# Number Theory

## 1704 Submissions

 viXra:1704.0393 [pdf] replaced on 2017-07-15 00:24:28

### AAFrempong Conjecture

Authors: A. A. Frempong
Comments: 2 Pages. Copyright © by A. A. Frempong

The above conjecture states that if A^x + B^y = C^z, where A, B, C, x, y, z are positive integers, x, y, z > 2, and A ≠ B ≠ C ≠ 2, then A, B and C cannot be the lengths of the sides of a triangle. This conjecture evolved when after proving the Beal conjecture algebraically (viXra:1702.0331), the author attempted to prove the same conjecture geometrically. A proof of the above conjecture may shed some light on the relationships between similar equations and the lengths of the sides of polygons. Counterexamples could be added to the exceptions.
Category: Number Theory

 viXra:1704.0392 [pdf] submitted on 2017-04-29 23:48:52

### Three Sequences of Primes Obtained from Poulet Numbers

Authors: Marius Coman
Comments: 2 Pages.

In this paper I make the following three conjectures: (I) The set of the primes which are the sum of three consecutive Poulet numbers is infinite; (II) The set of the primes which are partial sums of the sequence of Poulet numbers is infinite; (III) The set of the primes which are obtained concatenating four consecutive 2-Poulet numbers is infinite.
Category: Number Theory

 viXra:1704.0391 [pdf] submitted on 2017-04-29 23:52:47

### Three Sequences of Primes Obtained Using the Digital Root and the Digital Sum of a Prime

Authors: Marius Coman
Comments: 2 Pages.

In this paper I make the following three conjectures: (I) The set of the primes which are obtained concatenating to the left a prime with its digital sum is infinite; (II) The set of the primes which are obtained concatenating to the left a prime with its digital root is infinite; (III) The set of the primes which are equal to the sum of a prime p with the number obtained concatenating to the left p with its digital sum and the number obtained concatenating to the left p with its digital root is infinite.
Category: Number Theory

 viXra:1704.0306 [pdf] submitted on 2017-04-23 12:08:04

### Formula to Generate a Set of Poulet Numbers from a Poulet Number P and Its Factor D Lesser Than Sqr P

Authors: Marius Coman
Comments: 2 Pages.

In this paper I make the following observation: Let d be a factor (not necessarily prime) of the Poulet number P such that d < sqr P and m the least number such that m*d*(d – 1) > (P – 1)/2. Let n be equal to P – m*d*(d – 1). Then often exist a set of Poulet numbers Q such that Q mod(m*d*(d – 1)) = n. For example, for P = 2047 = 23*89 and d = 23, where d < sqr 2047, the least m such that m*23*22 > (P – 1)/2 is equal to 3 (1518 > 1023, while, for 2, 1012 < 1023); so, n = 2047 – 3*23*22 = 2047 – 1518 = 529 and indeed there exist a set of Poulet numbers Q such that Q mod 1518 = 529; the formula 1518*x + 529 gives the Poulet numbers 2047, 6601, 15709, 30889 (...) for x = 1, 4, 10, 20 (...).
Category: Number Theory

 viXra:1704.0296 [pdf] submitted on 2017-04-22 23:35:39

### Poulet Numbers Obtained Concatenating Two Primes P and P±30k

Authors: Marius Coman
Comments: 1 Page.

In a previous paper, “Poulet numbers in Smarandache prime partial digital sequence and a possible infinite set of primes” I conjectured that there exist an infinity of Poulet numbers which admit a deconcatenation in prime numbers. In this paper I conjecture that there exist an infinity of Poulet numbers which admit a deconcatenation in two prime numbers p and q where q = p + 30*k, where k integer.
Category: Number Theory

 viXra:1704.0295 [pdf] submitted on 2017-04-22 23:37:50

### Primes Obtained Concatenating Four Consecutive Numbers, the Largest One Being a Poulet Number

Authors: Marius Coman
Comments: 1 Page.

In this paper I conjecture that there exist an infinity of primes obtained concatenating four consecutive numbers, the largest one from them being a Poulet number. For example, 1726172717281729 is such a prime, obtained concatenating the numbers 1726, 1727, 1728 and 1729, where 1729 is a Poulet number (see the sequence A030471 in OEIS for primes which are concatenation of four consecutive numbers).
Category: Number Theory

 viXra:1704.0293 [pdf] submitted on 2017-04-23 04:09:15

### Poulet Numbers Which Can be Written as X^3±y^3

Authors: Marius Coman
Comments: 2 Pages.

It is well known the story of the Hardy-Ramanujan number, 1729 (also a Poulet number), which is the smallest number expressible as the sum of two cubes in two different ways, but I have not met yet, not even in OEIS, the sequence of the Poulet numbers which can be written as x^3±y^3, sequence that I conjecture in this paper that is infinite. I also conjecture that there are infinite Poulet numbers which are centered cube numbers (equal to 2*n^3 + 3*n^2 + 3*n + 1), also which are centered hexagonal numbers (equal to 3*n^2 + 3*n + 1).
Category: Number Theory

 viXra:1704.0274 [pdf] replaced on 2017-04-22 06:37:27

### Two Extreme Formulas for Converging to Pi,the Fastest and Most Painful

Authors: François Mendzina Essomba
Comments: 1 Page. extreme fomulas

I present in this small article two algorithms of calculation of pi, they are characterized by two extremities, one is the most convergent and the other the slowest of the imaginable formulas.
Category: Number Theory

 viXra:1704.0260 [pdf] submitted on 2017-04-20 07:53:36

### Fractals and Pi

Authors: Edgar Valdebenito
Comments: 37 Pages.

This note presents a collection of fractals related with constant pi
Category: Number Theory

 viXra:1704.0259 [pdf] submitted on 2017-04-20 07:57:06

### Question 2360 : Series For Pi

Authors: Edgar Valdebenito
Comments: 3 Pages.

This note presents some series for pi constant.
Category: Number Theory

 viXra:1704.0258 [pdf] submitted on 2017-04-20 08:02:54

### The Ramanujan-GÖLLNITZ-Gordon Continued Fraction

Authors: Edgar Valdebenito
Comments: 7 Pages.

In this research , the autor has detailed about: Numerical evaluation of the Ramanujan-Göllnitz-Gordon continued fraction.
Category: Number Theory

 viXra:1704.0240 [pdf] submitted on 2017-04-19 11:32:59

### Gelfond Constant

Authors: Edgar Valdebenito
Comments: 4 Pages.

This note presents some formulas related with Gelfond constant:exp(pi)
Category: Number Theory

 viXra:1704.0225 [pdf] submitted on 2017-04-17 17:28:46

### Conjecture on the Primes Obtained Concatenating Three Numbers, id Est a, B and A+b+n

Authors: Marius Coman
Comments: 2 Pages.

In this paper I make the following conjecture: For any n positive integer there exist an infinity of primes which can be deconcatenated in three numbers, i.e., from left to right, a, b and a + b + n. Examples: for n = 0, the least such prime is 101 (1 + 0 + 0 = 1); for n = 1, the least such prime is 113 (1 + 1 + 1 = 3); for n = 2, the least such prime is 103 (1 + 0 + 2 = 3); for n = 3, the least such prime is 137 (1 + 3 + 3 = 7); for n = 4, the least such prime is 127 (1 + 2 + 4 = 7); for n = 5, the least such prime is 139 (1 + 3 + 5 = 9); for n = 6, the least such prime is 107 (1 + 0 + 6 = 7); for n = 7, the least such prime is 3313 (3 + 3 + 7 = 13).
Category: Number Theory

 viXra:1704.0224 [pdf] submitted on 2017-04-17 17:31:06

### Conjecture on Primes Obtained Concatenating p, N and P+n, Where P and P+n Primes

Authors: Marius Coman
Comments: 2 Pages.

In this paper I make the following conjecture: For any n even there exist an infinity of primes which can be deconcatenated in three numbers, i.e., from left to right, p, n and p + n, where p and p + n are primes. Examples: for n = 2, the least such prime is 11213 (11 + 2 = 13); for n = 4, the least such prime is 347 (3 + 4 = 7); for n = 6, the least such prime is 11617 (11 + 6 = 17); for n = 8, the least such prime is 5813 (5 + 8 = 13); for n = 10, the least such prime is 31013 (3 + 10 = 13); for n = 12, the least such prime is 51217 (5 + 12 = 17); for n = 14, the least such prime is 51419 (5 + 14 = 19); for n = 16, the least such prime is 431659 (43 + 16 = 59).
Category: Number Theory

 viXra:1704.0210 [pdf] submitted on 2017-04-16 19:36:57

### ABC Conjecture–An Ambiguous Formulation

Authors: Zhang Tianshu
Comments: 13 Pages.

Due to exist forevermore uncorrelated limits of values of real number ε≥0, enable ABC conjecture to be able to be both proved and negated. In this article, we find a representative equality 1+2N(2N-2)=(2N-1)2 satisfying (2N-1)2>[Rad(1, 2N(2N-2), (2N-1)2)]1+ε, then both prove the ABC conjecture and negate the ABC conjecture according to two limits of values of ε.
Category: Number Theory

 viXra:1704.0196 [pdf] submitted on 2017-04-15 06:48:55

### New Approximation Algorithms of Pi, Accelerated Convergence Formulas from N = 100 to N = 2m

Authors: François Mendzina Essomba, Gael Dieudonné Essomba Essomba
Comments: 7 Pages. algorithm, convergence and approximation

We give algorithms for the calculation of pi. These algorithms can be easily developed in a linear manner and allows the calculation of pi with an infinite degree of convergence. Of course, the calculation of the second term passes through the first one, and it is necessary, as this type of algorithms, for a larger memory for calculations contrary to the formula BBP  whose execution corresponds to the order of the desired number. The advantage of our formulas in spite of the dificulty associated with extracting sin(x) lies in their degree of convergence, which is infinite, they prove the Borweins brothers hypothesis on the construction of algorithms At any speed as symbolized in our generic formula (8) of this paper. These formulas for the most part are totally new : We had found several other formulas of pi l
Category: Number Theory

 viXra:1704.0146 [pdf] replaced on 2017-08-19 09:22:03

### Discrete Mellin Convolution and Its Extensions, Perron Formula and Explicit Formulae

Authors: Jose Javier Garcia Moreta
Comments: 7 Pages.

In this paper we define a new Mellin discrete convolution, which is related to Perron's formula. Also we introduce new explicit formulae for arithmetic function which generalize the explicit formulae of Weil
Category: Number Theory

 viXra:1704.0129 [pdf] replaced on 2018-04-01 00:20:45

### Twin Primes Conjecture

Authors: Barry Foster
Comments: 2 Pages.

This attempt uses Bertrand’s postulate.
Category: Number Theory

 viXra:1704.0121 [pdf] submitted on 2017-04-10 07:46:25

### On The Prime Partition of n!

Authors: Shi-YuanDong
Comments: 2 Pages.

The prime partition of n!, On the Goldbach prime partition, and the algebraic sum of elements of prime.
Category: Number Theory

 viXra:1704.0114 [pdf] submitted on 2017-04-09 11:23:01

### A Condition on a Non-Collatz Number at the Boundary of a Successive Collatz Numbers Set

Authors: Abdelghaffar Slimane
Comments: 3 Pages. Academic use only

We give a condition that an odd number in the neighborhood of a successive collatz numbers set must verify to be a non-collatz number, and we use the result for odd numbers of the form 6k−1 at the boundary of a successive collatz numbers set.
Category: Number Theory

 viXra:1704.0110 [pdf] submitted on 2017-04-09 10:15:18

### Universal Evolution Model Based On Theory Of Natural Metric For Functions {Version –I}

Authors: Ramesh Chandra Bagadi
Comments: 10 Pages.

In this research investigation, the author has presented a novel scheme of Universal Evolution Model. This model can be also successfully used for forecasting purposes.
Category: Number Theory

 viXra:1704.0102 [pdf] submitted on 2017-04-09 00:26:53

### Poulet Numbers Which Can be Written as a Sum of Two Successive Primes Plus One

Authors: Marius Coman
Comments: 1 Page.

In this paper I make the following conjecture: there exist an infinity of Poulet numbers which can be written as a sum of two successive primes plus one (for the numbers that are the sum of two successive primes see the sequence A001043 in OEIS).
Category: Number Theory

 viXra:1704.0101 [pdf] replaced on 2017-05-11 09:44:47

### An Elementary Proof of Goldbach's Conjecture

Authors: Chongxi Yu
Comments: 22 Pages.

Prime numbers are the basic numbers and are crucially important. There are many conjectures concerning primes that have been challenging mathematicians for hundreds of years. Goldbach's conjecture is one of the oldest and most well-known unsolved problems in number theory and in all of mathematics. A kaleidoscope can produce an endless variety of colorful patterns and it looks like magic, but when you open one and examine it, it contains only very simple, loose, colored objects such as beads or pebbles and bits of glass. Humans are very easily cheated by 2 words, infinite and anything, because we never see infinite and anything, and so we always make a simple thing complex. Goldbach’s conjecture is about all very simple numbers, with the pattern of prime numbers similar to a “kaleidoscope” of numbers. If we divided all even numbers into 5 groups and primes into 4 groups, Goldbach’s conjecture becomes much simpler. Here we give a clear proof for Goldbach's conjecture based on the fundamental theorem of arithmetic, the prime number theorem, and Euclid's proof that the set of prime numbers is endless.
Category: Number Theory

 viXra:1704.0098 [pdf] submitted on 2017-04-08 09:49:00

### Conjecture on Poulet Numbers of the Form (Q+2^n)∙2^n+1 Where Q Prime

Authors: Marius Coman
Comments: 2 Pages.

In this paper I state the following conjecture: let P be a Poulet number and n the integer for which the number (P – 1)/2^n is odd; then there exist an infinity of Poulet numbers for which the number q = (P – 1)/2^n – 2^n is prime.
Category: Number Theory

 viXra:1704.0097 [pdf] submitted on 2017-04-08 10:06:47

### 算术中的多与少永远会造成二个质数的距离=2

Authors: Aaron Chau
Comments: 2 Pages.

Category: Number Theory

 viXra:1704.0093 [pdf] submitted on 2017-04-08 03:00:47

### A Finite Field Analogue for Appell Series F_{3}

Authors: Bing He
Comments: 16 Pages.

In this paper we introduce a finite field analogue for the Appell series F_{3} and give some reduction formulae and certain generating functions for this function over finite fields.
Category: Number Theory

 viXra:1704.0087 [pdf] submitted on 2017-04-07 09:13:10

### Squares of Primes that Can be Written as (p-Q-1)∙p-Q-1 Where P and Q Are Successive Primes

Authors: Marius Coman
Comments: 2 Pages.

In this paper I conjecture that there are an infinity of primes which can be written as sqr ((p – q – 1)*p – q – 1), where p and q are successive primes, p > q.
Category: Number Theory

 viXra:1704.0079 [pdf] submitted on 2017-04-06 23:53:45

### On Fermat's Equation and the General Case

Authors: C Sloane
Comments: 29 Pages. This paper is with several institutions and this submission is a time-stamp of authorship.

We discovered a beautiful symmetry to the equation x^n+y^n± z^n , first studied by Fermat, in a dependent variable t = x+y-z and the product (xyz) if we introduce a term we call the symmetric r = x^2+yz-xt-t^2. Once x^n+y^n± z^n is written in terms of powers of t, r and (xyz) we looked at the coefficient vs. exponent abstract space and found Lucas, Fibonacci and Convoluted Fibonacci sequences among other corollaries. We also found that 3 cases of a prime decomposition factor q of x^2+yz gave certain results for Fermat’s Last Theorem which could be eliminated if a forth case could also be solved. Intrigued by this, we then introduce partial congruence representations modulo a prime for this much harder forth case to find the ‘form’ of the solutions modulo q. The form of the solutions leads us to a cubic congruence method that solves the special and general cases. There are several pages and stages of the proofs where computer verification of the results is possible.
Category: Number Theory

 viXra:1704.0058 [pdf] submitted on 2017-04-05 08:41:51

### Universal Evolution Model Based On Theory Of Natural Metric For Functions And The Same As An Example Of A Universal Forecasting Model

Authors: Ramesh Chandra Bagadi
Comments: 9 Pages.

In this research investigation, the author has presented a novel scheme of Universal Evolution Model. This can also be used as a Universal Forecasting Model.
Category: Number Theory

 viXra:1704.0056 [pdf] submitted on 2017-04-05 10:40:16

### Universal Evolution Model And The Same As An Example Of A Universal Forecasting Model

Authors: Ramesh Chandra Bagadi
Comments: 10 Pages.

In this research investigation, the author has presented a novel scheme of Universal Evolution Model. Also, this model can be used as a Universal Forecasting Model.
Category: Number Theory

 viXra:1704.0029 [pdf] replaced on 2018-03-19 23:59:11

### An Introduction to the $n$-Irreducible Sequents and the $n$-Irreductible Number

Authors: A. Zaganidis
Comments: 19 Pages. I have chosen the category Number Theory since most of the consequences of the present article are inside the number theory.

In this work, we introduce the $n$-irreductible sequents and the $n$-irreductible numbers defined with the help of the second order logic. We give many concrete examples of $n$-irreductible numbers and $n$-irreductible sequents with the Peano's axioms and the axioms of the real numbers. Shortly, a sequent is $n$-irreductible iff the sequent is composed by some closed hypotheses and a $n$-irreductible formula (a close formula with one internal variable such that the formula is only true when we set that variable to the unique natural number $n$), and it does not exist some strict sub-sequent which are composed by some closed sub-hypotheses and some sub-$m$-irreductible formula with $m>1$. The definition is motivated by the intuition that \Nathypo do not carry natural numbers or "hidden natural numbers" except for the numbers $0$ and $1$, i.e., they can be used in a $n$-irreductible sequent. Moreover, we postulate at second order of logic that \Nathypo are not chosen randomly: \Nathypo are the only hypotheses which give the largest $n$-irreductible number $N_Z \NZ$. The Goldbach's conjecture, \poubelle{the Dubner's conjecture,}the Polignac's conjecture, the Firoozbakht's conjecture, the Oppermann's conjecture, the Agoh-Giuga conjecture, the generalized Fermat's conjecture and the Schinzel's hypothesis H are reviewed with this new (second order logic) $n$-irreductible axiom. Finally, two open questions remain: Can we prove that a natural number is not $n$-irreductible? If a $n$-irreductible number $n$ is found with a function symbol $f$ where its outputs values are only $0$ and $1$, can we always replace the function symbol $f$ by a another function symbol $\tilde{f}$ such that $\tilde{f}=1-f$ and the new sequent is still $n$-irreductible?
Category: Number Theory

 viXra:1704.0012 [pdf] submitted on 2017-04-02 07:11:13

### Universal Evolution Model Based On Theory Of Natural Metric For Functions

Authors: Ramesh Chandra Bagadi
Comments: 9 Pages.

In this research investigation, the author has presented a novel scheme of Universal Evolution Model.
Category: Number Theory