Number Theory

On Diophantine Equation Ax^4 + By^4 + Cz^4 + Dw^4 + Eu^4 = 0

Authors: Seiji Tomita
Comments: 6 Pages.

In this paper, we prove that there are infinitely many integer solutions of ax^4 + by^4 + cz^4 + dw^4 + eu^4 = 0 where a+b+c+d+e=0.
Category: Number Theory

An Original Method to Find Probable Prime Numbers

Authors: David Hill, Silvio Gabbianelli
Comments: 36 Pages. Distributed under Creative Commons Attribution-NoDerivatives 4.0 License

This paper presents an original method devised by David Hill for identifying probable prime numbers through a series of systematic steps involving division and rounding. The method begins with selecting any natural number, repeatedly dividing it by 2 until the result ends in a decimal of .5. Based on the parity of the original number, the resulting decimal is then rounded to the nearest even or odd number. This rounded number is either added to or subtracted from the original input number, often resulting in a prime number. While the method does not guarantee a prime in every case, it demonstrates a high success rate, particularly within the range of 2 to 100. An exception is made for generating the prime number 2 from the input number 1. To validate this method, two Python programs were utilized. One program tested integer numbers within a given range one by one, and the other produced a list of probable prime numbers found. Analysis of the results revealed that the method consistently found a higher number of primes than initially estimated. For example, starting in the range of 2 to 100 integers, the method found 34 primes compared to the estimated 25. This pattern held true across larger ranges, with the method finding up to 46.06% more primes out of the estimated in the range of 2 to 10,000,000.Additionally, the method identified the greatest prime numbers that extended significantly beyond the initial range limits. The trend line for the percentage increase in found primes suggested that the method becomes increasingly effective at identifying additional primes as the range expands. These findings suggest that the method has the potential to uncover a greater number of prime numbers than traditional estimation methods predict, providing a new approach to prime number discovery. This could indicate a deeper connection between composite numbers and primes through systematic division and balancing of odds and evens. Further research is needed to determine the method's effectiveness across larger and more varied ranges, but the initial results are promising.[Note: Silvio Gabbianelli created Python programs to test the method. This paper, written by him for me to present. The tests, conducted up to 10,000,000, yielded promising results.]
Category: Number Theory

On Degrees of Carry and Scholz's Conjecture

Authors: Theophilus Agama
Comments: 10 Pages.

Exploiting the notion of carries, we obtain improved upper bounds for the length of the shortest addition chains $iota(2^n-1)$ producing $2^n-1$. Most notably, we show that if $2^n-1$ has carries of degree at most $$kappa(2^n-1)=frac{1}{2}(iota(n)-lfloor frac{log n}{log 2}floor+sum limits_{j=1}^{lfloor frac{log n}{log 2}floor}{frac{n}{2^j}})$$ then the inequality $$iota(2^n-1)leq n+1+sum limits_{j=1}^{lfloor frac{log n}{log 2}floor}bigg({frac{n}{2^j}}-xi(n,j)bigg)+iota(n)$$ holds for all $nin mathbb{N}$ with $ngeq 4$, where $iota(cdot)$ denotes the length of the shortest addition chain producing $cdot$, ${cdot}$ denotes the fractional part of $cdot$ and where $xi(n,1):={frac{n}{2}}$ with $xi(n,2)={frac{1}{2}lfloor frac{n}{2}floor}$ and so on
Category: Number Theory

[Attempted] Proof of Fermat's Conjecture in Just a Few Lines

Authors: Atsu Dekpe
Comments: 9 Pages.

We present in this paper a formula for decomposing a power of an integer into a product of consecutive integers and its properties. We also discuss properties of some specific vectors (polynomials). By using these concepts, we provide simple proofs for both of Fermat's theorems. Furthermore, the proof of the great Fermat theorem is accessible to all students who have studied the notion of vector space.
Category: Number Theory

Invariant Polynomial

Authors: Ahcene Ait Saadi
Comments: 3 Pages. (Author name added to the article by viXra Admin as required; also, please cite and list scientific references)

For centuries, mathematicians have been studying polynomials, especially the zeros of polynomials. the theory of Galois states that we cannot find a general formula for solving equations greater than 4. In this article I study the invariant polynomials of degrees 6,10 and 12. When we make a variable change to these polynomials, they become two-square. Which allows us to solve equations of higher degree.
Category: Number Theory

Collatz Conjecture Proof with "Branches" of Tree

Authors: SeongJoo Han
Comments: 11 Pages.

The Collatz's Conjecture is still unsolved as ancestors said thatit is impossible. However, we did ?nd the 'In?nite Beautiful Branches' in"Collatzs's Tree". And the Branches of tree shows us the way to prove"Collatz is right" like blue sky in autumn.
Category: Number Theory

On the L-Functions from Generalized Riemann Hypothesis, Birch and Swinnerton-Dyer Conjecture, and the Prime Numbers from Polignac's and Twin Prime Conjectures

Authors: John Yuk Ching Ting
Comments: 64 Pages. LMFDB input on BSD conjecture, Riemann hypothesis, Polignac's and Twin prime conjectures

We analyze L-functions of elliptic curves (and apply Sign normalization) which support simplest version of Birch and Swinnerton-Dyer conjecture to be true. Dirichlet eta function (proxy function for Riemann zeta function as generating function for all nontrivial zeros) and Sieve of Eratosthenes (generating algorithm for all prime numbers) are essentially infinite series. We apply infinitesimals to their outputs. Riemann hypothesis asserts the complete set of all nontrivial zeros from Riemann zeta function is located on its critical line. It is proven to be true when usefully regarded as an Incompletely Predictable Problem. The complete set with derived subsets of Odd Primes contain arbitrarily large number of elements and satisfy Prime number theorem for Arithmetic Progressions, Generic Squeeze theorem and Theorem of Divergent-to-Convergent series conversion for Prime numbers. Having these theorems being satisfied, Polignac's and Twin prime conjectures are separately proven to be true when usefully regarded as Incompletely Predictable Problems.
Category: Number Theory

Other Relationship Between Prime Numbers and the Zeta Function

Authors: Khazri Bouzidi Fethi
Comments: 2 Pages. In French (Correction made by viXra Admin - Please conform!)

we will establish another relationship between the prime numbers and the classic zeta function of Riemman then we will prove that the sum of the inverses of the count function of the prime numbers is equal to a constant 1.48 for a well-chosen sequence of x.

on vas établir une autre relation entre les nombres premiers et la fonction zeta classique de Riemman ensuite on vas prouver que la somme des inverses de fonction de compte des nombres premiers est égal a la à une constante 1.48 pur une suite des x bien choisi.
Category: Number Theory

Estimating the Number of Primes Within a Limited Boundary

Authors: Junho Eom
Comments: 21 pages, 2 figures, 2 tables, 3 appendices

Within n^2, n boundaries were generated from the 1st to the nth, each containing n numbers. Primes less than n^2/2 were multiplied, intersected, and formed composites. At least one prime less than n or in the 1st boundary was used as a factor for the composites between n and n^2, or 2nd and nth boundaries, limiting the number of composites to (2n^2)/λ, where λ represented the wavelength of primes in the 1st boundary. Under these conditions, passively remaining numbers that were not connected to the wave of primes in the 1st boundary were all new primes between the 2nd and nth boundaries. Considering the cause-and-effect relationship among the primes less than n and the composites and new primes between 2nd and nth boundaries, the characteristics of composites could represent the characteristics of primes, and both were defined within a limited n^2 boundary. In this paper, these boundary characteristics were utilized to obtain the average number count per boundary, which led to obtaining the average number of primes per boundary. The average number of primes was multiplied by n boundaries with a coefficient of either β1 or β_√2, denoting the ratio of the number of primes. Using either β1 or β_√2, the number of primes was estimated between 10^6 and 10^28 and compared to the actual number of primes. Considering the relative error between β1 (Average 1.42%: maximum 2.92%, minimum 0.16%) or β_√2 (Ave. 0.37%: max. 0.96, min. 0.04%), it was concluded that the number of primes could be estimated with β_√2, allowing for a relative average error of 0.37%, in an equation of π(n^2)=π(n)∙n/β_√2, where 10^3 ≤ n ≤ 10^14, π(n) was the known number of primes within n, and β_√2 = ln(2√2∙n)/ln(n)+1.
Category: Number Theory

A Proof of the Collatz Conjecture

Authors: Xingyuan Zhang
Comments: 6 Pages.

In this paper, we had given a proof of the Collatz conjecture in elementary algebra. Since any given positive integer is conjectured to return to odd 1 in operations, we analyze continuous inverse operations starting with odd 1, it had proved that all of the inverse path numbers of a given non-triple is obtainable and any inverse operation path tends to infinity, we can get any odd and even, to do continuous forward operations for a positive integer obtained it will return to the odd 1 along the inverse operation paths.
Category: Number Theory

[attempted] Proof for Goldbach’s Conjecture Verification with Mathematical Induction Formula

Authors: Budee U. Zaman
Comments: 13 Pages.

This document forwards a freshly unearthed test of the Goldbach Conjecture, a longstanding enigma in the theory of numbers put forth byChristian Goldbach in 1742. In our point of view, we have been able to come up with a simple and yet stunning explanation on how numberswhich are divisible by 2 could be permanently expressed as the sum of two prime numbers. Through an extensive analysis, it will be seen that every other two numbers above 2 can always be expressed in that manner. Our evidence is based on fundamental theories of numbers and original methods that solve the problem without any difficulty. Consequently, understandingis not difficult at all. The pathway for further research in number theory has just been brought to light while at the same time indicatinghow vital determination and a variation of outlook are for any endeavour.
Category: Number Theory

The Strict Proof That the Riemann Zeta Function Equation Has No Non-Trivial Zeros

Authors: Xiaochun Mei
Comments: 13 Pages.

A standard method is proposed to prove strictly that the Riemann Zeta function equation has no non-trivial zeros. The real part and imaginary part of the Riemann Zeta function equation are separated completely. By comparing the real part and the imaginary part of Zeta function equation individually, a set of equation is obtained. It is proved that this equation set only has the solutions of trivial zeros. So the Riemann Zeta function equation has no non-trivial zeros. The Riemann hypothesis does not hold.
Category: Number Theory

Analysis and Improvement of Twin Prime Density Estimation

Authors: Bruce R. Nye
Comments: 3 Pages.

The ’twin prime conjecture’ was first proposed over 100 years ago. The work of Hardy and Littlewood still remains the dominant authority with respect to identifying twin prime density. The Hardy-Littlewood conjec- ture is paired with a counting function alongside the twin prime constant (0.660016). This process estimates twin prime count to ’x’, as the error is infinitely sieved to zero. The proposed limit represents a nuanced more precise approach to estimating the number of twin primes up to n, making this formula a technical improvement over the Hardy-Littlewood formula. By incorporating additional logarithmic terms and scaling factors, this for- mula refines the asymptotic estimate, offering deeper accuracy and deeper insights into the distribution of twin primes. This refinement is significant for both theoretical studies and practical applications in number theory, as it provides a more detailed and more accurate framework.
Category: Number Theory

Conditions for Convergence of the Sequence 1/(��^��|sin��|^��)

Authors: Yudai Sakuma
Comments: 4 Pages.

It is known that if the sequence 1/(��^��|sin��|^��) converges then ��(��)≤1+��/�� , but the convergence of this sequence has not been solved. In this study, the conditions for convergence of 1/(��^��|sin��|^��) were clarified by focusing on �� such that the value of |sin��| becomes explosively small. As a result, it was confirmed that ��(��)<1+��/�� is a sufficient condition for convergence of 1/(��^��|sin��|^��) . This is the same result as in the previous study, but because the method of proof is different, we succeeded in identifying a range of values for lim��→∞1/(��^��|sin��|^��) when ��(��)=1+��/�� .
Category: Number Theory

A New Understanding on the Problem That the Quintic Equation Has No Radical Solutions

Authors: Xiaochun Mei
Comments: 30 Pages. In Chinese

It is proved in this paper that Abel’s and Galois's proofs that the quintic equations have no radical solutions are invalid. Due to Abel’s and Galois's work about two hundred years ago, it was generally accepted that general quintic equations had no radical solutions. However, Tang Jianer etc. recently proves that there are radical solutions for some quintic equations with special forms. The theories of Abel and Galois can not explain these results. On the other hand, Gauss etc. proved the fundamental theorem of algebra. The theorem declared that there were n solutions for the n degree equations, including the radical and non-radical solutions. The theories of Abel and Galois contradicted with the fundamental theorem of algebra. Due to the reasons above, the proofs of Abel and Galois should be re-examined and re-evaluated. The author carefully analyzed the Abel’s original paper and found some serious mistakes. In order to prove that the general solution of algebraic equation he proposed was effective for the cubic equation, Abel took the known solution of cubic equation as a premise to calculate the parameters of his equation. Therefore, Abel’s proof is a logical circular argument and invalid. Besides, Abel confused the variables with the coefficients (constants) of algebraic equations. An expansion with 14 terms was written as 7 terms, 7 terms were missing.We prefer to consider Galois’s theory as a hypothesis rather than a proof. Based on that permutation group had no true normal subgroup, Galois concluded that the quintic equations had no radical solutions, but these two problems had no inevitable logic connection actually. In order to prove the effectiveness of radical extension group of automorphism mapping for the cubic and quartic equations, in the Galois’s theory, some algebraic relations among the roots of equations were used to replace the root itself. This violated the original definition of automorphism mapping group, led to the confusion of concepts and arbitrariness. For the general cubic and quartic algebraic equations, the actual solving processes do not satisfy the tower structure of the Galois’s solvable group. The resolvents of cubic and quartic equations are proved to have no the symmetries of Galois’s soluble group actually. It is invalid to use the solvable group theory to judge whether the higher degreeequation has a radical solution. The conclusion of this paper is that there is only the Sn symmetry for the n degree algebraic equations. (Truncated by viXra Admin to < 400 words)
Category: Number Theory

Prime Number Distribution Proving the Twin Prime and Goldbach Conjectures

Authors: Budee U. Zaman
Comments: 7 Pages.

The paper investigates the dispersion of prime numbers as well as the twin prime and goldbach’s conjectures. The initial key feature that primenumbers are never even (apart from 2) will be presented as the basis on which a new rule concerning their distribution can be developed. In that wise, this will help us to come up with a demonstration of why there exist an infinite number of odd pairs such that their difference is equal to 2. We also show that the Goldbach conjecture is true. This means that it is possible to write any even number greater than two as the sum of two prime numbers. The results contribute fresh knowledge concerning old mathematics subjects, especially those concerning the origins of prime numbers.
Category: Number Theory

The Inconsistency Problem of Riemann Zeta Function Equation

Authors: Xiaochun Mei
Comments: 16 Pages. In Chinese

Four basic problems are found in Riemann’s original paper proposed in 1859. The Riemann hypothesis becomes meaningless. 1. It is proved that on the real axis of complex plane, the Riemann Zeta function equation holds only at point Re(s)=1/2 (s=a+ib) . However, at this point, the Zeta function is infinite, rather than zero. At other points of real axis with a be not equal to zero and b=0 and , the two sides of function equation are contradictory. When one side is finite, another side may be infinite. 2. An integral item around the original point of coordinate system was neglected when Riemann deduced the integral form of Zeta function. The item was convergent when Re(s)>1 but divergent when Re(s)<1. The integral form of Zeta function does not change the divergence of its series form. Two reasons to cause inconsistency and infinite are analyzed. 3. A summation formula was used in the deduction of the integral form of Zeta function. The applicative condition of this formula is x>0. At point x=0 , the formula is meaningless. However, the lower limit of Zeta function’s integral is x=0, so the formula can not be used. 4. The formula of Jacobi function was used to prove the symmetry of Zeta function equation. The applicable condition of this formula is also x>0. Because the lower limit of integral in the deduction was , this formula can not be used too. The zero calculation of Riemann Zeta function is discussed at last. It is pointed out that because approximate methods are used, they are not the real zeros of strict Riemann Zeta function.
Category: Number Theory

Pi's Irrationality Using Maclaurin Polynomials

Authors: Timothy W. Jones
Comments: 7 Pages.

After reviewing Maclaurin series and the Alternating Series Estimation Theorem, we show how these can be combined with some arithmetic and algebraic observations to prove that pi is irrational.
Category: Number Theory

Is it Really that Difficult to Prove the Goldbach Conjecture?

Authors: Mary Anne Ji You, Óscar E. Chamizo Sánchez
Comments: 6 Pages.

The Goldbach conjecture, that is to say, every even number greater than 4 can be represented by the sum of two primes, is a simple and intractable statement that has been torturing mathematicians for more than 250 years. We wondered if the divide et impera method, so useful in programming and algorithmics, could provide some service here. The goal is simplify and separate the whole problem into three independent and fairly manegeable subproblems. An approach that, as far as I know, has not been tested before.
Category: Number Theory

Transcendence of Deformations of Polylogarithm Functions in Non-Zero Characteristic

Authors: David Adam
Comments: 15 Pages. In French

Let p be a prime number. In 2005, with the aim of reproving by Wade's method the transcendence for n ∈ Zp N of Γ∞(n), where Γ∞ is the Carlitz-Goss factorial, Yao introduces an uncountable class of deformations of Kochubei polylogarithm functions of which he shows the transcendence of their values u200bu200bin 1 (which is sufficient for his objective) and more generallyin 1/T^k (k ∈ N∗). In this present article, we answer Yao's question to extend this result into a non-zero algebraic. We prove this fact not only in the context of infinite place but also in that of finite places. We show an analogous result for the deformations of the polylogarithm function of Carlitz.
Category: Number Theory

Riemann Hypothesis Via Nicolas Criterion

Authors: Dmitri Martila
Comments: 3 Pages.

The Robin's Theorem with Nicolas criterion were used to prove the Riemann Hypothesis in a straightforward way.
Category: Number Theory

Empirical and Theoretical Validation of Beal's Conjecture

Authors: Óscar Reguera García
Comments: 8 Pages.

Beal's Conjecture posits that for any solution to the equation A^x + B^y = C^z with A, B, C beingpositive integers without common prime factors and x, y, z being integers greater than 2, A, B, and C must share at least one common prime factor. This study conducts a comprehensive empirical andtheoretical validation of the conjecture, using a combined theoretical analysis with computationalsimulations. No counterexamples were found in the extended range of 2 to 10,000 for A, B, C, and 3to 10 for x, y, z.
Category: Number Theory