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Prime Numbers: A Computational Perspective
Prime numbers beckon to the beginner, the basic notion of primality being accessible to a child. Yet, some of the simplest questions about primes have stumped humankind for millennia. In this book,…
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On the binary expansions of algebraic numbers
TLDR
By using number-theoretical analysis and extended computations, the transcendency of the Kempner--Mahler number {summation}{sub n{ge}0}1/2{sup 2{sup n}}, yet the authors can also handle numbers with a substantially denser occurrence of 1's.
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Computational strategies for the Riemann zeta function
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Discrete weighted transforms and large-integer arithmetic
TLDR
The concept of Discrete Weighted Transforms (DWTs) are introduced which substantially improve the speed of multiplication by obviating costly zero-padding of digits.
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On the Random Character of Fundamental Constant Expansions
TLDR
A theory to explain random behavior for the digits in the expansions of fundamental mathematical constants and proofs of base-2 normality for a collection of celebrated constants, including π, log 2, ζ(3), and others are proposed.
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On the $‘‘3x+1”$ problem
It is an open conjecture that for any positive odd integer m the function C(m) = (3m + 1)/2e(m) where e(m) is chosen so that C(m) is again an odd integer, satisfies Ce(m) = 1 for some h. Here we show…
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On the Evaluation of Euler Sums
TLDR
Various series expansions of ζ(r, s) for real numbers r and s are established, which generally involve infinitely many zeta values.
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Integrals of the Ising Class
TLDR
From an experimental-mathematical perspective, Ising-class integrals are analyzed, which involvedextreme-precision, multidimensional quadrature on intricate integrands and highly parallel computation was required.
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Random Generators and Normal Numbers
TLDR
An uncountable class of explicit normals that succumb to the PRNG approach is described, and b-normality is established for constants of the form Σ 1/(b mi c ni) for certain sequences (mi), (ni) of integers.
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A search for Wieferich and Wilson primes
TLDR
It is reported that there exist no new Wieferich primes p < 4 x 10 12 , and no new Wilson prime p < 5x 10 8 .
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