Tag: Number Theory

A Remarkable Pair of Sequences

The terms of the two integer sequences below are equal for all $latex {n}&fg=000000$ such that $latex {1<n<777{,}451{,}915{,}729{,}368}&fg=000000$,  but equality is violated for this enormous value and, intermittently, for larger values of $latex {n}&fg=000000$. Hypercube Tic-Tac-Toe The simple game of tic-tac-toe, or noughts and crosses, has been generalized in several ways. The number of cells in … Continue reading A Remarkable Pair of Sequences →

A Geometric Sieve for the Prime Numbers

In the time before computers (BC) various ingenious devices were invented for aiding the extensive calculations required in astronomy, navigation and commerce. In addition to calculators and logarithms, several nomograms were devised for specific applications, for example in meteorology and surveying. A Nomogram for Multiplication The graph of a parabola $latex {y=x^2}&fg=000000$ can be used … Continue reading A Geometric Sieve for the Prime Numbers →

Numerical Coincidences

A numerical coincidence is an equality or near-equality between different mathematical quantities which has no known theoretical explanation. Sometimes such equalities remain mysterious and intriguing, and sometimes theory advances to the point where they can be explained and are no longer regarded as surprising. Simple Examples A simple example is the near-equality between 2 cubed … Continue reading Numerical Coincidences →

Brun’s Constant and the Pentium Bug

Euclid showed by a deliciously simple argument that the number of primes is infinite. In a completely different manner, Euler confirmed the same result. Euler's conclusion followed from his demonstration that the sum of the reciprocals of the primes diverges: $latex \displaystyle \sum_{p\in\mathbb{P}} \frac{1}{p} = \infty &fg=000000$ Obviously, this could not happen if there were … Continue reading Brun’s Constant and the Pentium Bug →

Random Harmonic Series

We consider the convergence of the random harmonic series $latex \displaystyle R = \sum_{n=1}^{\infty}\frac{\sigma_{n}}{n} &fg=000000$ where $latex {\sigma_n\in\{-1,+1\}}&fg=000000$ is chosen randomly with probability $latex {1/2}&fg=000000$ of being either plus one or minus one. It follows from the Kolmogorov three-series theorem that the series is ``almost surely'' convergent. We are all familiar with the harmonic series … Continue reading Random Harmonic Series →

Lecture sans paroles: the factors of M67

In 1903 Frank Nelson Cole delivered an extraordinary lecture to the American Mathematical Society. For almost an hour he performed a calculation on the chalkboard without uttering a single word. When he finished, the audience broke into enthusiastic applause. Cole, an American mathematician born in 1861, was educated at Harvard. He lectured there and later … Continue reading Lecture sans paroles: the factors of M67 →

Waring’s Problem & Lagrange’s Four-Square Theorem

$latex \displaystyle \mathrm{num}\ = \square+\square+\square+\square &fg=000000$ Introduction We are all familiar with the problem of splitting numbers into products of primes. This process is called factorisation. The problem of expressing numbers as sums of smaller numbers has also been studied in great depth. We call such a decomposition a partition. The Indian mathematician Ramanujan proved … Continue reading Waring’s Problem & Lagrange’s Four-Square Theorem →

The Ups and Downs of Hailstone Numbers

Hailstones, in the process of formation, make repeated excursions up and down within a cumulonimbus cloud until finally they fall to the ground. We look at sequences of numbers that oscillate in a similarly erratic manner until they finally reach the value 1. They are called hailstone numbers. The Collatz Conjecture There are many simply-stated … Continue reading The Ups and Downs of Hailstone Numbers →