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Rules for calculating the periods of repeating decimals from rational fractions were given by James Whitbread Lee Glaisher in 1878. [5] For a prime p, the period of its reciprocal divides p − 1. [6] The sequence of recurrence periods of the reciprocal primes (sequence A002371 in the OEIS) appears in the 1973 Handbook of Integer Sequences.
For a general reciprocity law [1] pg 3, it is defined as the rule determining which primes the polynomial splits into linear factors, denoted {()}. There are several different ways to express reciprocity laws.
Then for , the expansion of () contains at least one term for each reciprocal of a positive integer with exactly prime factors (counting multiplicities) only from the set {+, +,}. It follows that the geometric series ∑ i = 0 ∞ ( x k ) i {\textstyle \sum _{i=0}^{\infty }(x_{k})^{i}} contains at least one term for each reciprocal of a ...
A prime quadruplet is a pair of two twin prime pairs, separated by a distance of 4 (the smallest possible distance). The first prime quadruplets are (5, 7, 11, 13), (11, 13, 17, 19), (101, 103, 107, 109). Brun's constant for prime quadruplets, denoted by B 4, is the sum of the reciprocals of all prime quadruplets:
The sum of the reciprocals of the Proth primes, of which there may be finitely many or infinitely many, is known to be finite, approximately 0.747392479. [2] The prime quadruplets are pairs of twin primes with only one odd number between them. The sum of the reciprocals of the numbers in prime quadruplets is approximately 0.8706.
Trump spoke shortly before he is set to meet with India’s Prime Minister Narendra Modi, who is visiting Washington, D.C. Addressing the reporters, Trump said that India is a “hard place to do ...
In the limit, the sum of the reciprocals of the primes < n and the function ln(ln n) are separated by a constant, the Meissel–Mertens constant (labelled M above). The Meissel–Mertens constant (named after Ernst Meissel and Franz Mertens), also referred to as the Mertens constant, Kronecker's constant (after Leopold Kronecker), Hadamard–de la Vallée-Poussin constant (after Jacques ...
During a Friday meeting with Prime Minister Shigeru Ishiba of Japan at the White House, Trump said he wanted to cut the US trade deficit with Japan, which in 2024 hit $68.5 billion. Trump didn't ...