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In mathematics and especially number theory, the sum of reciprocals generally is computed for the reciprocals of some or all of the positive integers (counting numbers)—that is, it is generally the sum of unit fractions. If infinitely many numbers have their reciprocals summed, generally the terms are given in a certain sequence and the first ''n'' of them are summed, then one more is included to give the sum of the first ''n''+1 of them, etc. If only finitely many numbers are included, the key issue is usually to find a simple expression for the value of the sum, or to require the sum to be less than a certain value, or to determine whether the sum is ever an integer. For an infinite series of reciprocals, the issues are twofold: First, does the sequence of sums diverge, meaning it eventually exceeds any given number, or does it converge, meaning there is some number that it gets arbitrarily close to without ever exceeding it? (A set of positive integers is said to be large if the sum of its reciprocals diverges, and small if it converges.) Second, if it converges, what is a simple expression for the value it converges to, is that value rational or irrational, and is that value algebraic or transcendental?〔Unless given here, references are in the linked articles.〕 ==Finitely many terms== *The optic equation requires the sum of the reciprocals of two positive integers ''a'' and ''b'' to equal the reciprocal of a third positive integer ''c''. All solutions are given by ''a'' = ''mn'' + ''m''2, ''b'' = ''mn'' + ''n''2, ''c'' = ''mn''. This equation appears in various contexts in elementary geometry. *The Fermat–Catalan conjecture concerns a certain Diophantine equation, equating the sum of two terms, each a positive integer raised to a positive integer power, to a third term that is also a positive integer raised to a positive integer power (with the base integers having no prime factor in common). The conjecture asks whether the equation has an infinitude of solutions in which the sum of the reciprocals of the three exponents in the equation must be less than 1. 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. *Moreover, Jozsef Kurschak proved in 1918 that the sum of the reciprocals of consecutive natural numbers (whether starting from 1 or not) is never an integer. *There are 215 non-consecutive sequences of four integers, counting re-arrangements as distinct, such that the sum of their reciprocals is 1. There are 14 of them if rearrangements are not allowed. *An Egyptian fraction is the sum of a finite number of reciprocals of positive integers. According to the proof of the Erdős–Graham problem, if the set of integers greater than one is partitioned into finitely many subsets, then one of the subsets can be used to form an Egyptian fraction representation of 1. *The Erdős–Straus conjecture states that for all integers ''n'' ≥ 2, the rational number 4/''n'' can be expressed as the sum of three reciprocals of integers. *The Fermat quotient with base 2, which is for odd prime ''p'', when expressed in mod ''p'' and multiplied by –2, equals the sum of the reciprocals mod ''p'' of the numbers lying in the first half of the range . *In any triangle, the sum of the reciprocals of the altitudes equals the reciprocal of the radius of the incircle (regardless of whether or not they are integers). *In a right triangle, the sum of the reciprocals of the squares of the altitudes from the legs (equivalently, of the squares of the legs themselves) equals the reciprocal of the square of the altitude from the hypotenuse. This holds whether or not the numbers are integers; there is a formula (see here) that generates all integer cases. *A triangle not necessarily in the Euclidean plane can be specified as having angles and Then the triangle is in Euclidean space if the sum of the reciprocals of ''p, q,'' and ''r'' equals 1, spherical space if that sum is greater than 1, and hyperbolic space if the sum is less than 1. *A harmonic divisor number is a positive integer whose divisors have a harmonic mean that is an integer. The first five of these are 1, 6, 28, 140, and 270. It is not known whether any harmonic divisor numbers are odd, but there are no odd ones less than1024. *When eight points are distributed on the surface of a sphere with the aim of maximizing the distance between them in some sense, the resulting shape corresponds to a square antiprism. Specific methods of distributing the points include, for example, minimizing the sum of all reciprocals of squares of distances between points. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「List of sums of reciprocals」の詳細全文を読む スポンサード リンク
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