Related papers: Testing Kak's Conjecture on Binary Reciprocal of P…
We pose 100 new conjectures on representations involving primes or related things, which might interest number theorists and stimulate further research. Below are five typical examples: (i) For any positive integer $n$, there exists…
A binary string representation of prime occurrences is a sequence of bits, where $1$ entries encode positions of prime numbers. This is a convenient representation for analysis of prime distribution, since it allows for application of a…
Prime reciprocals have applications in coding and cryptography and for generation of random sequences. This paper investigates the structural redundancy of prime reciprocals in base 10 in a manner that parallels an earlier study for binary…
In the paper, the occurrence of zeros and ones in the binary expansion of the primes is studied. In particular the statement in the title is established. The proof is unconditional.
We put a new conjecture on primes from the point of view of its binary expansions and make a step towards justification.
Prime numbers, whose properties are important subjects in mathematics, are also fundamental in computer science notably in IT security, Cryptocurrencies as Bitcoin and Blockchain, cryptography, Code theory notably Error detection codes,…
It is shown that the first $n$ prime numbers $p_1,...,p_n$ determine the next one by the recursion equation $$ p_{n+1} =\lim\limits_{s\to +\infty} [\prod\limits^n_{k=1} (1-\frac{1}{p^s_k}) \sum\limits^\infty_{j=1} \frac{1}{j^s} -1]^{-1/s}.…
Prime number multiplet classifications and patterns are extended to negative integers. The extension from prime numbers to single prime powers is also studied. Prime number septets at equal distance are given. It is also shown that each…
Jacobsthal's conjecture has been disproved by counterexample a few years ago. We continue to verify this conjecture on a larger scale. For this purpose, we implemented an extension of the Greedy Permutation Algorithm and computed the…
We prove that for any positive integer c and any s > 0 there are representations of c as a sum a+b of two coprime positive integers a, b, such that the respective radicals are all greater than K(s)R(c)^(1-s)c^2. For the reprasentations in…
A recent heuristic argument based on basic concepts in spectral analysis showed that the twin prime conjecture and a few other related primes counting problems are valid. A rigorous version of the spectral method, and a proof for the…
If $p_k$ is the k-th prime, the Firoozbakht conjecture states that the sequence $(p_k)^{1/k}$ is strictly decreasing. We use the table of first-occurrence prime gaps in combination with known bounds for the prime-counting function to verify…
Let $p_n$ be $n$th prime, and let $(S_n)_{n=1}^\infty:=(S_n)$ be the sequence of the sums of the first $2n$ consecutive primes, that is, $S_n=\sum_{k=1}^{2n}p_k$ with $n=1,2,\ldots$. Heuristic arguments supported by the corresponding…
In this paper, using the well known fact that the series of reciprocals of primes diverges, we obtain a general inequality for gaps of consecutive primes that holds for infinitely many primes. As it is shown the key ingredient for this…
Associate a unique numerical sequence called the modular signature with each positive integer, using modular residues of each integer under the prime numbers, and distinguishing between the core seed primes and non-core seed primes used to…
Some open questions related to prime reciprocal digit frequencies with potential applications to cryptography are presented.
An s-tuple of positive integers are k-wise relatively prime if any k of them are relatively prime. Exact formula is obtained for the probability that s positive integers are k-wise relatively prime.
A recent heuristic argument based on basic concepts in spectral analysis showed that the twin prime conjecture and a few other related primes counting problems are valid. A rigorous version of the spectral method, and a proof of the more…
We adopt an empirical approach to the characterization of the distribution of twin primes within the set of primes, rather than in the set of all natural numbers. The occurrences of twin primes in any finite sequence of primes are like…
Prime numbers are one of the most intriguing figures in mathematics. Despite centuries of research, many questions remain still unsolved. In recent years, computer simulations are playing a fundamental role in the study of an immense…