In this study we used an algebraic method that uses elementary algebra and binomial theorem. To create series. We also obtained several results in finite series. In this paper we prove the binary Goldbach conjecture.
It is derived simply by inverting the relation presented in the precursor paper with one of two approaches its generating function or the binomial theorem. This enables one to go from Lerch as a function of Hurwitz zetas of different orders , to Hurwitz as a function of Lerches.
A special case of this new functional equation is a relation between the Riemann's zeta function and the polylogarithm. In this work we used an algebraic method that uses elementary algebra. This relationship is the results that relate the solutions of congruence to the solution of the equations Category: Number Theory. Authors: Shazly Abdullah Comments: 9 Pages. To create series We used these series to study the prime numbers of the form , We found several characteristics.
In this paper we study an extension of the Euler totient function to the rationals and explore some applications. Authors: Richard Harris Comments: 2 Pages. An alternative, but equivalent, form of Riemann Hypothesis. Authors: Toshihiko Ishiwata Comments: 21 Pages. This paper is a trial to prove Riemann hypothesis according to the following process. We find that the value of F a that is the infinite series regarding a must be zero from the above infinite number of infinite series.
Authors: Jose R. Sousa Comments: 21 Pages. Sousa Comments: 14 Pages. Sousa Comments: 10 Pages. The literature has a formula for the polylogarithm at the negative integers, which utilizes the Stirling numbers of the second kind. Lastly, we present a new formula for the Hurwitz zeta function at the positive integers using this novelty. We present some novelties on the Riemann zeta function.
Authors: Hajime Mashima Comments: 8 Pages. This paper proves an inconsistency within ZFC by showing that a strengthened form of the strong Goldbach conjecture as well as its negation can be deduced.
The twin prime conjecture was proposed by Alfonse de Polignac in and has not been proven for nearly years. Since there is no mathematical model for prime numbers that can be completely and accurately represented, prime numbers are randomly distributed on the number axis, and twin prime numbers are also randomly distributed.
This paper proves the twin prime number conjecture through probability and statistics, and further discovers the twin prime number distribution theorem and the prime pair distribution theorem. In this paper we prove Lemoine's conjecture. Any even number can be decomposed into the sum of a larger prime number and a smaller odd number. The Goldbach conjecture can be proved by calculating whether the cumulative probability that the small odd number is a prime number is much greater than 1 and determining whether the prime number in many small odd numbers is inevitable.
In this paper, the Goldbach conjecture is proved by the method of probability and statistics, the Goldbach number theorem is further discovered, and a new method for the study of prime number distribution is created. The Riemann hypothesis is true. In this paper I present a solution for it in a very short and condensed way, making use of one of its equivalent problems. But as Carl Sagan once famously said, extraordinary claims require extraordinary evidence.
The evidence here is the newly discovered inversion formula for Dirichlet series. I present a method to solve the general cubic polynomial equation based on six years of research that started back in when, in the fifth grade, I first learned of Bhaskara's formula for the quadratic equation.
I was fascinated by Bhaskara's formula and naively thought I could replicate his method for the third degree equation, but only succeeded in , after countless failed attempts. The solution involves a simple transformation to form a cube and which, by chance, happens to reduce the degree of the equation from three to two which seems to be the case of all polynomial equations that admit solutions by means of radicals.
I also talk about my experiences trying to communicate these results to mathematicians, both at home and abroad. Authors: D. Janabi Comments: 2 Pages. In this short paper, we establish a number-theoretic conjecture about primes with a special property and give a hint for the proof. Derived from Dirichlet eta function [proxy for Riemann zeta function] are, in chronological order, simplified Dirichlet eta function and Dirichlet Sigma-Power Law.
Thus, it is proved that Riemann hypothesis is true whereby this function and law rigidly comply with Principle of Maximum Density for Integer Number Solutions. The geometrical-mathematical [unified] approach used in our proof is equivalent to the algebra-geometry [unified] approach of geometric Langlands program that was formalized by Professor Peter Scholze and Professor Laurent Fargues.
A succinct treatise on proofs for Polignac's and Twin prime conjectures is also outlined in this research paper. Authors: Vimosh Venugopal Comments: 14 Pages. The study details specified properties of whole numbers in conjunction with repetitive arrays and sequences. There prevails a common pattern for numbers when they exist in defined structures. The paper extends to the scope of progressions in regard to the specific number relationships and its reach in advanced mathematical studies.
The properties of numbers enumerated have its scope in the field of recreational mathematical theories as well.
The prime-number-formula at any distance from the origin has a systematic error, proportional to the square of the number of primes up to the square root of the distance. The proposed completion in the present paper eliminates by a quickly converging recursive formula the systematic error.
The remaining error is reduced to a symmetric dispersion, with standard deviation proportional to the number of primes at the square root of the distance.
Authors: Victor Sorokine Comments: 2 Pages. The third from the end digit in the sum of two equivalent Fermat's equalities with the last digits in the numbers A, B, C equal to a, b, c, and n-a, n-b, n-c, is equal to 1, and at the same time it is a single-valued function of the last digits. Authors: Michael Griffin Comments: 14 Pages. Fermat's Last Theorem is investigated on the set of Pythagorean triples using the ancient Greek formulas of Pythagoras, Euclid, and Plato. These are formulas used to derive natural number solutions of the Pythagorean theorem.
Authors: A. Frempong Comments: 6 Pages. This paper proves the original and the equivalent ABC conjectures. 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.
In this paper, the estimation formula of the number of primes in a given interval is obtained by using the prime distribution property. So twin prime conjecture is true. Thus Goldbach conjecture is true. 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. Submitted to The Ramanujan Journal. Comments welcome. Authors: Timothy W. Jones Comments: 2 Pages. We give a way to determine the irrationality of a certain type of series. Jankovic Comments: 5 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. 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 no odd perfect numbers exist by exploiting this concept.
Authors: Gaurav Krishna Comments: 10 Pages. 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.
Authors: Takamasa Noguchi Comments: 4 Pages. 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. Authors: Prakhar Rakhya Comments: 24 pages long. Authors: Kouji Takaki Comments: 5 Pages. 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.
Authors: Konstantinos Smpokos Comments: 5 Pages. In this article we will deal with the well known Gauss Lattice Point Problem. This problem is to find how many lattice points are inside in a circle with center 0.
This is an open problem. We will find an asymptotic formula for the number of such points. Authors: Prakhar Rakhya Comments: 18 Pages. Some new insights and relation between error term of prime counting function and Riemann zeta function using Prime zeta function.
Sobko Comments: 10 Pages. 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. Authors: Jaejin Lim Comments: 5 Pages.
These three proofs are interconnected, so they help prove it. If this is proved in this way, It implies that the problem can be proved in a new way of proof.
Sobko Comments: 53 Pages. 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 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.
Finally, we provide probabilistic proof of the Strong Goldbach Conjecture, by using the results described above. 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. Bass Comments: [Pages resized by viXra Admin. This paper presents a proof of the Collatz Conjecture by showing that the sequence of numbers generated cannot either diverge, converge to a single number, possess more than one stable oscillation, or alternate indefinitely.
The methods used are i Number pattern Recognition, ii basic analysis and iii simple interpretive logic. This paper presents a proof of the Goldbach Conjecture by comparing the distribution of prime numbers with the inverse distribution of odd composite numbers. Authors: Adarsha Chandra Comments: 17 Pages. The truth of the Collatz Conjecture follows immediately from the above. By applying basic mathematical principles, the author proves an equivalent ABC conjecture.
Authors: Philip Gibbs Comments: 12 Pages. Submitted to the journal 'Communications in Mathematics and Statistics. Keywords: Zeta function, non trivial zeros of Riemann zeta function, zeros of Dirichlet eta function inside the critical strip, definition of limits of real sequences.
Authors: Kurmet Sultan Comments: 1 Page. This is an abbreviated version of the main article. Proof of Goldbach's conjecture Category: Number Theory. Authors: Li Ke Comments: 7 Pages. Innovative ideas and methods. Sobko Comments: 15 Pages. We derive multiplicative and additive models with recurrent equations for generating sequences of prime numbers based on the reduced Sieve of Eratosthenes Algorithm and analyze their asymptotic behavior with the help of Riemann Zeta probability distribution.
This allows interpreting such sequences as realizations of random walks on set of natural numbers and on multiplicative semigroups generated by prime numbers, representing paths of stochastic dynamical systems. We analyze in this work an additive continuous-time probabilistic model of counting function of primes pi n in terms of diffusion approximation of non-Markov random walks. Computer modeling illustrates graphically an impressive fitting of trajectories for the original counting function, the calculated trend function, and the Brownian approximation.
Authors: Anass Massoudi Comments: 8 Pages. Authors: Philip Gibbs Comments: 8 Pages. A rational Diophantine m-tuple is a set of m rational numbers such that the product of any two is one less than a square. This parallel property implies a very strict geometric restriction which lead to two successful proofs of Riemann Hypothesis RH. One proof is from the contradictions which come from the trajectories of RZF, and the other proof is by applying Chauchy integral theorem to the trajectory of RZF.
We tried to provide sufficient graphs and videos for the understanding of the vector geometry properties of RZF and DEF. In appendix, we provided the source programs for analyzing vectors and suggested two other possible proofs of RH for further studies. Type equation here. By combination of finite number theory and quantum information the complete quantum information in the DNA genetic code has been made likely by Planat et al.
In the present contribution a varied quartic polynomial contrasting the polynomial used by Planat et al. Its roots are changed to more golden mean based ones in comparison to the Planat polynomial. As an outlook it should be emphesized that the connection between genetic code and resonance code of the DNA may lead us to a full understanding of how nature stores and processes compacted information and what indeed is consciousness linking everything with each other suggestedly mediated by all-pervasive dark constituents of matter respectively energy.
Authors: Robert Spoljaric Comments: 4 Pages. In this article we shall use the partial sums of the alternating harmonic series to a prove the harmonic series diverges, and b show that every harmonic number greater than is the sum of partial sums of the alternating harmonic series.
Authors: Li Ke Comments: 1 Page. Goldbach conjecture is known as the jewel in the crown of mathematics. At present, the research methods mainly include almost prime, exception set, three prime theorem and almost Goldbach problem.
But none of them solved the problem. In , Goldbach conjecture ushered in a new proof path — [new] sequence. This method is feasible in theory. So that tells us that the Goldbach conjecture is true. Original language Spanish translated into English. This text develops a new Primality Algorithm, this one obtains opposite results to Fermat's little theorem, since it uses similar mechanisms but applied to the analysis of patterns.
In Fermat's Theorem there are always Pseudoprimes hidden among the primes, which does not give certainty about the primality of an odd number analyzed, beyond the change of bases as happens with the Pseudoprime number In the Argentest algorithm, the opposite happens, the pseudoprimes do not pass the test, so we can confirm the primality of a number with absolute certainty and determination, but there is a percentage of primes that do not pass the test either, so we go to the change of base to re-analyze the patterns and confirm primality later.
For the first time, the article proposes a formula allowing to represent a cube of a natural number as a sum of cubes of three natural numbers. The article provides a test for twin primes and new tests for primes, which are radically different from the known tests. I present an algorithm that defines a function generator of sequences eventually periodic, with eligible cycle values and starting with any integer.
Jankovic Comments: 17 Pages. In this paper proof of the Polignac's Conjecture for gap equal to eight is going to be presented. It will be shown that consecutive primes with gap eight could be obtained through two stage sieve process, and that will be used to prove that infinitely many primes with gap eight exist.
The proof represents an simple extension of the recently presented proof that infinitely many sexy prime exist. The major contribution of this paper is presentation of all elementary modules that are necessary for the proof of Polgnac's conjecture in general case.
In the Argentest algorithm, the opposite happens, the pseudoprimes do not pass the test, so we can confirm the primality of a number with absolute certainty and determination, but there is a percentage of primes that do not pass the test either, so we go to the base change to re-analyze the patterns and confirm primality later Category: Number Theory. Note: This paper is written in French and English.
In the theory of solving equations with one unknown, Galois theory explains all, but this theory remains insufficient for equations with several unknowns, in this paper I show on the most famous example of these equations, « Fermat's equation », that the same ideas which allowed Galois his theory, also allow to justify Fermat's equation. This was conjectured by Christian Goldbach in and still remains unproved. GPMT is a 2-dimensional table of all possible pair of two numbers x, 2n — x , whose sum can be any even number 2n.
To functionally treat the sieve of Eratosthenes, we devised SFs that have sinusoidal symmetry and period properties. In this paper, the author proposes to establish, using relatively simple means, that the quantity of twin primes is infinite, and therefore that the conjecture concerning this notion is in fact a theorem.
We call k the generator of p. Twin primes have a common generator and therefore it makes sense to consider twin primes on the level of their generators. The present paper considers increasing sections of the number line containing prospective twin prime generators and gives a lower bound on the number of such generators in periodic subsections. We prove further that the prospective twin prime generators are asymptotically, uniformly distributed over those periodic subsections.
With these results the Twin Prime Conjecture finally can be proved. As there is no special primality test for Twin primes numbers. Argentest II is born, a personal research project that develops a new exclusive probabilistic primality test for Twin prime numbers.
Jankovic Comments: 13 Pages. In this paper proof of the Polignac's Conjecture for gap equal to six is going to be presented. Consecutive primes with gap six are known as sexy primes. The proof represents an extension of the proof of the twin prime conjecture.
It will be shown that sexy primes could be obtained through two stage sieve process, and that will be used to prove that infinitely many sexy primes exist. Authors: Zhang Tianshu Comments: 14 Pages. First, let us clarify certain of basic concepts related to the proof. After that, use the mathematical induction. Then again classify positive integers to prove one or even all of classes of positive integers on different levels. In addition, before the proof can begin, it is necessary to prepare several judging criteria to be used in the proof of each class of positive integers.
Authors: Zhang Tianshu Comments: 18 Pages. And then, by applying the odd-even relations of integers concerned in the symmetry, two inequalities are proved by the mathematical induction. Then again, other two inequalities are too proved by the method of splitting integers and then merging them. Authors: Zhang Tianshu Comments: 10 Pages.
For the unsolved kind, again divide it into 3 genera, and that formulate each of 2 genera therein into a sum of 3 unit fractions. For the unsolved genus, further divide it into 5 sorts, and that formulate each of 3 sorts therein into a sum of 3 unit fractions. For two unsolved sorts i. Authors: Brian Scannell Comments: 12 Pages. The nulls in antenna radiation patterns show the non-trivial Riemann zeta zeros.
Here we simulate antenna designs that show the zeros Category: Number Theory. This paper is a resume of my investigation on the Collatz conjecture. We take a look to the hypothesis of infinite systems of iterations or combined iterations. Authors: Shekhar Suman Comments: 11 Pages. Authors: Andrea Berdondini Comments: 6 Pages. In statistics, to evaluate the significance of a result, one of the most used methods is the statistical hypothesis test.
Using this theory, the fundamental problem of statistics can be expressed as follows: "A statistical data does not represent useful information, but becomes useful information only when it is shown that it was not obtained randomly".
Consequently, according to this point of view, among the hypotheses that perform the same prediction, we must choose the result that has a lower probability of being produced randomly.
Therefore, the fundamental aspect of this approach is to calculate correctly this probability value. This problem is addressed by redefining what is meant by hypothesis. The traditional approach considers the hypothesis as the set of rules that actively participate in the forecast. Instead, we consider as hypotheses the sum of all the hypotheses made, also considering the hypotheses preceding the one used.
Therefore, each time a prediction is made, our hypothesis increases in complexity and consequently increases its ability to adapt to a random data set. In this way, the complexity of a hypothesis can be precisely determined only if all previous attempts are known. Consequently, Occam's razor principle no longer has a general value, but its application depends on the information we have on the tested hypotheses.
Authors: Han Geurdes Comments: 6 Pages. A new attempt. This is rewritten after comments. Authors: Marcin Barylski Comments: 5 Pages. But do we need all primes to satisfy this conjecture? This work is devoted to selection of must-have primes and formulation of stronger version of LC with reduced set of primes. Authors: Marcin Barylski Comments: 10 Pages.
Hypothesis still remains open and is confirmed experimentally for bigger and bigger n. This work studies different approaches to finding the first confirmation of this conjecture in order to select the most effective confirmation method.
Authors: Jaykov Foukzon Comments: 43 Pages. External induction principle in nonstandard intuitionistic arithmetic were derived. Non trivial application in number theory is considered. The Goldbach-Euler theorem is obtained without any references to Catalan conjecture.
Authors: Marcin Barylski Comments: 3 Pages. This work is devoted to selection of must-have primes and formulation of stronger version of GSC with reduced set of primes. Authors: Marcin Barylski Comments: 4 Pages. This work is devoted to studies on sum of two prime numbers.
This work is devoted to studies on twin primes present in Goldbach partitions. This text develops and formulates the discovery of an unknown pattern for prime numbers, with amazing and calculable characteristics. Using a mechanism similar to the Collatz conjecture. In this paper we study the shortest addition chains of numbers of special forms.
Authors: Gaurav Krishna Comments: 11 Pages. The approach to the problem is in the reverse. Tried it the other way round, gets messier.
So just sharing the final simpler version. The paper contains tables, which are basically equations for sets of data and rearrangement of those tables is just playing with the underlying equations. But, up to date there is no valid proof of TPC. By using the periodicity of sinusoidal functions, we proved that TPC is true.
Authors: Dmitri Martila Comments: 6 Pages. In this short note, I provide a proof for the Riemann Hypothesis. You are free not to get enlightened about that fact. But please pay respect to new dispositions of the Riemann Hypothesis and research methods in this note. I start with Dr. Authors: Konstantinos Smpokos Comments: 9 Pages. Published in the Bulletin of the Hellenic Mathematical Society. Authors: Konstantinos Smpokos Comments: 11 Pages. In this article we generalize the Lehmer's totient problem in algebraic number fields.
We introduce the notion of a Lehmer number. Lehmer numbers are defined to be the natural numbers which obey the Lehmer's problem in the ring of algebraic integers of a number field.
Authors: Theophilus Agama Comments: 9 Pages. In this paper we develop the method of circle of partitions and associated statistics. As an application we prove conditionally the binary Goldbach conjecture.
We develop series of steps to prove the binary Goldbach conjecture in full. We end the paper by proving the binary Goldbach conjecture for all even numbers exploiting the strategies outlined.
In this thesis, we related FLT to two polynomial equations. What we found is that those two equations can not have equivalence properties in all four aspects which is enough to prove FLT. In this paper we are going to see two theoretical expressions in reference of canonical representations. Based in the classic definition of positive integers we can use some mathematical tools to define the subsets of composite and prime numbers in their canonical form. Authors: Antoni Cuenca Comments: 17 Pages.
We introduce a topology in the set of natural numbers via a subbase of open sets. With this topology, we obtain an irreducible hyperconnected space with no generic points. This fact allows proving that the cofinite intersections of subbasic open sets are always empty.
That implies the validity of the Twin Prime Conjecture. On the other hand, the existence of strictly increasing chains of subbasic open sets shows that the Polignac Conjecture is false for an infinity of cases. Authors: Suaib Lateef Comments: 6 pages. In this paper, we present an identity involving Tribonacci Numbers. We will prove this identity by extending the number of variables of Candido's identity to three. Authors: Bertrand Wong Comments: 15 Pages.
Published in an international mathematics journal. The primes, including the twin primes and the other prime pairs, are the building-blocks of the integers. It is a proof by contradiction, or, reductio ad absurdum, and it relies on an algorithm which will always bring in larger and larger primes, an infinite number of them.
However, the proof is also subtle and has been misinterpreted by some with one well-known mathematician even remarking that the algorithm might not work for extremely large numbers. A long unsettled related problem, the twin primes conjecture, has also aroused the interest of many researchers. The author has been conducting research on the twin primes for a long time and had published a paper on them in an international mathematics journal in Frempong Comments: 5 Pages.
The author will adhere to the wording of the original conjecture and not to any equivalent conjecture, since if one proves an equivalent conjecture, logically, one would also have to prove the equivalency, and otherwise, the proof of the original conjecture would be incomplete. As there is no special primality test for Sophie Germain primes and safe primes as is the case with Fermat primes and Mersenne primes. Argentest is born, a personal research project that develops a new exclusive deterministic primality test for Sophie Germain's prime numbers and safe prime numbers.
Authors: Pranjal Jain Comments: 12 Pages. We use numerical patterns as a guide towards the solution and explore an additional numerical pattern which shows a relation between decimal expansions and multiplicative inverses of powers of 3 modulo powers of Authors: Denis Gallet Comments: 6 Pages.
In this paper, i study particular values of Barnes G-function and we can simplify several integrals logarithm gamma. Authors: Alexey Ponomarenko Comments: 7 Pages. A lower bound is given for the number of primes in a special linear form less than N, under the assumption of the weakened Elliott-Halberstam conjecture. Sobko Comments: 24 Pages. A Recursive Algorithm described here generates consecutive sequences of Goldbach sets toward the proof of the Strong Goldbach Conjecture.
Approach suggested here is based on the fundamental principle of mathematical induction and uses rather elementary set-theoretical technique. It does not involve any sophisticated powerful tools and results of contemporary Number Theory, Algebraic Geometry, or Theory of Dynamical Systems with applications to measure preserving groups of transformations on the appropriate topological spaces. The main idea of this work is to develop a recursive algorithm toward building the sequence of consecutive Goldbach sets that represent solutions to the system of Goldbach equations in the intervals.
Validity of the algorithm is based on the proved in the article recurrent formula. Authors: Babacar Gueye Comments: 4 Pages. This article talk about Brocard's problem. This problem consist to find the solution of the diophancian equation n! Paul Erdos conjecture that the only solutions are the Brown numbers corresponding to the values 4, 5 and 7 of n.
Using here on fondamental result of number theory and studing some fonction between whitch the neperian logarithm we proof that there are any solution corresponding to a value of n greater than Authors: Kouji Takaki Comments: 3 Pages. In this paper we show under some special conditions that the natural density of Ulam numbers is zero.
Authors: Gorou Kaku Comments: 43 Pages. Authors: Simon Plouffe Comments: 6 Pages. An algorithm is presented allowing the compression of primes up to Authors: Shekhar Suman Comments: 7 Pages. Please mail me your feedback [Corrections are made by viXra Admin to comply with the rules of viXra. In this manuscript, we define a conformal map from the unit disc onto the semi plane. Authors: Isaac Mor Comments: 6 Pages.
I developed a very special summation function by using the floor function, which provides a characterization of twin primes and other related prime generalization. Dit is onderdeel van het Hilbert Book Model project. Ruimte kan worden bedekt met puntachtige objecten. Ruimte overdekt met een telbare verzameling puntachtige objecten gedraagt zich anders dan de ruimte die wordt bedekt door een ontelbare set puntachtige objecten.
This is part of the Hilbert Book Model Project. Space can be covered with point-like objects. Space covered by a countable set of point-like objects behaves differently from space that is covered by an uncountable set of point-like objects. On the divergent series De seriebus divergentibus , Euler treated of several divergent series. Hence m - n equals 2 times an integer, and so by definition of even, m - n is even. Directions for Writing Proofs of Universal Statements Writing proofs is similar to writing a computer program based on a set of specifications: organize your thoughts, declare your variables, document thoroughly [italicized brackets here] , and follow a logical progression.
Some general directions for proof writing:. Suppose m and n are odd numbers. Suppose p is a prime number. If p is prime [incorrect use of "if", the primeness of p is not in doubt as we have supposed it] , then p cannot be written as a product of two smaller positive integers.
Getting Proofs Started top Understanding the ideas of generalizing from the generic particular and the method of direct proof, allows one to write the beginnings of a proof even for a theorem not well understood. The beginning of a proof should be clearly marked and contain:. The names of the variables and state the kinds of objects they are. Supposition of the hypothesis of the if-then statement. What is to be shown. Example: Write the beginning of a proof for the statement " " G , if G is a complete, bipartite graph then G is connected.
To show: G is connected. Our final solution for the beginning of a proof is then: Proof: Suppose G is a particular but arbitrarily chosen complete, bipartite graph.
Showing that this statement is false is equivalent to showing that its negation is true. So by finding a value of x in D for which P x is true and Q x is false, we prove the negation of the original statement. The value of x for which the negation of a statement is true is called a counterexample.
To disprove a universally quantified conditional statement, one need only find a single counterexample. Proof by Contradiction top This technique is based on assuming the existence of elements in the domain that satisfy the hypothesis and not the conclusion, which then leads logically to a contradiction.
Sometimes the negation of a statement is easier to disprove leads to a contradiction than the original statement is to prove. In other words, show that the square root of 2 is irrational. This statement is a good candidate for proof by contradiction since we could not check all possible rational numbers to demonstrate that none has a square root of 2. So j 2 is even. This implies the j is even. Thus k 2 is even, and so k is even. We now have that both j and k are even and therefore have a common factor of 2.
This is a contradiction to our assumption. Hence the assumption is false and we have proven that there is no rational number whose square is 2, i.
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