منابع مشابه
Quotients of Gaussian Primes
It has been observed many times, both in the Monthly and elsewhere, that the set of all quotients of prime numbers is dense in the positive real numbers. In this short note we answer the related question: “Is the set of all quotients of Gaussian primes dense in the complex plane?” Quotient sets {s/t : s, t ∈ S} corresponding to subsets S of the natural numbers have been intensely studied in the...
متن کاملIrregularities in the Distribution of Primes and Twin Primes
The maxima and minima of sL(x)) — n(x), iR(x)) — n(x), and sL2(x)) — n2(x) in various intervals up to x = 8 x 10 are tabulated. Here n(x) and n2(x) are respectively the number of primes and twin primes not exceeding x, L(x) is the logarithmic integral, R(x) is Riemann's approximation to ir(x), and L2(x) is the Hardy-Littlewood approximation to ti"2(;c). The computation of the sum of inverses of...
متن کاملGaussian Mersenne and Eisenstein Mersenne primes
The Biquadratic Reciprocity Law is used to produce a deterministic primality test for Gaussian Mersenne norms which is analogous to the Lucas–Lehmer test for Mersenne numbers. It is shown that the proposed test could not have been obtained from the Quadratic Reciprocity Law and Proth’s Theorem. Other properties of Gaussian Mersenne norms that contribute to the search for large primes are given....
متن کاملA Stroll Through the Gaussian Primes
THE MOAT PROBLEM. One cannot walk to infinity on the real line if one uses steps of bounded length and steps on the prime numbers. This is simply a restatement of the classic result that there are arbitrarily large gaps in the primes. The proof is simple: a gap of size k is given by (k + 1)! + 2, (k + 1)! + 3, ... (k + 1)! + (k + 1). But the same problem in the complex realm is unsolved. More p...
متن کاملStudent-Faculty Seminar The Distribution of Primes
To understand the natural numbers, we must first understand the primes. Yet the behavior of the primes remains for the most part a mystery. For example, despite the efforts of many great mathematicians over hundreds of years, we still know very little about how the primes are distributed (scattered) throughout the natural numbers. The results and conjectures pertaining to the distribution of th...
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ژورنال
عنوان ژورنال: Journal of the European Mathematical Society
سال: 2021
ISSN: ['1435-9855', '1435-9863']
DOI: https://doi.org/10.4171/jems/1083