scholarly journals Considerations for Goldbach's Strong Conjecture

2021 ◽  
Author(s):  
Carleilton Severino Silva
Keyword(s):  
Number Theory ◽  
Prime Numbers ◽  
Weak Version ◽  
Strong Version ◽  
Leonhard Euler

Since 1742, the year in which the Prussian Christian Goldbach wrote a letter to Leonhard Euler with his Conjecture in the weak version, mathematicians have been working on the problem. The tools in number theory become the most sophisticated thanks to the resolution solutions. Euler himself said he was unable to prove it. The weak guess in the modern version states the following: any odd number greater than 5 can be written as the sum of 3 primes. In response to Goldbach's letter, Euler reminded him of a conversation in which he proposed what is now known as Goldbach's strong conjecture: any even number greater than 2 can be written as a sum of 2 prime numbers. The most interesting result came in 2013, with proof of weak version by the Peruvian Mathematician Harald Helfgott, however the strong version remained without a definitive proof. The weak version can be demonstrated without major difficulties and will not be described in this article, as it becomes a corollary of the strong version. Despite the enormous intellectual baggage that great mathematicians have had over the centuries, the Conjecture in question has not been validated or refuted until today.

scholarly journals Considerations for Goldbach's Strong Conjecture

2021 ◽  
Author(s):  
Carleilton Severino Silva
Keyword(s):  
Number Theory ◽  
Prime Numbers ◽  
Weak Version ◽  
Strong Version ◽  
Leonhard Euler

Since 1742, the year in which the Prussian Christian Goldbach wrote a letter to Leonhard Euler with his Conjecture in the weak version, mathematicians have been working on the problem. The tools in number theory become the most sophisticated thanks to the resolution solutions. Euler himself said he was unable to prove it. The weak guess in the modern version states the following: any odd number greater than 5 can be written as the sum of 3 primes. In response to Goldbach's letter, Euler reminded him of a conversation in which he proposed what is now known as Goldbach's strong conjecture: any even number greater than 2 can be written as a sum of 2 prime numbers. The most interesting result came in 2013, with proof of weak version by the Peruvian Mathematician Harald Helfgott, however the strong version remained without a definitive proof. The weak version can be demonstrated without major difficulties and will not be described in this article, as it becomes a corollary of the strong version. Despite the enormous intellectual baggage that great mathematicians have had over the centuries, the Conjecture in question has not been validated or refuted until today.

3. Prime-time mathematics

2020 ◽  
pp. 37-57
Author(s):  
Robin Wilson
Keyword(s):  
Number Theory ◽  
Prime Number ◽  
Prime Numbers ◽  
Prime Time ◽  
Leonhard Euler ◽  
Mersenne Primes ◽  
Fermat Primes

‘Prime-time mathematics’ explores prime numbers, which lie at the heart of number theory. Some primes cluster together and some are widely spread, while primes go on forever. The Sieve of Eratosthenes (3rd century BC) is an ancient method for identifying primes by iteratively marking the multiples of each prime as not prime. Every integer greater than 1 is either a prime number or can be written as a product of primes. Mersenne primes, named after French friar Marin de Mersenne, are prime numbers that are one less than a power of 2. Pierre de Fermat and Leonhard Euler were also prime number enthusiasts. The five Fermat primes are used in a problem from geometry.

Euler's contribution to number theory

2007 ◽  
Vol 91 (522) ◽  
pp. 453-461 ◽  
Author(s):  
Peter Shiu
Keyword(s):  
Eighteenth Century ◽  
Number Theory ◽  
Scientific Output ◽  
Scholarly Work ◽  
Long Period ◽  
Peter The Great ◽  
Complete Work ◽  
Leonhard Euler ◽  
The Subject ◽  
Historic Background

Individuals who excel in mathematics have always enjoyed a well deserved high reputation. Nevertheless, a few hundred years back, as an honourable occupation with means to social advancement, such an individual would need a patron in order to sustain the creative activities over a long period. Leonhard Euler (1707-1783) had the fortune of being supported successively by Peter the Great (1672-1725), Frederich the Great (1712-1786) and the Great Empress Catherine (1729-1791), enabling him to become the leading mathematician who dominated much of the eighteenth century. In this note celebrating his tercentenary, I shall mention his work in number theory which extended over some fifty years. Although it makes up only a small part of his immense scientific output (it occupies only four volumes out of more than seventy of his complete work) it is mostly through his research in number theory that he will be remembered as a mathematician, and it is clear that arithmetic gave him the most satisfaction and also much frustration. Gazette readers will be familiar with many of his results which are very well explained in H. Davenport's famous text [1], and those who want to know more about the historic background, together with the rest of the subject matter itself, should consult A. Weil's definitive scholarly work [2], on which much of what I write is based. Some of the topics being mentioned here are also set out in Euler's own Introductio in analysin infinitorum (1748), which has now been translated into English [3].

scholarly journals Is preregistration worthwhile?

2019 ◽  
Author(s):  
Aba Szollosi ◽  
David Kellen ◽  
Danielle Navarro ◽  
Rich Shiffrin ◽  
Iris van Rooij ◽  
...  
Keyword(s):  
Statistical Models ◽  
Statistical Tests ◽  
Error Rates ◽  
Statistical Techniques ◽  
Weak Version ◽  
Strong Version ◽  
Statistical Problems

Proponents of preregistration argue that, among other benefits, it improves the diagnosticity of statistical tests [1]. In the strong version of this argument, preregistration does this by solving statistical problems, such as family-wise error rates. In the weak version, it nudges people to think more deeply about their theories, methods, and analyses. We argue against both: the diagnosticity of statistical tests depend entirely on how well statistical models map onto underlying theories, and so improving statistical techniques does little to improve theories when the mapping is weak. There is also little reason to expect that preregistration will spontaneously help researchers to develop better theories (and, hence, better methods and analyses).

scholarly journals Skeptical Theism and the Threshold Problem

1970 ◽  
Vol 18 (1) ◽  
pp. 73-92
Author(s):  
Yishai A. Cohen
Keyword(s):  
Weak Version ◽  
Strong Version ◽  
Skeptical Theism ◽  
Cognitive Faculties ◽  
Threshold Problem

In this paper I articulate and defend a new anti-theodicy challenge to Skeptical Theism. More specifically, I defend the Threshold Problem according to which there is a threshold to the kinds of evils that are in principle justifiable for God to permit, and certain instances of evil are beyond that threshold. I further argue that Skeptical Theism does not have the resources to adequately rebut the Threshold Problem. I argue for this claim by drawing a distinction between a weak and strong version of Skeptical Theism, such that the strong version must be defended in order to rebut the Threshold Problem. However, the skeptical theist’s appeal to our limited cognitive faculties only supports the weak version.

Number Theory: A Very Short Introduction

2020 ◽  
Author(s):  
Robin Wilson
Keyword(s):  
Number Theory ◽  
Short Introduction ◽  
New Developments ◽  
Leonhard Euler

Number Theory: A Very Short Introduction explains the branch of mathematics primarily concerned with the counting numbers, 1, 2, 3, …. Long considered one of the most ‘beautiful’ areas of mathematics, number theory dates back over two millennia to the Ancient Greeks, but has seen some startling new developments in recent years. Trailblazers in the field include mathematicians Euclid of Alexandria, Pierre de Fermat, Leonhard Euler, and Carl Friedrich Gauss. Number theory has intrigued and attracted amateurs and professionals alike for thousands of years, appearing in both recreational contexts (puzzles) and practical concerns (cryptography). Some problems in number theory are easy, whereas others are notorious with no known solutions to date.

scholarly journals On exponential sums over prime numbers

1989 ◽  
Vol 46 (3) ◽  
pp. 423-437
Author(s):  
A. Sárközy ◽  
C. L. Stewart
Keyword(s):  
Number Theory ◽  
Exponential Sums ◽  
Prime Numbers ◽  
Consecutive Integers

AbstractIn this article we establish an estimate for a sum over primes that is the analogue of an estimate for a sum over consecutive integers which has proved to be very useful in applications of exponential sums to problems in number theory.

Index, sub-index and sub-factor of groups with interactions to number theory

2019 ◽  
Vol 19 (06) ◽  
pp. 2050101
Author(s):  
M. H. Hooshmand
Keyword(s):  
Number Theory ◽  
Direct Product ◽  
Prime Numbers ◽  
Additive Number Theory ◽  
Open Problems ◽  
Famous Conjecture ◽  
Future Directions ◽  
Left And Right ◽  
The Difference ◽  
Equivalent Conditions

This paper is the first step of a new topic about groups which has close relations and applications to number theory. Considering the factorization of a group into a direct product of two subsets, and since every subgroup is a left and right factor, we observed that the index conception can be generalized for a class of factors. But, thereafter, we found that every subset [Formula: see text] of a group [Formula: see text] has four related sub-indexes: right, left, upper and lower sub-indexes [Formula: see text], [Formula: see text] which agree with the conception index of subgroups, and all of them are equal if [Formula: see text] is a subgroup or normal sub-semigroup of [Formula: see text]. As a result of the topic, we introduce some equivalent conditions to a famous conjecture for prime numbers (“every even number is the difference of two primes”) that one of them is: the prime numbers set is index stable (i.e. all of its sub-indexes are equal) in integers and [Formula: see text]. Index stable groups (i.e. those whose subsets are all index stable) are a challenging subject of the topic with several results and ideas. Regarding the extension of the theory, we give some methods for evaluation of sub-indexes, by using the left and right differences of subsets. At last, we pose many open problems, questions, a proposal for additive number theory, and show some future directions of researches and projects for the theory.

The Legacy of Marin Mersenne: The Search for Primal Order and the Mentoring of Young Minds

2005 ◽  
Vol 98 (8) ◽  
pp. 525-529
Author(s):  
Jeffrey J. Wanko
Keyword(s):  
Number Theory ◽  
Prime Numbers ◽  
Regular Polygons ◽  
Perfect Numbers

The search for prime numbers has long held a great fascination for mathematicians and for mathematics enthusiasts. Whether as a mathematical recreation or as a serious study within number theory, this quest has resulted in some profound mathematical advances and in a few surprising results that held some unforeseen applications and connections to other areas. For example, the problems of finding perfect numbers and constructible regular polygons have both been simplified through the search for prime numbers (Bell 1937, Clawson 1996).

Number theory and other reminiscences of Viscount Cherwell

1988 ◽  
Vol 42 (2) ◽  
pp. 197-204 ◽  
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Fundamental Theorem ◽  
Prime Numbers ◽  
Christ Church ◽  
Active Interest ◽  
Fundamental Theorem Of Arithmetic ◽  
Elegant Proof ◽  
Theory Of Numbers

Lord Cherwell (i) was, of course, a very distinguished ex-perimental physicist but he had (like many others) a considerable active interest in the theory of numbers. I met him in 1930 when Christ Church, Oxford, elected me to a Senior (postgraduate) Scholarship and I migrated there from my original college. Cherwell’s first published work (2) in the theory of numbers was a very simple and elegant proof of the fundamental theorem of arithmetic, that any positive integer can be expressed as a product of prime numbers in just one way (apart from a possible rearrangement of the order of the factors). (A prime is a positive integer greater than 1 whose only factors are 1 and itself.) His proof is by the method of descent (used by Fermat, but not for this problem). Assume the fundamental theorem false and call any number that can be expressed as a product of primes in two or more ways abnormal.

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