Involving the Terms of a General Second Order Sequence of Numbers

Kunle Adegoke

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Horadam sequence; Fibonacci number; Lucas number; Lucas sequence; sum-mation identity; nested sum; multiple sum

Online: 5 September 2022 (07:07:39 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202108.0146.v20

A Complete Proof Of The Riemann Hypothesis Based On A New Expression Of $\xi(s)$

Weicun Zhang

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Riemann Hypothesis (RH); Proof; Completed zeta function

Online: 31 August 2022 (14:36:32 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202110.0087.v10

On the Additive and Subtractive Representation of Even Numbers from Primes

Ali Shehu, Jetmira Uka

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Sieve of Eratosthenes; Goldbach conjecture; Polignac conjecture; Twin Prime conjecture

Online: 30 August 2022 (03:11:25 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202110.0087.v9

On the Additive and Subtractive Representation of Even Numbers from Primes

Ali Shehu, Jetmira Uka

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Sieve of Eratosthenes; Goldbach conjecture; Polignac conjecture; Twin Prime conjecture

Online: 18 August 2022 (03:53:01 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202108.0146.v19

A Complete Proof Of The Riemann Hypothesis Based On A New Expression Of $\xi(s)$

Weicun Zhang

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Riemann Hypothesis (RH); Proof; Completed zeta function

Online: 15 August 2022 (11:18:55 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202110.0087.v8

On the Additive and Subtractive Representation of Even Numbers from Primes

Ali Shehu, Jetmira Uka

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Sieve of Eratosthenes; Goldbach conjecture; Polignac conjecture; Twin Prime conjecture

Online: 5 August 2022 (03:41:27 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202110.0087.v7

On the Additive and Subtractive Representation of Even Numbers from Primes

Ali Shehu, Jetmira Uka

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Sieve of Eratosthenes; Goldbach conjecture; Polignac conjecture; Twin Prime conjecture

Online: 22 July 2022 (03:38:39 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202110.0087.v6

On the Additive and Subtractive Representation of Even Numbers from Primes

Ali Shehu, Jetmira Uka

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Sieve of Eratosthenes; Goldbach conjecture; Polignac conjecture; Twin Prime conjecture

Online: 11 July 2022 (04:28:25 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202110.0087.v5

On the Additive and Subtractive Representation of Even Numbers from Primes

Ali Shehu, Jetmira Uka

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Sieve of Eratosthenes; Goldbach’s conjecture; Polignac’s conjecture; Twin Prime conjecture

Online: 30 June 2022 (03:50:17 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202206.0276.v1

Fermat’s Last Theorem Proved in Hilbert Arithmetic. III. The Quantum-Information Unification of Fermat’s Last Theorem and Gleason’s Theorem

Vasil Penchev

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: completeness; Gleason’s theorem; Fermat’s last theorem; Hilbert arithmetic; idempotency and hi-erarchy; Kochen and Specker theorem; nonstandard bijection; Peano arithmetic; quantum information

Online: 21 June 2022 (03:22:50 CEST)

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The previous two parts of the paper (correspondingly, https://philpapers.org/rec/PENFLT-2 and https://philpapers.org/rec/PENFLT-3) demonstrate that the interpretation of Fermat’s last theorem (FLT) in Hilbert arithmetic meant both in a narrow sense and in a wide sense can suggest a proof by induction in Part I and by means of the Kochen - Specker theorem in Part II. The same interpretation can serve also for a proof FLT based on Gleason’s theorem and partly similar to that in Part II. The concept of (probabilistic) measure of a subspace of Hilbert space and especially its uniqueness can be unambiguously linked to that of partial algebra or incommensurability, or interpreted as a relation of the two dual branches of Hilbert arithmetic in a wide sense. The investigation of the last relation allows for FLT and Gleason’s theorem to be equated in a sense, as two dual counterparts, and the former to be inferred from the latter, as well as vice versa under an additional condition relevant to the Gödel incompleteness of arithmetic to set theory. The qubit Hilbert space itself in turn can be interpreted by the unity of FLT and Gleason’s theorem. The proof of such a fundamental result in number theory as FLT by means of Hilbert arithmetic in a wide sense can be generalized to an idea about “quantum number theory”. It is able to research mathematically the origin of Peano arithmetic from Hilbert arithmetic by mediation of the “nonstandard bijection” and its two dual branches inherently linking it to information theory. Then, infinitesimal analysis and its revolutionary application to physics can be also re-realized in that wider context, for example, as an exploration of the way for physical quantity of time (respectively, for time derivative in any temporal process considered in physics) to appear at all. Finally, the result admits a philosophical reflection of how any hierarchy arises or changes itself only thanks to its dual and idempotent counterpart.

Preprint ARTICLE | doi:10.20944/preprints202206.0271.v1

Characterization of a Nonlinear Equality Composed of Multiple Products of Matrices and their Generalized Inverses

Yongge Tian

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: generalized inverse; matrix product; matrix equality; set inclusion; rank equality; block matrix

Online: 20 June 2022 (10:53:30 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202205.0207.v1

Special Spirals are Produced by the ROTASE Galactic Spiral Equations with the Sequential Prime Numbers

Hongjun Pan

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: prime numbers; galactic spiral equations; ROTASE model; phyllotaxis; sunflower seed head

Online: 16 May 2022 (11:04:46 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202202.0021.v3

Examination the Abc Conjecture Using Some Functions

Seiji Fujino

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Number theory; Diophantine equation; abc conjecture

Online: 13 May 2022 (04:17:21 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202205.0112.v1

Fermat’s Last Theorem Proved in Hilbert Arithmetic. II. Its Proof in Hilbert Arithmetic by the Kochen-Specker Theorem With or Without Induction

Vasil Penchev

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: arithmetic mechanics; Gleason’s theorem; Fermat’s last theorem (FLT); Hilbert arithmetic; Kochen and Specker’s theorem; Peano arithmetic; quantum information

Online: 9 May 2022 (09:54:41 CEST)

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The paper is a continuation of another paper (https://philpapers.org/rec/PENFLT-2) published as Part I. Now, the case of “n=3” is inferred as a corollary from the Kochen and Specker theorem (1967): the eventual solutions of Fermat’s equation for “n=3” would correspond to an admissible disjunctive division of qubit into two absolutely independent parts therefore versus the contextuality of any qubit, implied by the Kochen – Specker theorem. Incommensurability (implied by the absence of hidden variables) is considered as dual to quantum contextuality. The relevant mathematical structure is Hilbert arithmetic in a wide sense (https://dx.doi.org/10.2139/ssrn.3656179), in the framework of which Hilbert arithmetic in a narrow sense and the qubit Hilbert space are dual to each other. A few cases involving set theory are possible: (1) only within the case “n=3” and implicitly, within any next level of “n” in Fermat’s equation; (2) the identification of the case “n=3” and the general case utilizing the axiom of choice rather than the axiom of induction. If the former is the case, the application of set theory and arithmetic can remain disjunctively divided: set theory, “locally”, within any level; and arithmetic, “globally”, to all levels. If the latter is the case, the proof is thoroughly within set theory. Thus, the relevance of Yablo’s paradox to the statement of Fermat’s last theorem is avoided in both cases. The idea of “arithmetic mechanics” is sketched: it might deduce the basic physical dimensions of mechanics (mass, time, distance) from the axioms of arithmetic after a relevant generalization, Furthermore, a future Part III of the paper is suggested: FLT by mediation of Hilbert arithmetic in a wide sense can be considered as another expression of Gleason’s theorem in quantum mechanics: the exclusions about (n = 1, 2) in both theorems as well as the validity for all the rest values of “n” can be unified after the theory of quantum information. The availability (respectively, non-availability) of solutions of Fermat’s equation can be proved as equivalent to the non-availability (respectively, availability) of a single probabilistic measure as to Gleason’s theorem.

Preprint ARTICLE | doi:10.20944/preprints202112.0074.v2

Deriving Von Koch's Inequality Without Using the Riemann Hypothesis

Seiji Fujino

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Number theory; Von Koch's inequality; Riemann Hypothesis

Online: 6 May 2022 (12:45:00 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202202.0021.v2

Examination the Abc Conjecture Using Some Functions

Seiji Fujino

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Number theory; Diophantine equation; abc conjecture

Online: 6 May 2022 (05:26:56 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202204.0176.v1

Series of Floor and Ceiling Function—Part II: Infinite Series

Dhairya Shah, Manoj Sahni, Ritu Sahni, Ernesto León-Castro, Maricruz Olazabal-Lugo

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: ceiling function; floor function; Fibonacci Number; Generalised Dirichlet series; Lerch - Zeta Function; Hurwitz - Zeta function; Polylogarithm; Riemann-Zeta function

Online: 19 April 2022 (06:03:59 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202204.0159.v1

A Modified Piecewise Relaxation Method for Singular Initial Value Problems

Olumuyiwa Otegbeye, Shina Oloniiju, Gerald Marewo

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: relaxation; spectral methods; multi-domain

Online: 18 April 2022 (08:28:47 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202108.0146.v18

A Complete Proof Of The Riemann Hypothesis Based On A New Expression Of $\xi(s)$

Weicun Zhang

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Riemann Hypothesis (RH); Proof; Completed zeta function

Online: 4 April 2022 (10:55:39 CEST)

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Preprint ARTICLE | doi:10.20944/preprints202203.0025.v6

A Generalization of the Sum of Divisors Function

Darrell Cox, Sourangshu Ghosh, Eldar Sultanow

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: sum of divisors function; superabundant numbers; colossally abundant numbers; Riemann hypothesis

Online: 25 March 2022 (10:05:06 CET)

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Preprint ARTICLE | doi:10.20944/preprints202203.0025.v5

A Generalization of the Sum of Divisors Function

Darrell Cox, Sourangshu Ghosh, Eldar Sultanow

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: sum of divisors function; superabundant numbers; colossally abundant numbers; Riemann hypothesis

Online: 18 March 2022 (12:11:02 CET)

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Preprint ARTICLE | doi:10.20944/preprints202203.0025.v4

A Generalization of the Sum of Divisors Function

Darrell Cox, Sourangshu Ghosh, Eldar Sultanow

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: sum of divisors function; superabundant numbers; colossally abundant numbers; Riemann hypothesis

Online: 17 March 2022 (11:57:24 CET)

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Preprint ARTICLE | doi:10.20944/preprints202203.0183.v1

Fermat’s Last Theorem Proved in Hilbert Arithmetic. I. From the Proof by Induction to the Viewpoint of Hilbert Arithmetic

Vasil Penchev

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Fermat’s last theorem; Hilbert arithmetic; Kochen and Specker’s theorem; Peano arithmetic; quantum information; qubit Hilbert space

Online: 14 March 2022 (10:54:07 CET)

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Preprint ARTICLE | doi:10.20944/preprints202108.0146.v17

A Complete Proof Of The Riemann Hypothesis Based On A New Expression Of $\xi(s)$

Weicun Zhang

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Riemann Hypothesis (RH); Proof; Completed zeta function

Online: 8 March 2022 (11:00:07 CET)

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Preprint ARTICLE | doi:10.20944/preprints202203.0025.v3

A Generalization of the Sum of Divisors Function

Darrell Cox, Sourangshu Ghosh, Eldar Sultanow

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: sum of divisors function; superabundant numbers; colossally abundant numbers; Riemann hypothesis

Online: 7 March 2022 (10:53:32 CET)

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Preprint ARTICLE | doi:10.20944/preprints202203.0025.v2

A Generalization of the Sum of Divisors Function

Darrell Cox, Sourangshu Ghosh, Eldar Sultanow

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: sum of divisors function; superabundant numbers; colossally abundant numbers; Riemann hypothesis

Online: 3 March 2022 (10:14:28 CET)

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Preprint ARTICLE | doi:10.20944/preprints202203.0025.v1

A Generalization of the Sum of Divisors Function

Darrell Cox, Eldar Sultanow, Sourangshu Ghosh

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: sum of divisors function; superabundant numbers; colossally abundant numbers; Riemann hypothesis

Online: 1 March 2022 (14:43:37 CET)

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Preprint ARTICLE | doi:10.20944/preprints202105.0499.v3

3x + 1 Problem: Iterative Operations Neither Cause Loops nor Diverge to Infinity

Li Jiang

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: 3x + 1 problem; Collatz conjecture; Syracuse problem; iteration

Online: 24 February 2022 (08:38:01 CET)

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Preprint ARTICLE | doi:10.20944/preprints202108.0146.v16

The Riemann Hypothesis Is True If The Zeros Of $\xi(s)$ Have Strict Increasing Imaginary Parts

Weicun Zhang

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Riemann Hypothesis (RH); Proof; Completed zeta function

Online: 17 February 2022 (09:54:43 CET)

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Preprint ARTICLE | doi:10.20944/preprints202005.0331.v3

Multi-Objective Optimization of Periodical Objectives within a Process

Moses Oyaro Okello

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: optimization; multi-objective optimization; decision making; Time

Online: 3 February 2022 (17:30:32 CET)

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Multi-objective optimization (MOO) is an optimization involving minimization or maximization of several objective functions more than the conventional one objective optimization, which is useful in many fields. Many of the current methodologies addresses challenges and solutions that attempt to solve simultaneously several Objectives with multiple constraints subjoined to each. Often MOO are generally subjected to linear inequality, equality and or bounded constraint that prevent all objectives from being optimized at once. This paper reviews some recent articles in area of MOO and presents deep analysis of Random and Uniform Entry-Exit time of objectives. It further break down process into sub-process and then provide some new concepts for solving problems in MOO, which comes due to periodical objectives that do not stay for the entire duration of process lifetime, unlike permanent objectives which are optimized once for the entire process duration. A methodology based on partial optimization that optimizes each objective iteratively and weight convergence method that optimizes sub-group of objectives are given. Furthermore, another method is introduced which involve objective classification, ranking, estimation and prediction where objectives are classified based on their properties, and ranked using a given criteria and in addition estimated for an optimal weight point (pareto optimal point) if it certifies a coveted optimal weight point. Then finally predicted to find how far it deviates from the estimated optimal weight point. A Sample Mathematical Tri-Objectives and Real world Optimization was analyzed using partial method, ranking and classification method, the result showed that an objective can be added or removed without affecting previous or existing optimal solutions. Therefore suitable for handling time governed MOO. Although this paper presents concepts work only, it’s practical application are beyond the scope of this paper, however base on analysis and examples presented, the concept is worthy of igniting further research and application.

Preprint ARTICLE | doi:10.20944/preprints202202.0021.v1

Examination the abc conjecture using some functions.

Seiji Fujino

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Number theory; Diophantine equation; abc conjecture

Online: 1 February 2022 (21:29:47 CET)

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Preprint ARTICLE | doi:10.20944/preprints202110.0087.v4

On the Additive and Subtractive Representation of Even Numbers from Primes

Ali Shehu, Jetmira Uka

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Sieve of Eratosthenes; Goldbach’s conjecture; Polignac’s conjecture; Twin Prime conjecture

Online: 14 January 2022 (14:43:54 CET)

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Preprint ARTICLE | doi:10.20944/preprints202108.0146.v15

A Proof of the Riemann Hypothesis Based on a New Expression of the Completed Zeta Function

Weicun Zhang

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Riemann Hypothesis (RH); Proof; Completed zeta function

Online: 13 January 2022 (13:21:18 CET)

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Preprint ARTICLE | doi:10.20944/preprints202112.0429.v1

Sums and Partial Sums of Horadam Sequences: The Sum Formulas of $\sum_{k=0}^{n}x^{k}W_{k}$ and $ \sum_{k=n}^{n+m}x^{k}W_{k}$ via Generating Functions

Yüksel Soykan

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Horadam numbers; Horadam sequence; sum; partial sum; Fibonacci numbers; Lucas numbers

Online: 27 December 2021 (13:48:06 CET)

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Preprint ARTICLE | doi:10.20944/preprints202112.0423.v1

Sums Over Primes II

Rusen Li

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: sums over primes; generalized alternating hyperharmonic numbers; asymptotic formula; numbers with $k$-prime factors

Online: 27 December 2021 (11:10:43 CET)

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Preprint ARTICLE | doi:10.20944/preprints202112.0249.v4

Distribution of Composite Numbers and Determination of Prime Numbers

Ahmet Furkan Gocgen

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: composite number; composite numbers distribution; composite numbers generation; prime number

Online: 24 December 2021 (07:36:30 CET)

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Preprint ARTICLE | doi:10.20944/preprints202108.0146.v14

A Proof of the Riemann Hypothesis Based on a New Expression of the Completed Zeta Function

Weicun Zhang

Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Riemann Hypothesis (RH); Proof; Completed zeta function

Online: 21 December 2021 (12:30:36 CET)

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Computer Science and Mathematics