Related papers: Hitting a prime in 2.43 dice rolls (on average)

We investigate the distribution of the digits of quotients of randomly chosen positive integers taken from the interval $[1,T]$, improving the previously known error term for the counting function as $T\to+\infty$. We also resolve some…

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…

This paper introduces a new algorithm for the fundamental problem of generating a random integer from a discrete probability distribution using a source of independent and unbiased random coin flips. We prove that this algorithm, which we…

We give a detailed proof, in the identically distributed case, of a conjecture of Feige about the maximum probability that the sum of n independent non-negative integer valued random variables, each of mean 1, exceeds n. The general case is…

Imagine that there are two bins to which balls are added sequentially, and each incoming ball joins a bin with probability proportional to the p-th power of the number of balls already there. A general result says that if p>1/2, there…

We compute the expectation of the number of linear spaces on a random complete intersection in $p$-adic projective space. Here "random" means that the coefficients of the polynomials defining the complete intersections are sampled uniformly…

In this paper we present some observations about the well-known Goldbach conjecture. In particular we list and interpret some numerical results which allow us to formulate a relation between prime numbers and even integers. We can also…

For $a \neq 1$ and $p$ prime, we define numbers of the form $pa^2$ to be Square-Prime (SP) Numbers. For example, 75 = 3 $\cdot$ 25; 108 = 3 $\cdot$ 36; 45 = 5 $\cdot$ 9. These numbers are listed in the OEIS as A228056. We study the…

The rules of a game of dice are extended to a "hyper-die" with $n\in\mathbb{N}$ equally probable faces, numbered from 1 to $n$. We derive recursive and explicit expressions for the probability mass function and the cumulative distribution…

Let $F(\sigma)=\sum_{n=1}^\infty \frac{X_n}{n^\sigma}$ be a random Dirichlet series where $(X_n)_{n\in\mathbb{N}}$ are independent standard Gaussian random variables. We compute in a quantitative form the expected number of zeros of…

Simple divisibility rules are given for the 1st 1000 prime numbers.

We consider a simple dice game, which leads to an intriguing study of multinomial walks, with surprising and seemingly paradoxical properties. The winning and losing probabilities of a general version of the game are investigated via…

When numbers are added in base $b$ in the usual way, carries occur. If two random, independent 1-digit numbers are added, then the probability of a carry is $\frac{b-1}{2b}$. Other choices of digits lead to less carries. In particular, if…

We state a general purpose algorithm for quickly finding primes in evenly divided sub-intervals. Legendre's conjecture claims that for every positive integer $n$, there exists a prime between $n^2$ and $(n+1)^2$. Oppermann's conjecture…

We consider a card guessing game with complete feedback. An ordered deck of $n$ cards labeled $1$ up to $n$ is riffle-shuffled exactly one time. Given a value $p\in(0{,}1)\setminus\{\frac12\}$, the riffle shuffle is assumed to be…

We compute all irregular primes less than 2^31 = 2 147 483 648. We verify the Kummer-Vandiver conjecture for each of these primes, and we check that the p-part of the class group of Q(zeta_p) has the simplest possible structure consistent…

We are interested in classifying those sets of primes $\mathcal{P}$ such that when we sieve out the integers up to $x$ by the primes in $\mathcal{P}^c$ we are left with roughly the expected number of unsieved integers. In particular, we…

As long as people have studied mathematics, they have wanted to know how many primes there are. Getting precise answers is a notoriously difficult problem, and the first suitable technique, due to Riemann, inspired an enormous amount of…

We formulate, using heuristic reasoning, precise conjectures for the range of the number of primes in intervals of length $y$ around $x$, where $y\ll (\log x)^2$. In particular we conjecture that the maximum grows surprisingly slowly as $y$…

In this paper, a simple explanation for the Goldbach Conjecture is given. We have shown that the probability of violating the conjecture not only for the prime numbers, but also for any subset of natural numbers whose distribution is…