On the Expression of Composite Natural Numbers as Tensorial Products of Prime Natural Powers Vectors

This study describes how one can construct sets of composite natural numbers as tensorial products of the vectors created with the natural powers of prime numbers.

There Are Infinitely Many Mersnne Composite Numbers with Prime Exponents

By extending both arithmetical operations into finite sets of natural numbers, from the entire set of natural numbers successively deleting some residue classes modulo a prime, we invented a recursive sieve method or algorithm on natural numbers and their sets. The algorithm mechanically yields a sequence of sets, which converges to the set of all primes p such that 2p + 1 divides the Mersenne number Mp. The cardinal sequence corresponding to the sequence of sets is strictly increasing. So that we have captured enough usable structures, without any estimation, the existing theories of those structures allow us to prove an exact result: there are infinitely many Mersenne composite numbers with prime exponents Mp.

The Periodic Table of Primes

Over millennia, nobody has been able to predict where prime numbers sprout or how they spread. This study establishes the Periodic Table of Primes (PTP) using four prime numbers 2, 3, 5, and 7. We identify 48 integers out of a period 2×3×5×7=210 to be the roots of all primes as well as composites without factors of 2, 3, 5, and 7. Each prime, twin primes, or composite without factors of 2, 3, 5, and 7 is an offspring of the 48 integers uniquely allocated on the PTP. Three major establishments made in the article are the Formula of Primes, the Periodic Table of Primes, and the Counting Functions of Primes and Twin Primes.

Corresponding Prime Number Distribution Equation

The conjecture of twin prime numbers is a mathematical problem. Proving the twin prime conjecture using traditional modern number theory is extremely profound and complex. We propose an elementary research method for corresponding prime number, proved that the conjecture of twin prime numbers and obtain the corresponding prime distribution equation. According to the distribution rate of corresponding prime numbers, the distribution pattern of twin prime numbers was proved the distribution rate theorem. This is the distribution rate of prime numbers corresponding to composite numbers, which approaches the distribution rate of prime numbers corresponding to integers. Based on the corresponding prime distribution equation, obtain the twin prime inequality function. Then, the formula for calculating twin prime numbers was discussed. There is also the Hardy Littlewood conjecture. This provides a practical and feasible approach for studying the distribution of twin prime numbers.

On the Number of Primes in the Interval (x, 2x) by an Elementary Method

An elementary formula to know the number of primes in the interval (x, 2x) close to the exact figure for a fixed x is given here. A new elementary equation is derived (a relation between prime numbers and composite numbers distributed in the interval [1, 2x]). An elementary method to know the number of primes in a given magnitude is suitably placed in the form of a general formula, and we have proved it. The general formula is applied to the terms of the equation, and a tactical simplification of the terms gives rise to an expression whose verification envisages scope for its further studies.

Twin Prime Distribution Problem

The distribution of twin prime numbers is discussed. The research method of corresponding prime number distribution is proposed. The distribution of prime numbers corresponding to integers and composite numbers is discussed. Through the corresponding prime distribution rate of integers and composite numbers, it is found that the corresponding prime distribution rate of composite numbers approaches the corresponding prime distribution rate of integers. The distribution principle of corresponding prime number of composite number is proved. The twin prime distribution theorem is obtained. The number of twin prime numbers is thus obtained. It provides a practical way to study the conjecture of twin prime numbers.

Super Congruences Involving Alternating Harmonic Sums

Let p be an odd prime, the harmonic congruence such as , and many different variations and generalizations have been studied intensively. In this note, we consider the congruences involving the combination of alternating harmonic sums, where PP denotes the set of positive integers which are prime to p. And we establish the combinational congruences involving alternating harmonic sums for positive integer n=3,4,5.

Infinite Number of Twin Primes

This work is devoted to the theory of prime numbers. Firstly it introduced the concept of matrix primes, which can help to generate a sequence of prime numbers. Then it proposed a number of theorems, which together with theorem of Dirichlet, Siegel and Euler allow to prove the infinity of twin primes.

The Pattern of Prime Numbers

The prime numbers P≥5 obey a pattern that can be described by two forms or geometric progressions or that facilitates obtaining them sequentially, being possible also to calculate the quantity of primes that are in the geometric progressions as it is described in this document.

The Proof of Goldbach’s Conjecture on Prime Numbers

Goldbach’s Conjecture (“Every even positive integer strictly larger than 4 is the sum of two primes”) has remained unproven since 1742. This paper contains the proof that every positive composite integer n strictly larger than 3, is located at the middle of the distance between two primes, which implicitly proves Goldbach’s Conjecture for 2n as well.

Compositions of Generalized Derivations in Prime Rings

Let R be a prime ring with a non-central Lie ideal L. In the present paper we show that if the composition DG of two generalized derivations D and G is zero on L, then DG must be zero on R. And we get all the possibilities for the composition of a couple of generalized derivations to be zero on a non-central Lie ideal of a prime ring.

Remarks on Goldbach’s Conjecture on Prime Numbers

The oldest Goldbach’s Conjecture (“Every even positive integer strictly larger than 4 is the sum of two primes”) has remained unproven since 1742. The recent proof [1] connected Goldbach’s Conjecture with the fact that every positive composite integer n strictly larger than 3, is located at the middle of the distance between two primes. The present paper contains explicit additional and complementary details of the proof, insisting on the existence and the number of Goldbach’s representations of even positive integers as sums of pairs of primes.

Regular Decimations Result in Irregular Distribution of Primes

Purpose: Primes are notorious for their irregular distribution in natural numbers. Such a lack of regularity makes primes elusive. Many NP-hard problems are related to the irregular occurrence of primes in natural numbers. Methods: To extract the underlying regularity of prime distribution, author started from the complementary relationship between composites and primes, through the regular occurrence of composites to infer the regularity underlying primes. Results: Previously random-appearing occurrence of primes resulted from the regular periodic decimations of various frequencies and cycles set by primes. Conclusions: Primes are the survivors of natural numbers after periodic decimations caused by primes. This leads to a novel concise representation of the set of all primes using sine function, suggestive of periodicity for both primes and composites.

The Computing Formula of Number of Primes No More than Any Given Positive Integer

In this paper, we give out the formula of number of primes no more than any given n (n ∈ Z+, n > 2). At the same time, we also show the principle, derivation process of the formula and application examples, it is usually marked with π(n), which is: that is: where “[ ]” denotes taking integer. r = 1,2,3,4,5,6;sx = s1,s2,...,sj,sh;s1,s2,...,sj,,sh = 0,1,2,3,....As i ≥ 2, 2 ≤ sx ≤ i-1 (x=1,2,...,j,h).

Proof of three conjectures on congruences

This paper proves three conjectures on congruences involving central binomial coefficients or Lucas sequences.Let p be an odd prime and let a be a positive integer.It is shown that if p=1（mod 4）or a〉1then [3/4pa]∑k=0≡（2/pa）（mod p^2）where（—）denotes the Jacobi symbol.This confirms a conjecture of the second author.A conjecture of Tauraso is also confirmed by showing that p-1∑k=1 Lk/k^2≡0（mod p） provided p〉5.where the Lucas numbers Lo,L1,L2,...are defined by L_0=2,L1=1 and Ln＋1=Ln＋Ln-l（n=1,2,3,...）.The third theorem states that if p=5 then Fp^a-（p^a/5）mod p^3 can be determined in the following way： p^a-1∑k=0（-1）^k（2k k）≡（p^a/5）（1-2F p^a-（pa/5））（mod p^3）which appeared as a conjecture in a paper of Sun and Tauraso in 2010.

Small Modular Solutions to Fermat’s Last Theorem

The proof by Andrew Wiles of Fermat’s Last Theorem in 1995 resolved the existence question for non-trivial solutions in integers x,y,zto the equation xn+yn=znfor n>2. There are none. Surprisingly, there are infinitely many solutions if the problem is recast in terms of modular arithmetic. Over a hundred years ago Issai Schur was able to show that for any n there is always a sufficiently large prime p0such that for all primes p≥p0the congruence xn+yn≡zn(modp)has a non-trivial solution. Schur’s argument wasnon-constructive, and there is no systematic method available at present to construct specific examples for small primes. We offer a simple method for constructing all possible solutions to a large class of congruences of this type.

Pell方程组x^(2)-40y^(2)=1与y^(2)-Dz^(2)=9的公解

设D=2p_(1)⋯p_(s)(1≤s≤4),其中p_(1),⋯,p_(s)是互不相同的奇素数。主要利用奇偶分析、同余、递归序列以及Pell方程解的性质等初等方法,对Pell方程组x^(2)-40y^(2)=1与y^(2)-Dz^(2)=9的公解进行研究。得出当D≠2×7×103时,该方程组仅有平凡解(x,y,z)=(±19,±3,0);当D=2×7×103时,除了平凡解(x,y,z)=(±19,±3,0)外,还有非平凡解(x,y,z)=(±27379,±4329,±114)。研究结果丰富了这类Pell方程组整数解的研究内容。

Pell方程组x^(2)-33y^(2)=1与y^(2)-Dz^(2)=16的公解

利用奇偶分析、递归序列、同余和Pell方程的解的性质等一些初等方法,对D=2p1……ps(1≤s≤4),其中p1,…,ps是互不相同的奇素数时,Pell方程组x^(2)-33y^(2)=1与y^(2)-Dz^(2)=16的公解进行了研究.得到除开D=2×7×151,方程组有非平凡解(x,y,z)=(±48599,±8460,±184)这一基本情况之外,仅有平凡解(x,y,z)=(±23,±4,0),从而推进了这类Pell方程组整数解的研究.

关于Pell方程组x^(2)-39y^(2)=1与y^(2)-Dz^(2)=16的公解

利用同余、递归序列的方法以及Pell方程解的性质,证明了Pell方程组x^(2)-39y^(2)=1与y^(2)-Dz^(2)=16的公解,情况如下:(1)当D=2^(n)(n∈Z^(+))时,方程组只有平凡解(x,y,z)=(±25,±4,0);(2)当D=2p_(1)…p_(s)(1≤s≤4,p_(1),…,p_(s)是互异的奇素数)时,除开D=2×1249,方程组有非平凡解(x,y,z)=(±62425,±9996,±200)外,仅有平凡解(x,y,z)=(±25,±4,0).

ON HOMOGENEOUS ADDITIVE CONGRUENCES

In this paper, the additive equations of the type α_1λ_1~k+ … +α_sλ_s^k = 0 are studied, α_i'sbeing integers of an algebraic number field K of degree n. The main result is as follows: Ifs≥(2k)^(n+1) (or s≥cknlogk for 2 + k), the equation is solved nontrivially in any β-adic field,where β is a prime ideal of K.