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The sum of the reciprocals of the cubes of positive integers is called Apéry's constant ζ(3) , and equals approximately 1.2021 . This number is irrational, but it is not known whether or not it is transcendental. The reciprocals of the non-negative integer powers of 2 sum to 2 . This is a particular case of the sum of the reciprocals of any ...
The sum of the reciprocals of all perfect powers including duplicates (but not including 1) equals 1. The Erdős–Moser equation, + + + = (+) where m and k are positive integers, is conjectured to have no solutions other than 1 1 + 2 1 = 3 1.
The reciprocal function: y = 1/x.For every x except 0, y represents its multiplicative inverse. The graph forms a rectangular hyperbola.. In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x −1, is a number which when multiplied by x yields the multiplicative identity, 1.
Then if we denote the lengths of the parallel sides as a and b and half the length of the segment through the diagonal intersection as c, the sum of the reciprocals of a and b equals the reciprocal of c. [4] The special case in which the integers whose reciprocals are taken must be square numbers appears in two ways in the context of right ...
The harmonic number with = ⌊ ⌋ (red line) with its asymptotic limit + (blue line) where is the Euler–Mascheroni constant.. In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers: [1] = + + + + = =.
Pythagorean quadruples are sets of four integers such that the sum of the squares of the first three equals the square of the fourth. The Basel problem, solved by Euler in terms of , asked for an exact expression for the sum of the squares of the reciprocals of all positive integers.
A prime p (where p ≠ 2, 5 when working in base 10) is called unique if there is no other prime q such that the period length of the decimal expansion of its reciprocal, 1/p, is equal to the period length of the reciprocal of q, 1/q. [8]
Some reciprocals of primes that do not generate cyclic numbers are: 1 / 3 = 0. 3, which has a period (repetend length) of 1. 1 / 11 = 0. 09, which has a period of two. 1 / 13 = 0. 076923, which has a period of six. 1 / 31 = 0. 032258064516129, which has a period of 15. 1 / 37 = 0. 027, which has a period ...