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The purpose of this restriction is to preclude the known infinitude of solutions in which two exponents are 2 and the other exponent is any even number. 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.
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.
Since a = n(n + 1)/2, these formulae show that for an odd power (greater than 1), the sum is a polynomial in n having factors n 2 and (n + 1) 2, while for an even power the polynomial has factors n, n + 1/2 and n + 1.
In mathematics, the Bernoulli numbers B n are a sequence of rational numbers which occur frequently in analysis.The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for certain ...
The nth partial sum is given by a simple formula: = = (+). This equation was known to the Pythagoreans as early as the sixth century BCE. [5] Numbers of this form are called triangular numbers, because they can be arranged as an equilateral triangle.
The n th pronic number is the sum of the first n even integers, and as such is twice the n th triangular number [1] [2] and n more than the n th square number, as given by the alternative formula n 2 + n for pronic numbers. Hence the n th pronic number and the n th square number (the sum of the first n odd integers) form a superparticular ratio:
A taxicab number is the smallest integer that can be expressed as a sum of two positive third powers in n distinct ways. The Riemann zeta function is the sum of reciprocals of the positive integers each raised to the power s, where s is a complex number whose real part is greater than 1.
There exists a natural number N such that every even integer n larger than N is a sum of a prime less than or equal to n 0.95 and a number with at most two prime factors. Tomohiro Yamada claimed a proof of the following explicit version of Chen's theorem in 2015: [ 7 ]