Number Theory

Multivariate Circle of Partitions and the Squeeze Principle

Authors: Theophilus Agama
Comments: 5 Pages.

The goal of this paper is to extend the squeeze principle to circle of partitions with at least two resident points on their axes.
Category: Number Theory

Investigation on Brocard-Ramanujan Problem

Authors: Akash Shivaji Pawar
Comments: 7 Pages. 6 Lemma,2 Theorem

Exploring n! + 1 = m^2 for natural number solutions beyond n = 4, 5, 7 confirms no further solutions exist,validated by using GCD Linear Combination Theorem
Category: Number Theory

On the Nonexistence of Solutions to a Diophantine Equation Involving Prime Powers

Authors: Budee U. Zaman
Comments: 7 Pages.

This paper investigates the Diophantine equation pr + (p + 1)s = z2 Where p > 3, s ≥ 3 , z is an even integer. The focus of the study is to establish rigorous results concerning the existence of solutions within this specific parameter space. The main result presented in this paper demonstrates the absence of solutions under the stated conditions. The proof employs mathematical techniques to systematically address the case when the prime p exceeds 3, and the exponent s is equal to or greater than2, while requiring the solution to conform to the constraint of an even z. This work contributes to the understanding of the solvability of the given Diophantine equation and provides valuable insights into the interplay between prime powers and the resulting solutions.
Category: Number Theory

Euler’s Totient Function, Sum of Divisors and Primes

Authors: Rédoane Daoudi
Comments: (Note by viXra Admin: Future stub page/paper will not be accepted!)

Here I present a conjecture about Euler’s totient function, sum of divisors andprimes.
Category: Number Theory

On the Notion of Carries of Numbers 2^n-1 and Scholz Conjecture

Authors: Theophilus Agama
Comments: 16 Pages.

Applying the pothole method on the factors of numbers of the form $2^n-1$, we prove that if $2^n-1$ has carries of degree at most $$kappa(2^n-1)=frac{1}{2(1+c)}lfloor frac{log n}{log 2}floor-1$$ for $c>0$ fixed, then the inequality $$iota(2^n-1)leq n-1+(1+frac{1}{1+c})lfloorfrac{log n}{log 2}floor$$ holds for all $nin mathbb{N}$ with $ngeq 4$, where $iota(cdot)$ denotes the length of the shortest addition chain producing $cdot$. In general, we show that all numbers of the form $2^n-1$ with carries of degree $$kappa(2^n-1):=(frac{1}{1+f(n)})lfloor frac{log n}{log 2}floor-1$$ with $f(n)=o(log n)$ and $f(n)longrightarrow infty$ as $nlongrightarrow infty$ for $ngeq 4$ then the inequality $$iota(2^n-1)leq n-1+(1+frac{2}{1+f(n)})lfloorfrac{log n}{log 2}floor$$ holds.
Category: Number Theory

A Proof of the Wen-Yao Conjecture

Authors: David Adam
Comments: 24 Pages. In French

In this article, we characterize monomials in de facto values.Carlitz-Goss rielle defined on the complement of Fq (T) in a finite place which arealgebraic on Fq (T ). In particular, this confirms Wen-Yao's conjecturestated in 2003. This gives a necessary and sufficient condition on an en-p-adic tier so that the value of the Carlitz-Goss factorial in it is algebraic on Fq (T ). When restricted to rational arguments, we determinenot all algebraic relations between the values u200bu200btaken by this function, this which gives the counterpart for finite places of a result of Chang, Papanikolas, Thakur and Yu obtained in the case of infinite place.

Dans cette article, nous caractérisons les monômes en les valeurs de la facto-rielle de Carlitz-Goss définie sur le complété de Fq (T ) en une place finie qui sont algébriques sur Fq (T ). En particulier, cela confirme la conjecture de Wen-Yaoénoncée en 2003 . Celle-ci donne une condition necessaire et suffisante sur un en-tier p-adique pour que la valeur de la factorielle de Carlitz-Goss en celui-ci soit algébrique sur Fq (T ). Lorsque restreint aux arguments rationnels, nous détermi-nons toutes les relations algébriques entre les valeurs prises par cette fonction, cequi donne le pendant pour les places finies d’un résultat de Chang, Papanikolas, Thakur et Yu obtenu dans le cas de la place infinie.
Category: Number Theory

Complete Operations

Authors: Pith Peishu Xie
Comments: 27 Pages.

The Operator axioms have produced complete operations with real operators. Numerical computations have been constructed for complete operations. The classic calculator could only execute 7 operator operations: + operator operation(addition), - operator operation(subtraction), $times$ operator operation(multiplication), $div$ operator operation(division), ^{} operator operation(exponentiation), $surd$ operator operation(root extraction), log operator operation(logarithm). In this paper, we invent a complete calculator as a software calculator to execute complete operations. The experiments on the complete calculator could directly prove such a corollary: Operator axioms are consistent.
Category: Number Theory

A Proof of a Result of James Stirling

Authors: Hervé Gandran-Tomeng
Comments: 2 Pages.

A recent paper contains a proof of a result of James Stirling,$sum_{n=1}^inftyfrac{1}{n^2binom{2n}{n}}=frac{1}{3}sum_{n=1}^infty frac{1}{n^2}$What is following is another proof of this equality.
Category: Number Theory

Rediscovering Ramanujan

Authors: Edgar Valdebenito
Comments: 14 Pages.

In this note, we revisit Ramanujan-type series for 1/pi .
Category: Number Theory

Collatz Conjecture: a Countably Infinite Sequence

Authors: Vishesh Mangla
Comments: 2 Pages. CC BY-SA (Note by viXra admin: The article needs an abstract and scientific references are required

In this paper, I have explored the Collatz conjecture and presented a new result regarding the behavior of the sequence. The proof demonstrated that for natural numbers n subjected to the Collatz Algorithm, the sequence can potentially havea countably infinite number of terms.
Category: Number Theory

The Twin Prime Conjecture - An Analytical Approach

Authors: Patrick DiCarlo
Comments: 37 Pages.

This paper offers a proof of the twin prime conjecture. The basic strategy is to first establish that there is no highest prime number by calculating the rates at which the multiples of each successive prime preclude higher numbers from being prime, and then proving that this rate (in the aggregate) can never reach 100%. The same basic methodology is then used to show that there can also be no highest twin prime.
Category: Number Theory

One Theorem Complementary to the Fundamental Theorem of Arithmetic

Authors: Juan Elias Millas Vera
Comments: 2 Pages.

In this paper I want to show a complementary theorem of the Fundamental Theorem of Arithmetic. Using the delta notation (Δ) I was able to deduce a generic formula involving prime numbers and natural numbers.
Category: Number Theory

New Equivalent of the Riemann Hypothesis

Authors: Leonardo de Lima
Comments: 10 Pages.

In this article, it is demonstrated that if the zeta function does not have a sequence of zeros whose real part converges to 1, then it cannot have any zeros in the critical strip, showing that the Riemann Hypothesis is false.
Category: Number Theory

Cracking the Collatz Code: a Journey from Conjecture to Certainty Through Infinity

Authors: Eric Lough
Comments: 7 Pages.

The Collatz conjecture, a puzzle that has intrigued mathematicians for decades, has met its resolution through a simple combination of trial and error combined with a spreadsheet. While the primary goal was to definitively prove the conjecture, the exploration revealed unexpected insights into the underlying patterns of the Collatz sequence. This study introduces a novel approach to organizing and analyzing Collatz data, laying bare the formulas governing the sequence's behavior. The results present a conclusive proof of the Collatz conjecture and unveil a spectrum of sets and formulas reaching to infinity, each contributing to a deeper understanding of its workings. Comprehensive instructions are provided, encouraging collaboration and furthering the collective understanding of mathematical sequences.
Category: Number Theory

The Magic of Mirror Composite Numbers

Authors: Emilio Sánchez, Óscar E. Chamizo
Comments: 6 Pages.

In this paper, continuation and completion of some previous researches, we fully develop the new concept of mirror composite numbers. Mirror composite numbers are composite numbers of the form 2n-p for some n natural number and p prime. We shall show that the factorization of these numbers have interesting properties in order to face the Goldbach conjecture [2][3] by the divide et impera method.
Category: Number Theory

Proof for Specific Type of Continued Fraction

Authors: Isaac Mor
Comments: 9 Pages.

I am going to use telescoping series and then a proof by induction. I am using Lambert's continued fraction for the base case.
Category: Number Theory

Elliptic Curve

Authors: Ian Connell
Comments: 327 Pages.

The first version of this handbook was a set of notes of about 100 pages handed out to the class of an introductory course on elliptic curves given in the 1990 fall semester at McGill University in Montreal. Since then I haveadded to the notes, holding to the principle: If I look up a certain topic a year from now I want all the details right at hand, not in an "exercise", so if I’ve forgotten something I won’t waste time. Thus there is much that anordinary text would either condense, or relegate to an exercise. But at the same time I have maintained a solid mathematical style with the thought of sharing the handbook.
Category: Number Theory

A Proof of Riemann Hypothesis by Symmetry and Circular Properties of Riemann Zeta Function

Authors: Tae Beom Lee
Comments: 9 Pages.

Riemann zeta function(RZF) z(s) is a function of a complex variable s=x+iy. Riemann hypothesis(RH) states that all the non-trivial zeros of RZF lie on the critical line, 0.5+iy. The symmetricity of RZF zeros implies that if z(a+ib)=0,0<a<0.5, then z(1-a+ib)=0, too. The graphs of RZF are similar to the graphs of circles with non-uniform radius and argument. These two, symmetry and circular properties of RZF, are the basis of our proof.
Category: Number Theory