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

The game of memory is played with a deck of n pairs of cards. The cards in each pair are identical. The deck is shuffled and the cards laid face down. A move consists of flipping over first one card then another. The cards are removed from…

Alice and Bob take turns (with Alice playing first) in declaring numbers from the set $[1,2N]$. If a player declares a number that was previously declared, that player looses and the other player wins. If all numbers are declared without…

Let $a_0\in\{0,\dots,9\}$. We show there are infinitely many prime numbers which do not have the digit $a_0$ in their decimal expansion. The proof is an application of the Hardy-Littlewood circle method to a binary problem, and rests on…

Two possibly unfair $n$-sided dice, both labelled $1, 2, \ldots, n$, are rolled, and the sum is recorded. How should the dice's sides be weighted so that the resulting sum is closest to the uniform distribution on $2, 3, \ldots, 2n$? We…

The usual $n$-in-a-row game is a positional game in which two player alternately claim points in $\bb{Z}^2$ with the winner being the first player to claim $n$ consecutive points in a line. We consider a variant of the game, suggested by…

We prove that the Random-Edge simplex algorithm requires an expected number of at most 13n/sqrt(d) pivot steps on any simple d-polytope with n vertices. This is the first nontrivial upper bound for general polytopes. We also describe a…

In this note, we approximate the average of prime powers in the decomposition of $n!$ into prime numbers.

Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, it is proved that, for $0.989<\gamma<1$, there exist infinitely many primes $p$ such that…

We harness both human ingenuity and the power of symbolic computation to study the number of coin tosses until reaching $n$ Heads or $m$ Tails. We also talk about the closely related problem of reaching $n$ Heads and $m$ Tails. This paper…

We introduce and analyze a natural game formulated as follows. In this one-person game, the player is given a random permutation $A=(a_1,\dots, a_n)$ of a multiset $M$ of $n$ reals that sum up to $0$, where each of the $n!$ permutation…

The author proves that for $0.9985 < \gamma < 1$, there exist infinitely many primes $p$ such that $[p^{1/\gamma}]$ has at most 5 prime factors counted with multiplicity. This gives an improvement upon the previous results of…

We present a simple, closed formula which gives all the primes in order. It is a simple product of integer floor and ceiling functions.

We prove that every odd number $N$ greater than 1 can be expressed as the sum of at most five primes, improving the result of Ramar\'e that every even natural number can be expressed as the sum of at most six primes. We follow the circle…

We consider a two player simultaneous-move game where the two players each select any permissible $n$-sided die for a fixed integer $n$. A player wins if the outcome of his roll is greater than that of his opponent. Remarkably, for $n>3$,…

In the game of Matching Pennies, Alice and Bob each hold a penny, and at every tick of the clock they simultaneously display the head or the tail sides of their coins. If they both display the same side, then Alice wins Bob's penny; if they…

The Prime Number Theorem states that the number of primes in $\{1,\ldots,x\}$, denoted $\pi(x)$, is approximately $\frac{x}{\ln(x)}$. In this paper, we investigate the distribution of primes for domains other than $\N$. First we look at…

We obtain an upper bound for the number of pairs $ (a,b) \in {A\times B} $ such that $ a+b $ is a prime number, where $ A, B \subseteq \{1,...,N \}$ with $|A||B| \, \gg \frac{N^2}{(\log {N})^2}$, $\, N \geq 1$ an integer. This improves on a…

Let $\mathbb{A}[n]$ be the group of $n$-torsion points of a commutative algebraic group $\mathbb{A}$ defined over a number field $F$. For a prime ideal $\mathfrak{p}$, we let $N_{\mathfrak{p}}(\mathbb{A}[n])$ be the number of…

Consider a gambling game in which we are allowed to repeatedly bet a portion of our bankroll at favorable odds. We investigate the question of how to minimize the expected number of rounds needed to increase our bankroll to a given target…

Delay games are two-player games of infinite duration in which one player may delay her moves to obtain a lookahead on her opponent's moves. For $\omega$-regular winning conditions it is known that such games can be solved in…