Related papers: Hitting a prime by rolling a die with infinitely m…

Let $\lfloor t\rfloor$ denote the integer part of $t\in\mathbb{R}$ and $\|x\|$ the distance from $x$ to the nearest integer. Suppose that $1/2<\gamma_2<\gamma_1<1$ are two fixed constants. In this paper, it is proved that, whenever $\alpha$…

Mean-payoff games on timed automata are played on the infinite weighted graph of configurations of priced timed automata between two players, Player Min and Player Max, by moving a token along the states of the graph to form an infinite…

We study a sequential coin-flipping game in which a player starts with~$n$ coins, each landing heads independently with probability~$p$. In each round the player flips all remaining coins and must set aside at least one coin showing heads;…

Numerical computations suggest that each point on a certain optimized shape called the ideal trefoil is in contact with two other points. We consider sequences of such contact points, such that each point is in contact with its predecessor…

This work proposes a proof of the simplest cubic primes counting problem. It shows that the subset of primes {p = n^3 + 2 is prime : n => 1} is an infinite subset of primes. Further, the expected order of magnitude of the cubic primes…

In 1970, Statistics giant, Bradley Efron, amazed the world by coming up with a set of four dice, let's call them A,B,C,D, whose faces are marked with [0,0,4,4,4,4], [3,3,3,3,3,3],[2,2,2,2,6,6],[1,1,1,5,5,5] respectively, where die A beats…

We speculate on the distribution of primes in exponentially growing, linear recurrence sequences $(u_n)_{n\geq 0}$ in the integers. By tweaking a heuristic which is successfully used to predict the number of prime values of polynomials, we…

We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the Elliott-Halberstam conjecture, we prove that…

This paper is concerned with asymptotic behaviour of a repeated game of "odds and evens", with strategies of both players represented by finite automata. It is proved that, for every $n$, there is an automaton with $2^n \cdot…

We proved that any even number not less than 6 can be expressed as the sum of two old primes, $2n=p_i+p_j$

Consider the following game between a random player R and a deterministic player D. There is a pile of n elements at the beginning. The rules for playing are as follows: In each turn of R, if the pile contains exactly m elements, R removes…

The prime number graph is the set of points $(n,p_n)$ where $p_n$ denotes the $n^{\rm th}$ prime. Let $L(n)$ be the minimum number of straight line segments needed to cover the first $n$ points in this set. Let $B(n)$ be the largest number…

Two players alternate tossing a biased coin where the probability of getting heads is p. The current player is awarded alpha points for tails and alpha+beta for heads. The first player reaching n points wins. For a completely unfair coin…

You play the following game: you start out with $n$ coins that all have probability $p$ to land heads. You toss all of them and you then need to set aside at least one of them, which will not be tossed again. Now you repeat the process with…

This recreational mathematics article shows that the game of Snakes and Ladders is intransitive: square 69 has a winning edge over 79, which in turn beats 73, which beats 69. Analysis of the game is a nice illustration of Markov chains,…

For any $m = 3 \left( 2n + 1 \right) with \ n \in \mathbb{N^*} ,$ the prime counting function $\pi(m) = 4 + \left \vert A_4(m) \right \vert + 2 \left \vert A_6(m) \right \vert $ where $A_6(m) $ and $ A_4(m) $ are the sets of Twin Primes and…

Given a set of $m$ points and a set of $n$ lines in the plane, we consider the problem of computing the faces of the arrangement of the lines that contain at least one point. In this paper, we present an $O(m^{2/3}n^{2/3}+(n+m)\log n)$ time…

We use the backward Kolmogorov equation approach to understand the apparently paradoxical feature that the mean waiting time to encounter distinct fixed-length sequences of heads and tails upon repeated fair coin flips can be different. For…

We find an upper bound for the sum $\sum_{x<n\leq 2x}\textbf{1}_{\mathbb{P}}(n+h_{i_{1}})\cdots\textbf{1}_{\mathbb{P}}(n+h_{i_{m+1}})w_{n}$, where $(h_{i_{1}},...,h_{i_{m+1}})$ is any $(m+1)$-tuple of elements in the admissible set…

Many people have flipped coins but few have stopped to ponder the statistical and physical intricacies of the process. We collected $350{,}757$ coin flips to test the counterintuitive prediction from a physics model of human coin tossing…