An interesting question, known as the Gaussian moat problem, asks whether it is possible to walk to infinity on Gaussian primes with steps of bounded length. Our work examines a similar situation in the real quadratic integer ring $\mathbb{Z}[\sqrt{2}]$ whose primes cluster near the asymptotes $y = \pm x/\sqrt{2}$ as compared to Gaussian primes, which cluster near the origin. We construct a probabilistic model of primes in $\mathbb{Z}[\sqrt{2}]$ by applying the prime number theorem and a combinatorial theorem for counting the number of lattice points whose absolute values of their norms are at most $r^2$. We then prove that it is impossible to walk to infinity if the walk remains within some bounded distance from the asymptotes. Lastly, we perform a few moat calculations to show that the longest walk is likely to stay close to the asymptotes; hence, we conjecture that there is no walk to infinity on $\mathbb{Z}[\sqrt{2}]$ primes with steps of bounded length.
An interesting open problem in number theory asks whether it is possible to walk to infinity on primes, where each term in the sequence has one more digit than the previous. In this paper, we study its variation where we walk on the Fibonacci sequence. We prove that all walks starting with a Fibonacci number and the following terms are Fibonacci numbers obtained by appending exactly one digit at a time to the right have a length of at most two. In the more general case where we append at most a bounded number of digits each time, we give a formula for the length of the longest walk.
We study analogies between the rational integers on the real line and the Gaussian integers on other lines in the complex plane. This includes a Gaussian analog of Bertrands Postulate, the Chinese Remainder Theorem, and the periodicity of divisibility. We also computationally investigate the distribution of Gaussian primes along these lines and leave the reader with several open problems.
One aspect of Chebyshev's bias is the phenomenon that a prime number, $ q $, modulo another prime number, $ p$, experimentally seems to be slightly more likely to be a nonquadratic residue than a quadratic residue. We thought it would be interesting to model this residue bias as a "random" walk using Legendre symbol values as steps. Such a model would allow us to easily visualize the bias. In addition, we would be able to extend our model to other number fields. In this report, we first outline underlying theory and some motivations for our research. In the second section, we present our findi…
We generalize the Gaussian moat problem to the finitely many imaginary quadratic fields exhibiting unique factorization in their integers. We map this problem to the Euclidean minimum spanning tree problem and use Delaunay triangulation and Kruskal's algorithm to find moats in these fields in $O(|\mathbf{P}| \log |\mathbf{P}|)$ time, where $\mathbf{P}$ denotes the set of primes we are searching through. Our algorithm uses symmetry and boundaries to correctly and efficiently identify all moats, in order of increasing length, along with the farthest prime reached for each moat, in an underlying …
In recent years, computer simulations are playing a fundamental role in unveiling some of the most intriguing features of prime numbers. In this work, we define an algorithm for a deterministic walk through a two-dimensional grid that we refer to as Prime Walk. The walk is constructed from a sequence of steps dictated by and dependent on the sequence of last digits of the primes. Despite the apparent randomness of this generating sequence, the resulting structure -- both in 2d and 3d -- created by the algorithm presents remarkable properties and regularities in its pattern that we proceed to a…
Emmanuel Kowalski and William Sawin proved, using a deep independence result of Kloosterman sheaves, that the polygonal paths joining the partial sums of the normalized classical Kloosterman sums S(a,b0;p)/p^{1/2} converge in the sense of finite distributions to a specific random Fourier series, as a varies over (Z/pZ)^*, b0 is fixed in (Z/pz)* and p tends to infinity among the odd prime numbers. This article considers the case of S(a,b0;p^n)/p^{n/2}, as a varies over (Z/p^nZ)^*, b0 is fixed in (Z/p^nZ)^*, p tends to infinity among the odd prime numbers and n>=2 is a fixed integer. A convergen…
For the old question whether there is always a prime in the interval [kn, (k+1)n] or not, the famous Bertrand's postulate gave an affirmative answer for k=1. It was first proved by P.L. Chebyshev in 1850, and an elegant elementary proof was given by P. Erdos in 1932. M. El Bachraoui used elementary techniques to prove the case k=2 in 2006. This paper gives a proof of the case k=3, again without using the prime number theorem or any deep analytic result. In addition we give a lower bound for the number of primes in the interval [3n, 4n], which shows that as n tends to infinity, the number of pr…
In this paper, we develop Furstenberg's proof of infinity of primes, and prove several results about prime divisors of sequences of integers, including the celebrated Schur's theorem. In particular, we give a simple proof of a classical result which says that a non-degenerate linear recurrence sequence of integers of order k>1 has infinitely many prime divisors.
The "self-power" map $x \mapsto x^x$ modulo $m$ and its generalized form $x \mapsto x^{x^n}$ modulo $m$ are of considerable interest for both theoretical reasons and for potential applications to cryptography. In this paper, we use $p$-adic methods, primarily $p$-adic interpolation, Hensel's lemma, and lifting singular points modulo $p$, to count fixed points and rooted closed walks of equations related to these maps when $m$ is a prime power. In particular, we introduce a new technique for lifting singular solutions of several congruences in several unknowns using the left kernel of the Jacob…
The aim of this article is to investigate how various Riemann Hypotheses would follow only from properties of the prime numbers. To this end, we consider two classes of $L$-functions, namely, non-principal Dirichlet and those based on cusp forms. The simplest example of the latter is based on the Ramanujan tau arithmetic function. For both classes we prove that if a particular trigonometric series involving sums of multiplicative characters over primes is $O(\sqrt{N})$, then the Euler product converges in the right half of the critical strip. When this result is combined with the functional eq…
Let $A$ be a subset of primes up to $x$. If we assume $A$ is well-distributed (in the Siegel-Walfisz sense) in any arithmetic progressions to moduli $q\leqslant(\log x)^c$ for any $c>0$, then the sumset $A+A$ has density 1/2 in the natural numbers as $x$ tends to infinity, which also yields almost all even positive integers could be represented as the sums of two primes in $A$ as $x$ tends to infinity. This result, improving the previous results in such special case, could be compared with the classical estimation for the exceptional set of binary Goldbach problem.
In this note, we show that there are many infinity positive integer values of $n$ in which, the following inequality holds $$ \left\lfloor{1/2}(\frac{(n+1)^2}{\log(n+1)}-\frac{n^2}{\log n})-\frac{\log^2 n}{\log\log n}\right\rfloor\leqπ\big((n+1)^2\big)-π(n^2). $$
If p is a prime and n a positive integer, let v(n) denote the exponent of p in n, and u(n)=n/p^{v(n)} the unit part of n. If k is a positive integer not divisible by p, we show that the p-adic limit of (-1)^{pke} u((kp^e)!) as e goes to infinity is a well-defined p-adic integer, which we call z_k. In terms of these, we give a formula for the p-adic limit of binom{a p^e +c, b p^e +d) as e goes to infinity, which we call binom(a p^\infty +c, b p^\infty +d). Here a \ge b are positive integers, and c and d are integers.
Let $\mathbb{Z}^2$ be the two-dimensional integer lattice. For an integer $k\geq 1$, a non-zero lattice point is $k$-free if the greatest common divisor of its coordinates is a $k$-free number. We consider the proportions of $k$-free and twin $k$-free lattice points on a path of an $α$-random walker in $\mathbb{Z}^2$. Using the second-moment method and tools from analytic number theory, we prove that these two proportions are $1/ζ(2k)$ and $\prod_{p}(1-2p^{-2k})$, respectively, where $ζ$ is the Riemann zeta function and the infinite product takes over all primes.
Given a sequence $(\mathfrak{X}_i, \mathscr{K}_i)_{i=1}^\infty$ of Markov chains, the cut-off phenomenon describes a period of transition to stationarity which is asymptotically lower order than the mixing time. We study mixing times and the cut-off phenomenon in the total variation metric in the case of random walk on the groups $\mathbb{Z}/p\mathbb{Z}$, $p$ prime, with driving measure uniform on a symmetric generating set $A_p \subset \mathbb{Z}/p\mathbb{Z}$.
We introduce a new arithmetic function $a(n)$ defined to be the number of random multiplications by residues modulo $n$ before the running product is congruent to 0 modulo $n$. We give several formulas for computing the values of this function and analyze its asymptotic behavior. We find that it is closely related to $P_1(n)$, the largest prime divisor of $n$. In particular, $a(n)$ and $P_1(n)$ have the same average order asymptotically. Furthermore, the difference between the functions $a(n)$ and $P_1(n)$ is $o(1)$ as $n$ tends to infinity on a set with density approximately $0.623$. On the o…
The sequence of the primes $p$ for which a variety over $\mathbb{Q}$ has no $p$-adic point plays a fundamental role in arithmetic geometry. This sequence is deterministic, however, we prove that if we choose a typical variety from a family then the sequence has random behavior. We furthermore prove that this behavior is modelled by a random walk in Brownian motion. This has several consequences, one of them being the description of the finer properties of the distribution of the primes in this sequence via the Feynman-Kac formula.
