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In number theory, a polite number is a positive integer that can be written as the sum of two or more consecutive positive integers. A positive integer which is not polite is called impolite . [ 1 ] [ 2 ] The impolite numbers are exactly the powers of two , and the polite numbers are the natural numbers that are not powers of two.
The n-th harmonic number, which is the sum of the reciprocals of the first n positive integers, is never an integer except for the case n = 1. Moreover, József Kürschák proved in 1918 that the sum of the reciprocals of consecutive natural numbers (whether starting from 1 or not) is never an integer.
That is, = (+) This is the difference-of-squares formula, which can be useful for mental arithmetic: for example, 47 × 53 can be easily computed as 50 2 − 3 2 = 2500 − 9 = 2491. A square number is also the sum of two consecutive triangular numbers.
A pronic number is a number that is the product of two consecutive integers, that is, a number of the form (+). [1] The study of these numbers dates back to Aristotle.They are also called oblong numbers, heteromecic numbers, [2] or rectangular numbers; [3] however, the term "rectangular number" has also been applied to the composite numbers.
In mathematics, summation is the addition of a sequence of numbers, called addends or summands; the result is their sum or total.Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined.
Any integer can be uniquely represented [4] as a sum of negafibonacci numbers in which no two consecutive negafibonacci numbers are used. For example: For example: −11 = F −4 + F −6 = (−3) + (−8)
The Goldbach conjecture for practical numbers, a prime-like sequence of integers, was stated by Margenstern in 1984, [33] and proved by Melfi in 1996: [34] every even number is a sum of two practical numbers.
If an odd perfect number exists, it will have at least nine distinct prime factors. [4] Non-intersecting chords between four points on a circle. 9 is the sum of the cubes of the first two non-zero positive integers + which makes it the first cube-sum number greater than one. [5]