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The sum of the reciprocal of the primes increasing without bound. The x axis is in log scale, showing that the divergence is very slow. The red function is a lower bound that also diverges.
The sum of the reciprocals of the primes of the form 4n + 1 is divergent. By Fermat's theorem on sums of two squares , it follows that the sum of reciprocals of numbers of the form a 2 + b 2 , {\displaystyle \ a^{2}+b^{2}\ ,} where a and b are non-negative integers, not both equal to 0 , diverges, with or without repetition.
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] For example, 3 is the only prime with period 1, 11 is the only prime with period 2, 37 is the only prime ...
Because it is a divergent series, it should be interpreted as a formal sum, an abstract mathematical expression combining the unit fractions, rather than as something that can be evaluated to a numeric value. There are many different proofs of the divergence of the harmonic series, surveyed in a 2006 paper by S. J. Kifowit and T. A. Stamps. [13]
The reciprocals of prime numbers produce a divergent series (so the set of primes is "large"; see divergence of the sum of the reciprocals of the primes): 1 2 + 1 3 + 1 5 + 1 7 + 1 11 + 1 13 + ⋯ → ∞ . {\displaystyle {1 \over 2}+{1 \over 3}+{1 \over 5}+{1 \over 7}+{1 \over 11}+{1 \over 13}+\cdots \rightarrow \infty .}
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 ...
Prime number theorem; ... Divergence of the sum of the reciprocals of the primes; V. Vantieghems theorem; Vinogradov's theorem; W. Wilson's theorem; Wolstenholme's ...
The convergence of the sum of reciprocals of twin primes follows from bounds on the density of the sequence of twin primes. Let π 2 ( x ) {\displaystyle \pi _{2}(x)} denote the number of primes p ≤ x for which p + 2 is also prime (i.e. π 2 ( x ) {\displaystyle \pi _{2}(x)} is the number of twin primes with the smaller at most x ).