Browse Classroom Capsules and Notes

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Proof without Words: Sum of Squares

The sum of squares formula is proved visually.

Proof without Words: Sum of Integers

A visual derivation of the formula for the sum of the first \(n\) integers is presented.

Integral Solutions to the Equation \(x^2 + y^2 + z^2 = u^2\): A Geometrical Approach

Six identities, each of which gives infinitely many (but not all) integral solutions to the equation in the title, are shown to be special cases of a more general identity.

A Certain Type of Number Expressible as the Sum of Two Squares

Integers equaling the sum of the squares of their contiguous parts (a generalization of two-digit numbers which are the sum of the squares of their digits) are discussed, with several examples...

Permutable Primes

The article discusses prime numbers which remain prime after the permutation of the digits.

Proofs without Words: Count the Dots

This article contains a geometric demonstration of an identity for integers.

A Number-Theoretic Sum

The author presents a new Identity for integer functions and their inversion formulas.

Why Study Equations over Finite Fields?

The article discusses the reason for studying finite fields.

Smith Numbers

A Smith number is a composite number the sum of whose digits is the sum of all the digits of all its prime factors. The author reviews the history of these numbers, starting with the proof that...

Proof without Words: A Triangular Identity

A geometric demonstration of a recursive identity for triangular numbers