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Lambek–Moser Theorem
The Lambek–Moser theorem is a mathematical description of partitions of the natural numbers into two Complement (set theory), complementary sets. For instance, it applies to the partition of numbers into even number, even and odd number, odd, or into prime number, prime and non-prime (one and the composite numbers). There are two parts to the Lambek–Moser theorem. One part states that any two monotonic function, non-decreasing integer functions that are inverse, in a certain sense, can be used to split the natural numbers into two complementary subsets, and the other part states that every complementary partition can be constructed in this way. When a formula is known for the natural number in a set, the Lambek–Moser theorem can be used to obtain a formula for the number not in the set. The Lambek–Moser theorem belongs to combinatorial number theory. It is named for Joachim Lambek and Leo Moser, who published it in 1954, and should be distinguished from an unrelated theor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive integers Some authors acknowledge both definitions whenever convenient. Sometimes, the whole numbers are the natural numbers as well as zero. In other cases, the ''whole numbers'' refer to all of the integers, including negative integers. The counting numbers are another term for the natural numbers, particularly in primary education, and are ambiguous as well although typically start at 1. The natural numbers are used for counting things, like "there are ''six'' coins on the table", in which case they are called ''cardinal numbers''. They are also used to put things in order, like "this is the ''third'' largest city in the country", which are called ''ordinal numbers''. Natural numbers are also used as labels, like Number (sports), jersey ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square (algebra)
In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2, and is denoted by a superscript 2; for instance, the square of 3 may be written as 32, which is the number 9. In some cases when superscripts are not available, as for instance in programming languages or plain text files, the notations ''x''^2 ( caret) or ''x''**2 may be used in place of ''x''2. The adjective which corresponds to squaring is '' quadratic''. The square of an integer may also be called a '' square number'' or a ''perfect square''. In algebra, the operation of squaring is often generalized to polynomials, other expressions, or values in systems of mathematical values other than the numbers. For instance, the square of the linear polynomial is the quadratic polynomial . One of the important properties of squaring, for numbers as well as in many other mathematical systems, is that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof Without Words
In mathematics, a proof without words (or visual proof) is an illustration of an identity (mathematics), identity or mathematical statement which can be demonstrated as self-evident by a diagram without any accompanying explanatory text. Such proofs can be considered more Mathematical beauty, elegant than formal or rigour, mathematically rigorous proofs due to their self-evident nature. When the diagram demonstrates a particular case of a general statement, to be a proof, it must be generalisable. A proof without words is not the same as a mathematical proof, because it omits the details of the logical argument it illustrates. However, it can provide valuable intuitions to the viewer that can help them formulate or better understand a true proof. Examples Sum of odd numbers The statement that the sum of all positive parity (mathematics), odd numbers up to 2''n'' − 1 is a square number, perfect square—more specifically, the perfect square ''n''2—can be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Samuel Beatty (mathematician)
Samuel Beatty (1881–1970) was dean of the Faculty of Mathematics at the University of Toronto, taking the position in 1934. Early life Beatty was born in 1881. In 1915, he graduated from the University of Toronto with a PhD and a dissertation entitled ''Extensions of Results Concerning the Derivatives of an Algebraic Function of a Complex Variable'', with the help of his adviser, John Charles Fields. He was the first person to receive a PhD in mathematics from a Canadian university. In 1925 he was elected a Fellow of the Royal Society of Canada. In 1926, he published a problem in the ''American Mathematical Monthly'', which formed the genesis for the Beatty sequence. University of Toronto Beatty was dean of the Faculty of Mathematics at the University of Toronto, taking the position in 1934. The famous mathematician Richard Brauer was recruited by Beatty in 1935. He invited Harold Scott MacDonald Coxeter to the University of Toronto with a position as an assistant professor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime-counting Function
In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number . It is denoted by (unrelated to the number ). A symmetric variant seen sometimes is , which is equal to if is exactly a prime number, and equal to otherwise. That is, the number of prime numbers less than , plus half if equals a prime. Growth rate Of great interest in number theory is the growth rate of the prime-counting function. It was conjectured in the end of the 18th century by Gauss and by Legendre to be approximately \frac where is the natural logarithm, in the sense that \lim_ \frac=1. This statement is the prime number theorem. An equivalent statement is \lim_\frac=1 where is the logarithmic integral function. The prime number theorem was first proved in 1896 by Jacques Hadamard and by Charles de la Vallée Poussin independently, using properties of the Riemann zeta function introduced by Riemann in 1859. Proof ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Viggo Brun
Viggo Brun (13 October 1885 – 15 August 1978) was a Norwegian professor, mathematician and number theorist. Contributions In 1915, he introduced a new method, based on Legendre's version of the sieve of Eratosthenes, now known as the '' Brun sieve'', which addresses additive problems such as Goldbach's conjecture and the twin prime conjecture. He used it to prove that there exist infinitely many integers ''n'' such that ''n'' and ''n''+2 have at most nine prime factors, and that all large even integers are the sum of two numbers with at most nine prime factors. He also showed that the sum of the reciprocals of twin primes converges to a finite value, now called Brun's constant: by contrast, the sum of the reciprocals of all primes is divergent. He developed a multi-dimensional continued fraction algorithm in 1919–1920 and applied this to problems in musical theory. He also served as praeses of the Royal Norwegian Society of Sciences and Letters in 1946. Biography ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thue–Morse Sequence
In mathematics, the Thue–Morse or Prouhet–Thue–Morse sequence is the binary sequence (an infinite sequence of 0s and 1s) that can be obtained by starting with 0 and successively appending the Boolean complement of the sequence obtained thus far. It is sometimes called the fair share sequence because of its applications to fair division or parity sequence. The first few steps of this procedure yield the strings 0, 01, 0110, 01101001, 0110100110010110, and so on, which are the prefixes of the Thue–Morse sequence. The full sequence begins: :01101001100101101001011001101001.... The sequence is named after Axel Thue, Marston Morse and (in its extended form) Eugène Prouhet. Definition There are several equivalent ways of defining the Thue–Morse sequence. Direct definition To compute the ''n''th element ''tn'', write the number ''n'' in binary. If the number of ones in this binary expansion is odd then ''tn'' = 1, if even then ''tn'' = 0. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Representation
A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method for representing numbers that uses only two symbols for the natural numbers: typically "0" (zero) and "1" (one). A ''binary number'' may also refer to a rational number that has a finite representation in the binary numeral system, that is, the quotient of an integer by a power of two. The base-2 numeral system is a positional notation with a radix of 2. Each digit is referred to as a bit, or binary digit. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by almost all modern computers and computer-based devices, as a preferred system of use, over various other human techniques of communication, because of the simplicity of the language and the noise immunity in physical implementation. History The modern binary number system was studied in Europe in the 16th and 17th centuries by Thomas Harriot, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Odious Number
In number theory, an odious number is a positive integer that has an odd Hamming weight, number of 1s in its Binary number, binary expansion. Nonnegative integers that are not odious are called evil numbers. In computer science, an odious number is said to have Parity bit#Parity, odd parity. Examples The first odious numbers are: Properties If a(n) denotes the nth odious number (with a(0)=1), then for all n, a(a(n))=2a(n). Every positive integer n has an odious multiple that is at most n(n+4). The numbers for which this bound is tight are exactly the Mersenne numbers with even exponents, the numbers of the form n = 2^-1, such as 3, 15, 63, etc. For these numbers, the smallest odious multiple is exactly n(n+4) = 2^+2^-3. Related sequences The odious numbers give the positions of the nonzero values in the Thue–Morse sequence. Every power of two is odious, because its binary expansion has only one nonzero bit. Except for 3, every Mersenne prime is odious, because its binary exp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Evil Number
In number theory, an evil number is a non-negative integer that has an even number of 1s in its binary expansion. These numbers give the positions of the zero values in the Thue–Morse sequence, and for this reason they have also been called the Thue–Morse set. Non-negative integers that are not evil are called odious numbers. Examples The first evil numbers are: :0, 3, 5, 6, 9, 10, 12, 15, 17, 18, 20, 23, 24, 27, 29, 30, 33, 34, 36, 39 ... Equal sums The partition of the non-negative integers into the odious and evil numbers is the unique partition of these numbers into two sets that have equal multisets of pairwise sums. As 19th-century mathematician Eugène Prouhet showed, the partition into evil and odious numbers of the numbers from 0 to 2^k-1, for any k, provides a solution to the Prouhet–Tarry–Escott problem of finding sets of numbers whose sums of powers are equal up to the kth power. In computer science In computer science, an evil number is said to have even ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parity (mathematics)
In mathematics, parity is the Property (mathematics), property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not.. For example, −4, 0, and 82 are even numbers, while −3, 5, 23, and 69 are odd numbers. The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers with decimals or fractions like 1/2 or 4.6978. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the parity of zero is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwise it is even—as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
