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[15] viXra:2110.0182 [pdf] submitted on 2021-10-31 03:15:13
Authors: A. A. Frempong
Comments: 6 Pages. Copyright © by A. A. Frempong
This paper proves the original and the equivalent ABC conjectures. The hypothesis for the original conjecture is basically the same as the hypothesis for the equivalent conjectures, and this hypothesis states that there exists only finitely many triples (A, B, C) of coprime positive integers, with A + B = C. The conclusion for the original conjecture would be that the product, d, of the distinct prime factors of A, B and C, is usually not much smaller than C. This conclusion would be interpreted as |C – d| < ε, where ε is a positive real number. The conclusions for the equivalent conjectures would be the following: 1. C > rad(d)^(1 + ε), 2. C< K(rad(d))^(1 + ε), where K is a constant, and K is a function of ε, a positive real number, 3. q(a, b, c) = (logC/(log(rad(d)))) > 1 + ε; However, for the equivalent conjectures, one will apply the conclusions containing the constant K, since their solutions for ε can readily be applied in the epsilon-delta proofs in this paper. One will also introduce the constant K into q(a, b, c) = (logC/(log(rad(d)))) > 1 + ε to obtain (logC/(log(rad(d)))) < K(1 + ε). Thus, the conclusions to be used for the equivalent conjectures are 1, C< K(rad(d))^(1 + ε), equivalently, {(logC – logK -log(rad(d)))/(log(rad(d)))} < ε;
2. {(logC/(log(rad(d)))} < K(1 + ε), equivalently, {(logC– Klog(rad(d)))/(Klog(rad(d)))} < ε. Let H = | A + B – C |. Then |H| < δ (δ being a positive real number) would be the hypothesis; and let |L| < ε be the conclusion for the original conjecture with L = C – d. For the equivalent conjectures, let L < ε be the equivalent conclusion, where L = (log C– logK – log(rad(d)))/(log(rad(d))); L = (logC– Klog(rad(d)))/(Klog(rad(d))). It has been proved that if
| A + B – C | < δ, then for the original conjecture, |L| < ε; and for the equivalent conjectures,
L < ε.
Category: Number Theory
[14] viXra:2110.0178 [pdf] replaced on 2021-11-10 03:33:24
Authors: Yong Zhao, Jianqin Zhou
Comments: 16 Pages.
In this paper, the estimation formula of the number of primes in a given interval is obtained by using the prime distribution property. For any prime pairs $p>5$ and $ q>5 $, construct a disjoint infinite set sequence $A_1, A_2, \ldots, A_i. \ldots $, such that the number of prime pairs ($p_i$ and $q_i $, $p_i-q_i = p-q $) in $A_i $ increases gradually, where $i>0$. So twin prime conjecture is true. We also prove that for any even integer $m>2700$, there exist more than 10 prime pairs $(p,q)$, such that $p+q=m$. Thus Goldbach conjecture is true.
Category: Number Theory
[13] viXra:2110.0165 [pdf] submitted on 2021-10-27 05:28:45
Authors: Pal Doroszlai, Horacio Keller
Comments: 12 Pages.
The union of arithmetic progressions of primes reflected over a point at any distance from the origin,
results the double density of occupation of integer positions by the series of multiples of primes. It is
shown, that the number of free positions left by the double density of occupation has a lower limit function.
These free positions represent equidistant primes satisfying Goldbach's conjecture. Herewith may be
proved as well, that at any distance from the origin, within the section equal to the square root of the
distance, there is a prime. Therefore the series of primes represent a continuum and may be integrated.
Further it may be proved, that the number of any two primes, with a given even number as difference
between them, is unlimited. Thus, the number of twin primes is unlimited as well.
Category: Number Theory
[12] viXra:2110.0163 [pdf] replaced on 2021-11-27 05:54:04
Authors: Abdelmajid Ben Hadj Salem
Comments: 4 Pages. Comments welcome.
In this paper about the abc conjecture, assuming the conjecture c<R^2 is true, we give an elementary proof that the abc conjecture is false.
Category: Number Theory
[11] viXra:2110.0149 [pdf] submitted on 2021-10-25 11:16:50
Authors: Timothy W. Jones
Comments: 2 Pages.
We give a way to determine the irrationality of a certain type of series.
Category: Number Theory
[10] viXra:2110.0133 [pdf] replaced on 2021-11-02 07:09:15
Authors: Marko V. Jankovic
Comments: 6 Pages.
In this paper, a novel approximation of the prime counting function, based on modified Eulerian logarithmic
integral, is going to be presented. Proposed approximation reduces the approximation error without increase of
computational complexity when it is compared to approximation based on Eulerian logarithmic integral. Experimental
results were used to support the claim. Combining proposed method with Riemannian approximation of prime counting
function it is possible to design the new approximation function that outperforms Riemannian approximation for all values
that were analyzed.
Category: Number Theory
[9] viXra:2110.0120 [pdf] replaced on 2021-10-24 07:10:49
Authors: Theophilus Agama
Comments: 5 Pages. This is a completely revised version.
In this note, we introduce the notion of the disc induced by an arithmetic function and apply this notion to the odd perfect number problem. We show that under certain special local condition an odd perfect number exists by exploiting this concept.
Category: Number Theory
[8] viXra:2110.0119 [pdf] submitted on 2021-10-20 05:58:05
Authors: Gaurav Krishna
Comments: 10 Pages.
3n+1 & n/2 we have multiplicative operation and inverse multiplicative operation. But in the case of odd element we have an extra additive operation involved. This makes any sort of analysis very difficult as it is not know how to combine additive and multiplicative operations together in a series of transformation. If we had known how to solve additive and multiplicative operations together in a series of transformations, primes would have been much easier to deal with.
In order to deal with this limitation;
We create a function that gives same results that the transformations would yield without really applying them. The function shall be represented as [r_b].
We define the relationships between [r_b] & n and various forms of [r_b]
We analyze the conditions for failure of the conjecture and test them using [r_b]
Category: Number Theory
[7] viXra:2110.0114 [pdf] submitted on 2021-10-19 00:46:17
Authors: Takamasa Noguchi
Comments: 4 Pages.
We have created a handy tool that allows you to calculate the q^k power root of a easily and quickly.
However, the calculation may require a primitive root, and if the calculation requires a primitive root and you do not know the primitive root, please use the Tonelli-Shanks algorithm.
Category: Number Theory
[6] viXra:2110.0087 [pdf] submitted on 2021-10-17 11:54:52
Authors: Ramesh Chandra Bagadi
Comments: 1 Page.
In this research investigation, the author proposes Fibonacci Type Series Using Prime Sequence.
Category: Number Theory
[5] viXra:2110.0079 [pdf] submitted on 2021-10-16 09:17:38
Authors: Timothy W. Jones
Comments: 8 Pages.
We develop an aspect of decimal representation of rational numbers and use it to prove a family of series converges to an irrational number.
Category: Number Theory
[4] viXra:2110.0047 [pdf] submitted on 2021-10-11 19:54:59
Authors: Gregory M. Sobko
Comments: 10 Pages.
An additive model of random walks on set of natural numbers is applied to analyze the probability distribution of gaps, that is differences d = p’- p, between consecutive prime numbers p and p'. The well-known fact is that gaps between consecutive primes can be as small as 2
(for twin primes) and arbitrary large.
This work is concerns with sets DP(d)
of primes with gaps d (called d-primes),
where d is an even number. For DP(2) we have
a set of twin primes, with the unproven conjecture that DP(2) is an infinite set. We provide some
statistical analysis for the frequency distribution of d-primes. The main result of this work is the proof that DP(d) is infinite set for every even d. The proof is based on modified Cramér’s probabilistic model for the distribution of prime numbers. This method has been discussed in detail in the author’s previous preprint [1].
Category: Number Theory
[3] viXra:2110.0033 [pdf] submitted on 2021-10-07 21:27:25
Authors: Jaejin Lim
Comments: 5 Pages.
In this paper, we prove Twin prime conjecture and Goldbach’s conjecture. We do this in three stages by turns; one is ‘Application Principle of Mathematical Induction’, another is ‘Proof of Twin prime conjecture’ and the other is ‘Proof of Goldbach’s conjecture’. These three proofs are interconnected, so they help prove it. Proofs of Twin prime conjecture and Goldbach’s conjecture are proved by Application Principle of Mathematical Induction. And Twin prime conjecture is based on Goldbach’s conjecture. So, we can get the result, Twin prime and Goldbach’s conjecture are true. The reason why we could get the result is that I use twin prime’s characteristic that difference is 2 and apply this with Application Principle of Mathematical Induction. If this is proved in this way, It implies that the problem can be proved in a new way of proof.
Category: Number Theory
[2] viXra:2110.0022 [pdf] submitted on 2021-10-05 02:19:14
Authors: Gregory M. Sobko
Comments: 53 Pages.
This work is concerned with probabilistic concepts of ’randomness’ and ‘independence’ relevant to classical number-theoretic problems. Basic properties of divisibility for natural numbers are interpreted in terms of probability spaces and appropriate probability distributions on classes of congruence. We analyze and demonstrate the importance of Zeta probability distribution, proving that probabilistic independence of coprime factors for randomly chosen natural numbers is equivalent to the fact that a random variable representing these numbers must follow Zeta probability distribution. We show that the probabilistic Cramér’s model for the asymptotic distribution of primes is validated by the proven here its asymptotic equivalence to Zeta probability distribution, in agreement with
the Prime Number Theorem.
We prove the exact formula for a Zeta distributed random variable to represent a prime number. It is used to generate and analyze the corresponding multiplicative and additive random walks on semigroups generated by primes and natural numbers, respectively.
We prove that Cramér’s model for the distribution
of primes, modified as a generalized predictable
non-stationary Bernoulli process with dependent terms is asymptotically a process with pairwise independent Bernoulli variables. Finally, we provide probabilistic proof of the Strong Goldbach Conjecture, by using the results described above.
Category: Number Theory
[1] viXra:2110.0015 [pdf] submitted on 2021-10-05 21:51:49
Authors: Andrea Berdondini
Comments: 2 Pages.
The uncertainty of the statistical data is determined by the value of the probability of obtaining an equal or better result randomly. Since this probability depends on all the actions performed, two fundamental results can be deduced. Each of our random and therefore unnecessary actions always involves an increase in the uncertainty of the phenomenon to which the statistical data refers. Each of our non-random actions always involves a decrease in the uncertainty of the phenomenon to which the statistical data refers.
Category: Number Theory
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