Browse Classroom Capsules and Notes

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A Curious Way to Test for Primes

Primality testing via repeated differentiation of certain functions is the topic of this paper.

Golden Matrix Ring Mod \(p\)

The author uses the golden matrix ring \(Z(A)\) generated by \(A= \left[ \begin{array}{cccc} 0 & 1 \\ 1 & 1 \end{array} \right] \) to prove certain identities involving Fibonacci...

Proof Without Words: Fibonacci Tiles

The author gives geometrical proofs of a number of identities for the Fibonacci numbers.

Fibonacci-Like Sequences and Pell Equations

Solving Pell Equations using Fibonacci-like sequences

Sums of Consecutive Integers

Characterize positive integers which can be written as a sum of consecutive integers

Pythagorean Triples with Square and Triangular Sides

Describing Pythagorean Triples with one square side and a triangular side. Their number is infinite.

Dirichletino

The author presents a simple proof of a special case of Dirichlet`s celebrated theorem on primes in arithmetic progressions.

A Curious Way to Test for Primes Explained

By providing increasingly simpler test functions, this note places in context a primality test developed by Dennis P. Walsh ("A curious test for primes," this Magazine 80(4), October...

Proof Without Words: Every Octagonal Number Is the Difference of Two Squares

This paper offers a visual illustration of the fact that every octagonal number is the difference of two squares.

A Graph Theoretic Summation of the Cubes of the First \(n\) Integers

A combinatorial proof of the sum of the cubes of the first \(n\) integers is presented, by counting edges in complete bipartite graphs.