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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.
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.
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:
Gauss published the first and second proofs of the law of quadratic reciprocity on arts 125–146 and 262 of Disquisitiones Arithmeticae in 1801. In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it ...
The Legendre symbol ( a / p ) is only defined for odd primes p. It obeys the same rules as the Jacobi symbol (i.e., reciprocity and the supplementary formulas for ( −1 / p ) and ( 2 / p ) and multiplicativity of the "numerator".) Problem: Given that 9907 is prime, calculate ( 1001 / 9907 ).
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 ...
The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes. Gauss, Carl Friedrich (1986), Disquisitiones Arithemeticae (Second, corrected edition), translated by Clarke, Arthur A.,