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Newman's Conjecture
In mathematics, specifically in number theory, Newman's conjecture is a conjecture about the behavior of the Partition function (number theory), partition function modulo any integer. Specifically, it states that for any integers and such that 0\le r\le m-1, the value of the partition function p(n) satisfies the congruence p(n)\equiv r\pmod for infinitely many non-negative integers . It was formulated by mathematician Morris Newman in 1960. It is unsolved as of 2020. History Oddmund Kolberg was probably the first to prove a related result, namely that the Partition function (number theory), partition function takes both even and odd values infinitely often. The proof employed was of elementary nature and easily accessible, and was proposed as an exercise by Newman in the American Mathematical Monthly. 1 year later, in 1960, Newman proposed the conjecture and proved the cases m=5 and 13 in his original paper, and m=65 two years later. Ken Ono, an American mathematician, made f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Bruijn–Newman Constant
The de Bruijn–Newman constant, denoted by Λ and named after Nicolaas Govert de Bruijn and Charles M. Newman, is a mathematical constant defined via the zero of a function, zeros of a certain function (mathematics), function ''H''(''λ'', ''z''), where ''λ'' is a real number, real parameter and ''z'' is a complex number, complex variable. More precisely, :H(\lambda, z):=\int_^ e^ \Phi(u) \cos (z u) d u, where \Phi is the super-exponential function, super-exponentially decaying function :\Phi(u) = \sum_^ (2\pi^2n^4e^ - 3 \pi n^2 e^ ) e^ and Λ is the unique real number with the property that ''H'' has only real zeros if and only if ''λ'' ≥ Λ. The constant is closely connected with Riemann hypothesis, Riemann's hypothesis concerning the zeros of the Riemann zeta function, Riemann zeta-function: since the Riemann hypothesis is equivalent to the claim that all the zeroes of ''H''(0, ''z'') are real, the Riemann hypothesis is equivalent to the conjecture tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of Complex analysis, analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Erdős
Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in discrete mathematics, graph theory, number theory, mathematical analysis, approximation theory, set theory, and probability theory. Much of his work centered around discrete mathematics, cracking many previously unsolved problems in the field. He championed and contributed to Ramsey theory, which studies the conditions in which order necessarily appears. Overall, his work leaned towards solving previously open problems, rather than developing or exploring new areas of mathematics. Erdős published around 1,500 mathematical papers during his lifetime, a figure that remains unsurpassed. He firmly believed mathematics to be a social activity, living an itinerant lifestyle with the sole purpose of writing mathematical papers with other mathem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coprime Integers
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. One says also '' is prime to '' or '' is coprime with ''. The numbers 8 and 9 are coprime, despite the fact that neither considered individually is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of a reduced fraction are coprime, by definition. Notation and testing Standard notations for relatively prime integers and are: and . In their 1989 textbook ''Concrete Mathematics'', Ronald Graham, Donald Knuth, and Oren Patashnik proposed that the notation a\perp b be used to indicate that and are relatively prime and that the term "prime" be used instead of coprime (as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ken Ono
Ken Ono (born March 20, 1968) is a Japanese-American mathematician who specializes in number theory, especially in integer partitions, modular forms, umbral moonshine, the Riemann Hypothesis and the fields of interest to Srinivasa Ramanujan. He is the Marvin Rosenblum Professor of Mathematics at the University of Virginia. Early life and education Ono was born on March 20, 1968 in Philadelphia, Pennsylvania. He is the son of mathematician Takashi Ono, who emigrated from Japan to the United States after World War II. His older brother, immunologist and university president Santa J. Ono, was born while Takashi Ono was in Canada working at the University of British Columbia, but by the time Ken Ono was born the family had returned to the US for a position at the University of Pennsylvania. In the 1980s, Ono attended Towson High School, but he dropped out. He later enrolled at the University of Chicago without a high school diploma. There he raced bicycles, and he was a member of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Monthly
''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content. In this the ''American Mathematical Monthly'' fulfills a different role from that of typical mathematical research journals. The ''American Mathematical Monthly'' is the most widely read mathematics journal in the world according to records on JSTOR. Tables of contents with article abstracts from 1997–2010 are availablonline The MAA gives the Lester R. Ford Awards annually to "authors of articles of expository excellence" published in the ''American Mathematical Monthly''. Editors *2022– ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition Function (number Theory)
In number theory, the partition function represents the number of possible partitions of a non-negative integer . For instance, because the integer 4 has the five partitions , , , , and . No closed-form expression for the partition function is known, but it has both asymptotic expansions that accurately approximate it and recurrence relations by which it can be calculated exactly. It grows as an exponential function of the square root of its argument. The multiplicative inverse of its generating function is the Euler function; by Euler's pentagonal number theorem this function is an alternating sum of pentagonal number powers of its argument. Srinivasa Ramanujan first discovered that the partition function has nontrivial patterns in modular arithmetic, now known as Ramanujan's congruences. For instance, whenever the decimal representation of ends in the digit 4 or 9, the number of partitions of will be divisible by 5. Definition and examples For a positive integer , is the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Important examples Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, ''b'', and ''c'' can satisfy the equation ''a^n + b^n = c^n'' for any integer value of ''n'' greater than two. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of '' Arithmetica'', where he claimed that he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
65 (number)
65 (sixty-five) is the natural number following 64 and preceding 66. In mathematics Sixty-five is the 23rd semiprime and the 3rd of the form (5.q). It is an octagonal number. It is also a Cullen number. Given 65, the Mertens function returns 0. This number is the magic constant of a 5x5 normal magic square: \begin 17 & 24 & 1 & 8 & 15 \\ 23 & 5 & 7 & 14 & 16 \\ 4 & 6 & 13 & 20 & 22 \\ 10 & 12 & 19 & 21 & 3 \\ 11 & 18 & 25 & 2 & 9 \end. This number is also the magic constant of n-Queens Problem for n = 5. 65 is the smallest integer that can be expressed as a sum of two distinct positive squares in two ways, 65 = 82 + 12 = 72 + 42. It appears in the Padovan sequence, preceded by the terms 28, 37, 49 (it is the sum of the first two of these). There are only 65 known Euler's idoneal numbers. 65 is a Stirling number of the second kind, the number of ways of dividing a set of six objects into four non-empty subsets. 65 = 15 + 24 + 33 + 42 + 51. 65 is the length of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
40 (number)
40 (forty) is the natural number following 39 and preceding 41. Though the word is related to "four" (4), the spelling "forty" replaced "fourty" in the course of the 17th century and is now the standard form. In mathematics *Forty is a composite number, a refactorable number, an octagonal number, and—as the sum of the first four pentagonal numbers: 1 + 5 + 12 + 22 =40—it is a pentagonal pyramidal number. Adding up some subsets of its divisors (e.g., 1, 4, 5, 10, and 20) gives 40; hence, 40 is a semiperfect number. *Given 40, the Mertens function returns 0. 40 is the smallest number with exactly nine solutions to the equation Euler's totient function \varphi (x)=n. *Forty is the number of -queens problem solutions for n=7. *Forty is a repdigit in ternary (1111, ''i.e.'', 3^ + 3^ + 3^ + 3^, or, in other words, \frac ) and a Harshad number in decimal. In science *The atomic number of zirconium. *Negative forty is the unique temperature at which the Fahrenheit and Celsius ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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