scholarly journals Conjecture in Additive Twin Primes Numbers Theory

2013 ◽  
Vol 5 ◽  
pp. 27-30
Author(s):  
Ibrahima Gueye
Keyword(s):  
Prime Numbers ◽  
Additive Theory ◽  
Twin Primes ◽  
Theory Of Numbers

For two millennia, the prime numbers have continued to fascinate mathematicians. Indeed, a conjecture which dates back to this period states that the number of twin primes is infinite. In 1949 Clement showed a theorem on twin primes. For the record, the theorem of Clement has quickly been known to be ineffective in the development of twin primes because of the factorial. This is why I thought ofusing the additive theory of numbers to find pairs of twin primes from the first two pairs of twin primes. What I have formulated as a conjecture. In same time i presentmy idea about the solution of the Goldbach’s weak conjecture.

Magic squares of squares over a finite field

2021 ◽  
pp. 111-122
Author(s):  
Stewart Hengeveld ◽  
Giancarlo Labruna ◽  
Aihua Li
Keyword(s):  
Finite Field ◽  
Prime Numbers ◽  
Similar Question ◽  
Magic Squares ◽  
Content Type ◽  
Magic Square ◽  
Twin Primes ◽  
Quadratic Residues ◽  
Perfect Squares ◽  
Open Question

A magic square M M over an integral domain D D is a 3 × 3 3\times 3 matrix with entries from D D such that the elements from each row, column, and diagonal add to the same sum. If all the entries in M M are perfect squares in D D , we call M M a magic square of squares over D D . In 1984, Martin LaBar raised an open question: “Is there a magic square of squares over the ring Z \mathbb {Z} of the integers which has all the nine entries distinct?” We approach to answering a similar question when D D is a finite field. We claim that for any odd prime p p , a magic square over Z p \mathbb Z_p can only hold an odd number of distinct entries. Corresponding to LaBar’s question, we show that there are infinitely many prime numbers p p such that, over Z p \mathbb Z_p , magic squares of squares with nine distinct elements exist. In addition, if p ≡ 1 ( mod 120 ) p\equiv 1\pmod {120} , there exist magic squares of squares over Z p \mathbb Z_p that have exactly 3, 5, 7, or 9 distinct entries respectively. We construct magic squares of squares using triples of consecutive quadratic residues derived from twin primes.

On a problem in the additive theory of numbers

1932 ◽  
Vol 34 (1) ◽  
pp. 637-644 ◽  
Author(s):  
C. J. A. Evelyn ◽  
E. H. Linfoot
Keyword(s):  
Additive Theory ◽  
Theory Of Numbers

Note on a Result in the Additive Theory of Numbers

1938 ◽  
Vol s2-43 (1) ◽  
pp. 142-151 ◽  
Author(s):  
H. Davenport ◽  
H. Heilbronn
Keyword(s):  
Additive Theory ◽  
Theory Of Numbers

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 Brocard conjecture

2021 ◽  
Author(s):  
Jan Feliksiak
Keyword(s):  
Lower Bound ◽  
Great Degree ◽  
Common Ground ◽  
Prime Numbers ◽  
Upper Bounds ◽  
Lower Upper ◽  
Twin Primes ◽  
True False

The Brocard conjecture asserts that the number of primes, within the interval, between the squares of two subsequent primes is greater than or equal to 4. Although the number of primes within this interval varies to a great degree, there is a common ground, which makes it possible to settle this old conundrum. Three bounds are developed: the least lower bound and the lower/upper bounds. The least lower bound is implemented to prove the conjecture. The lower/upper bounds exploit the shortest such interval, namely between the twin primes. This has been done in order to establish the bounds, on the smallest number of primes within that interval. The research objective was not only to provide a true/false answer, but to clarify some aspects of the distribution of prime numbers within this interval as well.

ON A PROBLEM IN THE ADDITIVE THEORY OF NUMBERS (VI)

1933 ◽  
Vol os-4 (1) ◽  
pp. 309-314 ◽  
Author(s):  
C. J. A. EVELYN ◽  
E. H. LINFOOT
Keyword(s):  
Additive Theory ◽  
Theory Of Numbers

On Waring's problem for cubes

1991 ◽  
Vol 109 (2) ◽  
pp. 229-256 ◽  
Author(s):  
Jörg Brüdern
Keyword(s):  
Singular Series ◽  
Natural Numbers ◽  
Additive Theory ◽  
Waring’S Problem ◽  
Waring's Problem ◽  
Quantitative Form ◽  
Image Position ◽  
Theory Of Numbers

A classical conjecture in the additive theory of numbers is that all sufficiently large natural numbers may be written as the sum of four positive cubes of integers. This is known as the Four Cubes Problem, and since the pioneering work of Hardy and Littlewood one expects a much more precise quantitative form of the conjecture to hold. Let v(n) be the number of representations of n in the proposed manner. Then the expected formula takes the shapewhere (n) is the singular series associated with four cubes as familiar in the Hardy–Littlewood theory.

A problem in additive number theory

1988 ◽  
Vol 103 (1) ◽  
pp. 27-33 ◽  
Author(s):  
Jörg Brüdern
Keyword(s):  
Number Theory ◽  
Additive Number Theory ◽  
Interesting Problem ◽  
Natural Numbers ◽  
Additive Theory ◽  
Additive Number ◽  
Image Position ◽  
Theory Of Numbers

The determination of the minimal s such that all large natural numbers n admit a representation asis an interesting problem in the additive theory of numbers and has a considerable literature, For historical comments the reader is referred to the author's paper [2] where the best currently known result is proved. The purpose here is a further improvement.

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