# Browse Classroom Capsules and Notes

You can filter the list below by selecting a subject category from the drop-down list below, for example by selecting 'One-Variable Calculus'. Then click the 'APPLY' button directly, or select a subcategory to further refine your results.

Displaying 11 - 20 of 133

Andrew Wiles proves that Fermat's Last Theorem is false for integer exponents larger than $2$.  Using the Gelfond-Schneider Theorem on transcendental numbers, the author generalizes...

Using a result on periodic continued fractions, the author presents a rational function method of approximating square roots that is faster than Newton's method.

A real matrix is called square-palindromic if, for every base $b$, the sum of the squares of its rows, columns, and four sets of diagonals (as described in the article) are unchanged when...

When is the average of sums of powers of integers itself a sum of the ﬁrst $n$ integers raised to a power? We provide all solutions when averaging two sums, and provide some conditions...

In a classic pouring problem, given two unmarked jugs with capacities $m$ and $n$ pints, where $m$ and $n$ are relatively prime integers, and an...

In Sam Loyd's classical Courier Problem, a courier goes around an army while both travel at constant speeds. If the army travels its length during the time the courier makes his trip, how...

This note is on the digital (floating-point) representation in various arithmetic bases of the reciprocal of an integer $m$. An algorithm is given to change the representation of $1/m... A covering system is a system of \(k$ arithmetic progressions whose union includes all integers. This paper presents upper bounds on the number of consecutive integers which need to be...

The number $2$ is a quadratic residue mod $p$ if $p = 8k + 1$ or $p = 8k + 7$, but not if $p = 8k + 3$ or $p = 8k + 5$. This is proved by a simple counting argument, assuming the...

The author presents three solutions to a problem concerning the terms of a certain linear recurrence modulo prime numbers.