[22] **viXra:1907.0345 [pdf]**
*submitted on 2019-07-17 08:31:03*

**Authors:** Edgar Valdebenito

**Comments:** 3 Pages.

We give a formula for Pi.

**Category:** Number Theory

[21] **viXra:1907.0303 [pdf]**
*submitted on 2019-07-17 05:02:05*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This Relative formula shows the relationship between e and π without i.

**Category:** Number Theory

[20] **viXra:1907.0288 [pdf]**
*submitted on 2019-07-15 08:52:01*

**Authors:** Igor Hrnčić

**Comments:** 29 Pages.

In this manuscript we use the Perron formula to connect zeta evaluated on the root free halfplane to zeta evaluated on the critical strip. This is possible since the Perron formula is of the form f(s)=O f(s+w) with O being an integral operator. The variable s+w is on the root free halfplane, and yet s can be on the critical strip. Hence, the Perron formula serves as a form of a functional equation that connects the critical strip with the root free halfplane. Then, one simply notices that in the Perron formula, the left hand side converges only conditionally, whilst the right hand side converges absolutely. This, of course, cannot be, since the left side of an equation is always equal to the right side. This contradiction when examined in detail disproves the Riemann hypothesis. This method is employed on an arbitrary distribution of zeta roots as well, concluding that zeta has a root arbitrarily close to the vertical line passing through unity.

**Category:** Number Theory

[19] **viXra:1907.0276 [pdf]**
*submitted on 2019-07-15 14:34:10*

**Authors:** Sitangsu Maitra

**Comments:** 2 page

Proof of Goldbach's strong conjecture by unique path in a prime selection system.

**Category:** Number Theory

[18] **viXra:1907.0221 [pdf]**
*submitted on 2019-07-13 10:26:58*

**Authors:** Kamal Barghout

**Comments:** 5 Pages. The manuscript is not to be copied or used in whole or part. The manuscript is copyrighted.

In this note I will show how Beal’s conjecture can be used to study abc conjecture. I will first show how Beal’s conjecture was proved and derive the necessary steps that will lead to further understand the abc conjecture hoping this will aid in proving it. In short, Beal’s conjecture was identified as a univariate Diophantine polynomial identity derived from the binomial identity by expansion of powers of binomials, e.g. the binomial〖 (λx^l+γy^l )〗^n; λ,γ,l,n are positive integers. The idea is that upon expansion and reduction to two terms we can cancel the gcd from the identity equation which leaves the coefficient terms coprime and effectively describes the abc conjecture. To further study the abc terms we need to specifically look for criterion upon which the general property of abc conjecture that states that if the two numbers a and b of the conjecture are divisible by large powers of small primes, a+b tends to be divisible by small powers of large primes which leads to a+b be divisible by large powers of small primes. In this note I only open the door to investigate related possible criterions that may lead to further understand the abc conjecture by expressing it in terms of binomial expansions as Beal’s conjecture was handled.

**Category:** Number Theory

[17] **viXra:1907.0206 [pdf]**
*submitted on 2019-07-12 23:13:57*

**Authors:** Toshiro Takami

**Comments:** 10 Pages.

In the Riemann zeta function, when the value of the nontrivial zero is zero, the value of the real part of the function is negative from 0 to 0.5, but the value of the real part of the function is 0.5 to 1 I found it to be positive.
We also found that the positive and negative of the imaginary part also interchanged with the real part 0.5.
This tendency is seen as a tendency near the non-trivial zero value, but becomes less and less as it deviates from the non-trivial zero value.
We present and discuss the case of four non-trivial zero values. This seems to be an important finding and will be announced here.

**Category:** Number Theory

[16] **viXra:1907.0191 [pdf]**
*submitted on 2019-07-12 02:40:19*

**Authors:** Labib Zakaria

**Comments:** 12 Pages. Hopefully this is obvious from the abstract & a quick overview of the paper, but this is not meant to be an immensely technical paper. It is simply meant to be so that people can nurture an appreciation for math. Constructive criticism appreciated.

There exist many algorithms to test the primality of positive natural numbers both proved and unproved, as well as in base 10 and outside base 10. Once the primality of a number has been determined, natural questions are $(1)$ what the unique prime factors of it are and $(2)$ their degree, according to the fundamental theorem of arithmetic.
These questions can prove to be useful in beginning to analyze the properties of the number by allowing us to determine the number of (proper) divisors of a number as well as their sum and product. In regards to $(1)$, there are many algorithms that could be applied to determine these prime factors through modular arithmetic algorithms. We will be tackling this question in base 10 specifically by constructing functions as curious mathematicians.

**Category:** Number Theory

[15] **viXra:1907.0171 [pdf]**
*submitted on 2019-07-11 00:49:20*

**Authors:** Surajit Ghosh

**Comments:** 19 Pages.

Riemann hypothesis stands proved in three diﬀerent ways.To prove Riemann hypothesis from the functional equation concept of Delta function is introduced similar to Gamma and Pi function. Zeta values are renormalised to remove the poles of zeta function. Extending sum to product rule fundamental formula of numbers are deﬁned which then helps proving other prime conjectures namely goldbach conjecture, twin prime conjecture etc.

**Category:** Number Theory

[14] **viXra:1907.0154 [pdf]**
*submitted on 2019-07-09 18:42:44*

**Authors:** Viktor Kalaj

**Comments:** 10 Pages. This paper is rather succinct; it deals with a contradiction while testing the Riemann Zeta function valid on 0 < Re(s) < 1

In this paper, we summarize results of a contradiction while testing the Riemann Hypothesis

**Category:** Number Theory

[13] **viXra:1907.0126 [pdf]**
*submitted on 2019-07-09 01:25:02*

**Authors:** Darrin Taylor

**Comments:** Pages.

In base 3, the presence of leading 1s during division has a one to one correlation with the 3n+1 operation.
This is because dividing a leading 1 in base 3 is the only way to lose a digit and 3n+1 shifts are the only way to gain a digit. Total digit length doesn't change around a loop so they must equal each other.
Because the leading 1 pattern among a series of divides only has 2 segments either 1->2 or 1->2->1 there are a limited number of patterns that can make up a loop. Naming 1->2->(next segments leading 1) as segment A Naming 1->2->1->(next segments leading 1) as segment B We can see that A is 2 divides and 1 non localized shift while B is 3 divides and 2 non localized shifts. The pattern ABB...ABB descends because 8 divides and 5 shifts descends for numbers larger than 1000 and lower then 1000 have been numerically disqualified previously.
So the sequence BBB must exist at least once in every loop. BBB implies ABBB or BBBB if BBBB then "expel" a B which ascends and keep searching for the segment before the sequence. Once ABBB is found this implies AABBB or BABBB and AABBB is disproven as not possible. Once BABBB is known this implies ABABBB or BBABBB and ABABBB is disproven. Once BBABBB is known this implies ABBABBB or BBBABBB and ABBABBB is disproven. Once BBBABBB is found we can "expel" ABBB which ascends and BBB(ABBB) becomes BBB and we are back where we started. Once the entire loop has been traversed this way the sequence has expelled only (B) or (ABBB) and the remaining sequence is BBB and all of these ascend. Loops must have ascending and descending segments for a total non ascending and non descending but this loop always ascends.
Thus it cannot be a loop and no loops of As and Bs can exist as those with fewer Bs than ABBABBABB…..always descend and adding a single B makes it always ascend. And As and Bs are the only possible segments to add.

**Category:** Number Theory

[12] **viXra:1907.0109 [pdf]**
*submitted on 2019-07-06 06:57:31*

**Authors:** Victor Sorokine

**Comments:** 4 Pages.

В ПЕРВОМ СЛУЧАЕ каждое число (А) заменяется на сумму (A'+A°n) последней цифры и остатка. После раскрытия биномов в равенстве Ферма все члены объединятся в два слагаемых: E=A'^n+B'^n-C'^n с третьей цифрой E''', которая в одном из n-1 эквивалентных равенств Ферма равна 2, и остаток D с третьей цифрой D''', равной либо 0, либо n-1, и, следовательно, третья цифра в числе A^n+B^n-C^n не равна 0.
ВО ВТОРОМ СЛУЧАЕ (например A=A°n^k, но (BС)'≠0, ) после преобразования 3kn-значного окончания числа B в 1 и оставления в числах А, В, С лишь последних значащих цифр простейшие расчёты показывают, что (3kn-2)-я цифра числа A^n+B^n-C^n нулю не равна и не меняется после восстановления всех остальных цифр в числах A, B, C, т.к. является функцией только последней цифры числа A°.

**Category:** Number Theory

[11] **viXra:1907.0108 [pdf]**
*replaced on 2019-07-17 12:48:59*

**Authors:** Simon Plouffe

**Comments:** 77 Pages.

Conference held in Montréal at the ACA 2019, ETS.

**Category:** Number Theory

[10] **viXra:1907.0091 [pdf]**
*submitted on 2019-07-05 13:23:11*

**Authors:** Viktor Kalaj

**Comments:** 11 Pages. Notify me, the author, Viktor Kalaj, if this paper is in anyway difficult to read by the print (font, size, etc.)

This paper deals with a proposed contradiction to the Riemann Hypothesis. We see by a deductive approach the necessity of no zeroes for the entire critical strip, including for the critical line.

**Category:** Number Theory

[9] **viXra:1907.0089 [pdf]**
*submitted on 2019-07-05 17:23:01*

**Authors:** Viktor Kalaj

**Comments:** 1 Page. Minor typo correction in my paper "A technical procedure for the Riemann Hypothesis".

There was a minor typographical error in my paper entitled "A technical procedure for the Riemann Hypothesis". It does not affect the technical procedure of the paper.

**Category:** Number Theory

[8] **viXra:1907.0088 [pdf]**
*submitted on 2019-07-05 17:28:12*

**Authors:** Viktor Kalaj

**Comments:** A minor typographical correction to my 11-page paper

I made a typographical error that is now corrected. There is no change in the flow of the paper entitled "A technical procedure for the Riemann Hypothesis".

**Category:** Number Theory

[7] **viXra:1907.0087 [pdf]**
*replaced on 2019-07-17 02:03:38*

**Authors:** Toshiro Takami

**Comments:** 6 Pages.

In my previous paper “Consideration of the Riemann hypothesis” c=0.5 and x is non- trivial zero value, and it was described that it converges to almost 0, but a serious proof in mathematical expression could not be obtained.
In this paper, we give a proof of mathematical expression.
“the non-trivial zero values of all positive infinity and negative infinity lie on the real value 0.5” I am here mathematically proved.

**Category:** Number Theory

[6] **viXra:1907.0063 [pdf]**
*submitted on 2019-07-04 01:20:32*

**Authors:** Predrag Terzic

**Comments:** 4 Pages.

General,deterministic,unconditional,polynomial time primality test is introduced.

**Category:** Number Theory

[5] **viXra:1907.0055 [pdf]**
*submitted on 2019-07-03 10:09:12*

**Authors:** Http://vixra.org/author/andrew_w_ivashenko

**Comments:** 1 Page.

Decomposition of integer powers of a mersenne number into binomial coefficients

**Category:** Number Theory

[4] **viXra:1907.0046 [pdf]**
*submitted on 2019-07-02 08:37:34*

**Authors:** Edgar Valdebenito

**Comments:** 2 Pages.

We give some integrals for Pi.

**Category:** Number Theory

[3] **viXra:1907.0045 [pdf]**
*submitted on 2019-07-02 08:40:14*

**Authors:** Edgar Valdebenito

**Comments:** 1 Page.

This note presents two identities for Pi.

**Category:** Number Theory

[2] **viXra:1907.0037 [pdf]**
*submitted on 2019-07-02 16:29:32*

**Authors:** Toshiro Takami

**Comments:** 6 Pages.

In my previous paper “Consideration of the Riemann hypothesis” c=0.5 and x is non- trivial zero value, and it was described that it converges to almost 0, but a serious proof in mathematical expression could not be obtained.
It is impossible to make c = 0.5 exactly like this. c can only be 0.5 and its edge.
It is considered that “when the imaginary value increases to infinity, the denominator of the number becomes infinity and shifts from 0.5 to 0”.

**Category:** Number Theory

[1] **viXra:1907.0018 [pdf]**
*submitted on 2019-07-01 23:59:43*

**Authors:** Simon Plouffe

**Comments:** 58 Pages.

Une revue historique du nombre Pi faite à l'IUT de Nantes.
A presentation of Pi made at Université de Nantes (IUT) on April 25 2019.

**Category:** Number Theory