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).

1. What is number theory?

2020 ◽  
pp. 1-13
Author(s):  
Robin Wilson
Keyword(s):  
Number Theory ◽  
Historical Context ◽  
Building Blocks ◽  
Prime Numbers ◽  
Sum Of Squares ◽  
Whole Numbers ◽  
Perfect Numbers

‘What is number theory?’ puts number theory in its historical context, from the Pythagoreans to the present, explaining integers (whole numbers), prime numbers (the building blocks of number theory) squares and cubes, and perfect numbers (numbers whose factors add up to the number itself). How long can gaps between prime numbers be? Is there a formula for producing perfect numbers? Which primes can be expressed as a sum of squares? Other questions arise when we start adding primes.

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.

9. Aftermath

2020 ◽  
pp. 142-150
Author(s):  
Robin Wilson
Keyword(s):  
Credit Cards ◽  
Prime Numbers ◽  
Exciting Field ◽  
Perfect Numbers ◽  
Modern Mathematics

The Aftermath returns to some of the problems posed early in the book. The questions solved include: In which years does February have four Sundays? How many shuffles are needed to restore the order of the cards in a pack with two Jokers? How do prime numbers keep our credit cards secure? Are there infinitely many primes with final digit 9? Integers, squares and cubes, prime numbers and perfect numbers are revisited, with instructions for solving some of the ancient puzzles that inspired generations of mathematicians, and concluding with some of the unsolved problems in this exciting field of modern mathematics.

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.

Digital Sums of Perfect Numbers and Triangular Numbers

1969 ◽  
Vol 62 (3) ◽  
pp. 179-182
Author(s):  
Robert W. Prielipp
Keyword(s):  
Number Theory ◽  
Elementary Level ◽  
Present Evidence ◽  
Triangular Numbers ◽  
Perfect Numbers

BEFORE a theorem can be proved it must first be discovered. One of the areas of mathematics in which some very intriguing results can be uncovered, even at a relatively elementary level, is that of number theory. We proceed to present evidence to substantiate the preceding remark.

Even Perfect Numbers—An Update

1981 ◽  
Vol 74 (6) ◽  
pp. 460-463
Author(s):  
Stanley J. Bezuszka
Keyword(s):  
Number Theory ◽  
Problem Solving ◽  
History Of Mathematics ◽  
Independent Study ◽  
History Of ◽  
Perfect Numbers ◽  
Excellent Resource

Do you have students who are computer buffs, always looking for a new problem to program efficiently? Do you have students who do independent study projects? If so, motivate them with this topic that is rich in the history of mathematics and number theory—perfect numbers. They provide an excellent resource for theoretical as well as computerized problem solving.

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.

scholarly journals THE QUOTIENT SET OF -GENERALISED FIBONACCI NUMBERS IS DENSE IN

2017 ◽  
Vol 96 (1) ◽  
pp. 24-29 ◽  
Author(s):  
CARLO SANNA
Keyword(s):  
Number Theory ◽  
Prime Number ◽  
Algebraic Number ◽  
Fibonacci Numbers ◽  
Prime Numbers ◽  
Algebraic Number Theory ◽  
Initial Values ◽  
Quotient Set ◽  
Successive Term

The quotient set of $A\subseteq \mathbb{N}$ is defined as $R(A):=\{a/b:a,b\in A,b\neq 0\}$. Using algebraic number theory in $\mathbb{Q}(\sqrt{5})$, Garcia and Luca [‘Quotients of Fibonacci numbers’, Amer. Math. Monthly, to appear] proved that the quotient set of Fibonacci numbers is dense in the $p$-adic numbers $\mathbb{Q}_{p}$ for all prime numbers $p$. For any integer $k\geq 2$, let $(F_{n}^{(k)})_{n\geq -(k-2)}$ be the sequence of $k$-generalised Fibonacci numbers, defined by the initial values $0,0,\ldots ,0,1$ ($k$ terms) and such that each successive term is the sum of the $k$ preceding terms. We use $p$-adic analysis to generalise the result of Garcia and Luca, by proving that the quotient set of $k$-generalised Fibonacci numbers is dense in $\mathbb{Q}_{p}$ for any integer $k\geq 2$ and any prime number $p$.

scholarly journals Some interesting tasks from the classical number theory

2020 ◽  
pp. 150-188
Author(s):  
Zurab Agdgomelashvili ◽  
Keyword(s):  
Number Theory ◽  
Prime Number ◽  
Prime Numbers ◽  
Natural Numbers ◽  
Prime Divisors ◽  
Problem Class

The article considers the following issues: – It’s of great interest for p and q primes to determine the number of those prime number divisors of a number 1 1 pq A p    that are less than p. With this purpose we have considered: Theorem 1. Let’s p and q are odd prime numbers and p  2q 1. Then from various individual divisors of the 1 1 pq A p    number, only one of them is less than p. A has at least two different simple divisors; Theorem 2. Let’s p and q are odd prime numbers and p  2q 1. Then all prime divisors of the number 1 1 pq A p    are greater than p; Theorem 3. Let’s q is an odd prime number, and p N \{1}, p]1;q] [q  2; 2q] , then each of the different prime divisors of the number 1 1 pq A p    taken separately is greater than p; Theorem 4. Let’s q is an odd prime number, and p{q1; 2q1}, then from different prime divisors of the number 1 1 pq A p    taken separately, only one of them is less than p. A has at least two different simple divisors. Task 1. Solve the equation 1 2 1 z x y y    in the natural numbers x , y, z. In addition, y must be a prime number. Task 2. Solve the equation 1 3 1 z x y y    in the natural numbers x , y, z. In addition, y must be a prime number. Task 3. Solve the equation 1 1 z x y p y    where p{6; 7; 11; 13;} are the prime numbers, x, y  N and y is a prime number. There is a lema with which the problem class can be easily solved: Lemma ●. Let’s a, b, nN and (a,b) 1. Let’s prove that if an  0 (mod| ab|) , or bn  0 (mod| ab|) , then | ab|1. Let’s solve the equations ( – ) in natural x , y numbers: I. 2 z x y z z x y          ; VI. (x  y)xy  x y ; II. (x  y)z  (2x)z  yz ; VII. (x  y)xy  yx ; III. (x  y)z  (3x)z  yz ; VIII. (x  y) y  (x  y)x , (x  y) ; IV. ( y  x)x y  x y , (y  x) ; IX. (x  y)x y  xxy ; V. ( y  x)x y  yx , (y  x) ; X. (x  y)xy  (x  y)x , (y  x) . Theorem . If a, bN (a,b) 1, then each of the divisors (a2  ab  b2 ) will be similar. The concept of pseudofibonacci numbers is introduced and some of their properties are found.

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