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For an integer n>=2, let lpf(n) denote the least prime factor of n. A pair of integers (x,y) is called a twin peak if 1. x<y, 2. lpf(x)=lpf(y), 3. For all z, x<z<y implies ...

The unknot, also called the trivial knot (Rolfsen 1976, p. 51), is a closed loop that is not knotted. In the 1930s Reidemeister first proved that knots exist which are ...

The one-dimensional wave equation is given by (partial^2psi)/(partialx^2)=1/(v^2)(partial^2psi)/(partialt^2). (1) In order to specify a wave, the equation is subject to ...

n weighings are sufficient to find a bad coin among (3^n-1)/2 coins (Steinhaus 1999, p. 61). vos Savant (1993) gives an algorithm for finding a bad ball among 12 balls in ...

The abc conjecture is a conjecture due to Oesterlé and Masser in 1985. It states that, for any infinitesimal epsilon>0, there exists a constant C_epsilon such that for any ...

The simple continued fraction representations of e given by [2; 1, 2, 1, 1, 4, 1, 1, 6, ...] (OEIS A003417). This continued fraction is sometimes known as Euler's continued ...

The number of ways a set of n elements can be partitioned into nonempty subsets is called a Bell number and is denoted B_n (not to be confused with the Bernoulli number, ...

The game of billiards is played on a rectangular table (known as a billiard table) upon which balls are placed. One ball (the "cue ball") is then struck with the end of a ...

Consider the probability Q_1(n,d) that no two people out of a group of n will have matching birthdays out of d equally possible birthdays. Start with an arbitrary person's ...

Buffon's needle problem asks to find the probability that a needle of length l will land on a line, given a floor with equally spaced parallel lines a distance d apart. The ...