[13] **viXra:1901.0227 [pdf]**
*replaced on 2019-01-17 08:32:12*

**Authors:** James Edwin Rock

**Comments:** 5 pages. This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Collatz sequences are formed by applying the Collatz algorithm to any positive integer. If it is even repeatedly divide by two until it is odd, then multiply by three and add one to get an even number and vice versa. If the Collatz conjecture is true eventually you always get back to one. A connected Collatz Structure is created, which contains all positive integers exactly once. The terms of the Collatz Structure are joined together via the Collatz algorithm. Thus, every positive integer forms a Collatz sequence with unique terms terminating in the number one.

**Category:** Number Theory

[12] **viXra:1901.0193 [pdf]**
*replaced on 2019-01-13 18:40:10*

**Authors:** Toshiro Takami

**Comments:** 50 Pages.

I calculated it by looking for a counter example, but I can not determine whether this is a counter example or just a normal zero point and post it here.
It is a different value from the previous counter example. I searched for points of higher value, but I could not find it for some reason.
The point this time is near the previous point, is the number (the number axis) distorted only in this part? It can not be determined.
zeta[0.4999977+i393939944.25715353678841792735]= -3.372108136572006... × 10^-19 + 4.002018173119188... × 10^-13 i
and
zeta[0.50001314+i393939946.4889505702488576920]= -7.000197154138805... × 10^-19 - 2.848659916217643... × 10^-12 i

**Category:** Number Theory

[11] **viXra:1901.0191 [pdf]**
*replaced on 2019-01-16 15:31:22*

**Authors:** Toshiro Takami

**Comments:** 12 Pages.

It presents counter exsample which is close to 0.5 of 5 Riemann expectations but not 0.5.
Regardless of how they are calculated, they are all found in the same area for some reason. I could not find it in other areas.
This is considered only because the number axis is distorted in this area.
Somewhere on the net there is a memory that reads the mathematician's view that "there are countless zero points in the vicinity of 0.5".
The value I put out is hand calculation using WolfranAlpha,
It seems that it is necessary to strictly correct by supercomputer.
zeta[0.5000866+i393939939.3731193515534038924]= -1.60917723458557... × 10^-18 - 1.428779604546702... × 10^-11 i
and
zeta[0.4999977+i393939944.25715353678841792735]= -3.372108136572006... × 10^-19 + 4.002018173119188... × 10^-13 i
and
zeta[0.50001314+i393939946.4889505702488576920]= -7.000197154138805... × 10^-19 - 2.848659916217643... × 10^-12 i
and
zeta[0.4999944+i393939958.90878694741368323631]= 9.30660314868779... × 10^-19 + 1.342928180878699... × 10^-12 i
and
zeta[0.4999964+i393939964.659437163857861]= -5.914628349384624... × 10^-16 + 6.504227267123851... × 10^-13 i
and

**Category:** Number Theory

[10] **viXra:1901.0188 [pdf]**
*submitted on 2019-01-14 01:42:48*

**Authors:** Toshiro Takami

**Comments:** 60 Pages.

Riemann hypothesis
Further, three more counter example?

**Category:** Number Theory

[9] **viXra:1901.0155 [pdf]**
*submitted on 2019-01-11 06:22:54*

**Authors:** Edgar Valdebenito

**Comments:** 1 Page.

This note presents three trigonometric identities.

**Category:** Number Theory

[8] **viXra:1901.0116 [pdf]**
*submitted on 2019-01-10 02:36:15*

**Authors:** Quang Nguyen Van

**Comments:** 2 Pages.

The equation a^5 + b^5 = c^2 has no solution in integer. However, related to Fermat- Catalan conjecture, the equation a^5 + b^5 = 2c^2 has a solution in integer. In this article, we give a parametric equation of the equation a^5 + b^5 = 2c^2.

**Category:** Number Theory

[7] **viXra:1901.0108 [pdf]**
*submitted on 2019-01-08 11:13:12*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 5 Pages. A Proof of ABC conjecture is submitted to the Journal of Number Theory (2019). Comments Welcome.

In this paper, we assume that the ABC conjecture is true, then we give a proof that Beal conjecture is true. We consider that Beal conjecture is false then we arrive to a contradiction. We deduce that the Beal conjecture is true.

**Category:** Number Theory

[6] **viXra:1901.0104 [pdf]**
*submitted on 2019-01-08 18:01:42*

**Authors:** Toshiro Takami

**Comments:** 5 Pages.

I tried to prove that(Riemann hypothesis), but I realized that I can not prove how I did it.
When we calculate by the sum method of (1) we found that the nontrivial zero point will never converge to zero.
Calculating ζ(2), ζ(3), ζ(4), ζ(5) etc. by the method of the sum of (1) gives the correct calculation result.
This can be considered because convergence is extremely slow in the case of complex numbers, but there is no tendency to converge at all.
Rather, it tends to diffuse.
In other words, it is inevitable to conclude that Riemann's hypothesis is a mistake.
We will fundamentally completely erroneous ones, For 150 years, We were trying to prove it.

**Category:** Number Theory

[5] **viXra:1901.0101 [pdf]**
*submitted on 2019-01-09 00:16:39*

**Authors:** Johnny E. Magee

**Comments:** 7 Pages.

We identify equivalent restatements of the Brocard-Ramanujan diophantine equation, $(n! + 1) = m^2$; and employing the properties and implications of these equivalencies, prove that for all $n > 7$, there are no values of $n$ for which $(n! + 1)$ can be a perfect square.

**Category:** Number Theory

[4] **viXra:1901.0058 [pdf]**
*replaced on 2019-01-08 17:53:16*

**Authors:** Toshiro Takami

**Comments:** 17 Pages.

2^s/(2^-1)*3^s/(3^-1)*5^s/(5^s-1)*7^s/(7^s-1)………
Whether the above equation converges to 0 was verified.
Convergence is extremely slow, and divergence tendency was rather rather abundant when the prime number was 1000 or more.
It was thought that the above equation could possibly be an expression that can be composed only of real numbers.

**Category:** Number Theory

[3] **viXra:1901.0046 [pdf]**
*submitted on 2019-01-04 11:35:20*

**Authors:** Nazihkhelifa

**Comments:** 4 Pages.

Relation between
The Euler Totient,
the counting prime formula
and the prime generating Functions
The theory of numbers is an area of mathematis hiih eals ith
the propertes of hole an ratonal numbers... In this paper I ill
intro uie relaton bet een Euler phi funiton an prime iountnn
an neneratnn formula, as ell as a ioniept of the possible
operatons e ian use ith them. There are four propositons hiih
are mentone in this paper an I have use the efnitons of these
arithmetial funitons an some Lemmas hiih refeit their
propertes, in or er to prove them

**Category:** Number Theory

[2] **viXra:1901.0030 [pdf]**
*submitted on 2019-01-03 17:17:16*

**Authors:** Robert C. Hall

**Comments:** 24 Pages.

The Benford's Law Summation test consists of adding all numbers that begin with a particular first or first two digits and determining its distribution with respect to these first or first two digits numbers. Most people familiar with this test believe that the distribution is a uniform distribution for any distribution that conforms to Benford's law i.e. the distribution of the mantissas of the logarithm of the data set is uniform U[0,1). The summation test that results in a uniform distribution is true for an exponential function (geometric progression) but not true for a data set that conforms to a Log Normal distribution even when the Log Normal distribution itself closely approximates a Benford's Law distribution.

**Category:** Number Theory

[1] **viXra:1901.0007 [pdf]**
*submitted on 2019-01-01 16:19:56*

**Authors:** Rachid Marsli

**Comments:** 11 Pages.

In this work, we show a sufficient and necessary condition for an integer of the form
(z^n-y^n)/(z-y)to be divisible by some perfect nth power p^n, where p is an odd prime. We also show how to construct such integers. A link between
the main result and Fermat’s last theorem is discussed. Other related ideas, examples and applications are provided.

**Category:** Number Theory