Browse Classroom Capsules and Notes

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Proof without Words: Sum of Products of Consecutive Integers

A geometric demonstration of an identity for the sum of products of consecutive integers.

Proof without Words: The Sum of a Positive Number and Its Reciprocal Is at Least Two (four proofs)

Four geometric demonstrations of an inequality: the sum of a positive number and its reciprocal is at least \(2\)

Coin Flipping, Dynamical Systems, and the Fibonacci Numbers

This article links two seemingly unrelated problems, one in probability and the other in dynamical systems, and shows they are actually one involving Fibonacci numbers.

A Fibonacci-Like Sequence of Composite Numbers

The author finds a pair of positive integers that generates a generalized Fibonacci sequence containing no prime numbers.

On the Rational Solutions of \( x^y = y^x \)

The author provides a new way of finding all rational solutions of \( x^y = y^x \) as well as a historical survey of this problem.

Quadratic Residues and the Frobenius Coin Problem

An odd prime \(p\) has \((p-1)/2\) quadratic residues mod \(p\), and for relatively prime \(p\) and \(q\) there are \((p-1)(q-1)/2\) non-representable Frobenius numbers. The author discusses a...

Proof Without Words: Alternating Sums of Squares of Odd Numbers

A visual solution for an alternating sum of odd squares is presented.

Nonexistence of a Composition Law

It is known that every positive odd integer can be expressed in the form \(x^2 + y^2 + 2z^2\) for some integers \(x\), \(y\) and \(z\). The authors discuss the non-existence of certain...

On Candido's Identity

The authors discuss the existence of functions from the nonnegative reals to the nonnegative reals that satisfy the functional equation underlying Candido`s identity.

Polynomial Congruences and Density

The author discusses the distribution of solutions to a polynomial congruence as the modulus varies.