134 articles
9 (number)
9 (nine) is the natural number following 8 and preceding 10: the largest single digit, the square of 3, and the smallest odd composite number, basis of casting out nines.
90 (number)
90 (ninety), the natural number following 89 and preceding 91, is an abundant pronic number with twelve divisors, best known as the degree measure of a right angle.
Abundant number
An abundant number, or excessive number, is a positive integer whose proper divisors add up to more than the number itself, the smallest being 12.
Almost integer
An almost integer is a real number very close to, but not equal to, an integer. The flagship example, e^(π√163), differs from 262537412640768744 by less than 10^-12.
Amicable numbers
Amicable numbers, or amicable pairs, are two numbers each equal to the sum of the other's proper divisors; the smallest pair, 220 and 284, was known to Thābit ibn Qurra in the ninth century.
Bell number
The Bell number B(n) is the number of ways to partition a set of n labeled elements into non-empty subsets; the sequence begins 1, 1, 2, 5, 15.
Bell numbers
The Bell numbers, named for mathematician Eric Temple Bell, count the partitions of a set of n labeled elements, beginning 1, 1, 2, 5, 15, 52.
Bernoulli number
The Bernoulli numbers are a sequence of rational numbers introduced by Jacob Bernoulli to sum powers of integers, appearing in the Euler–Maclaurin formula and the Riemann zeta function.
Billion
Billion, also called one billion, is the number 1,000,000,000 (a thousand million) in American and British English, though older European usage gives it the value 10^12.
Bose–Einstein statistics
Bose–Einstein statistics, also called the Bose–Einstein distribution, describes identical particles with integer spin (bosons), any number of which may occupy the same quantum state; Bose introduced it in 1924.
Catalan number
The Catalan numbers, also called the Catalan sequence, form an integer sequence beginning 1, 1, 2, 5, 14, 42 that counts balanced parentheses and triangulations.
Crank of a partition
The crank of a partition, also called Dyson's crank, is a partition statistic defined by Andrews and Garvan in 1988 that gives a combinatorial explanation of Ramanujan's three partition congruences.
Cube (algebra)
The cube of a number, or cubic number, is the number multiplied by itself twice, written a3; cubing gives the volume of a cube with edge a.
Cyclic number
A cyclic number is an integer whose multiples are cyclic permutations of its own digits; 142857, the repeating decimal block of 1/7, is the classic example.
Digital root
The digital root, also called the repeated digital sum, is the single digit left after repeatedly summing an integer's digits; it equals the remainder modulo 9.
Double factorial
The double factorial of a nonnegative integer n, written n!!, is the product of all positive integers up to n with the same parity as n.
Fermat number
A Fermat number is an integer of the form 2^(2^n) + 1, named after Pierre de Fermat. Only the first five, 3 through 65537, are known to be prime.
Fibonacci sequence
The Fibonacci sequence, also known as the Fibonacci series, is the integer sequence 0, 1, 1, 2, 3, 5, 8, 13, in which each term is the sum of the two before it.
Gelfond's constant
Gelfond's constant is the transcendental number e^π, approximately 23.1407, proven transcendental in 1934 by Aleksandr Gelfond and independently by Theodor Schneider in 1935, settling Hilbert's seventh problem.
Generalizations of Fibonacci numbers
Generalizations of Fibonacci numbers, including the k-bonacci or k-step Fibonacci numbers, are integer sequences that extend the Fibonacci recurrence by summing more preceding terms, giving tribonacci and tetranacci numbers.
Glaisher's theorem
Glaisher's theorem is an 1883 result stating that partitions into parts not divisible by d equal partitions with no part repeated d or more times.
Googol
A googol, ten duotrigintillion, is the number 10 to the 100th power, named in 1938 by nine-year-old Milton Sirotta, whose uncle Edward Kasner popularized it.
Graham's number
Graham's number is an enormous integer, built from Knuth's up-arrow notation, that arose as a Ramsey theory upper bound and was popularized by Martin Gardner in 1977.
Happy number
A happy number is a positive integer that reaches 1 when repeatedly replaced by the sum of the squares of its digits, for example 7, 13, and 19.
Harmonic number
A harmonic number is the sum of the reciprocals of the first n positive integers, a partial sum of the divergent harmonic series that grows like the natural logarithm.
Harshad number
A Harshad number, also called a Niven number, is a positive integer divisible by the sum of its digits; 1729 qualifies because 1729 = 91 × 19, the sum of its digits.
Highly composite number
A highly composite number, also called an anti-prime or maximally divisible number, is a positive integer with more divisors than any smaller one, first analyzed by Ramanujan in 1915.
Jack function
The Jack function, also called the Jack polynomial, is a symmetric polynomial depending on a partition and a parameter, introduced by Henry Jack in 1970.
Jacobsthal numbers
The Jacobsthal numbers are the sequence 0, 1, 1, 3, 5, 11, ... defined by the recurrence J(n) = J(n−1) + 2J(n−2), named for the German mathematician Ernst Erich Jacobsthal.
Kaprekar number
A Kaprekar number is a positive integer whose square can be split into two parts that add up to the original number, such as 45.