Browse Classroom Capsules and Notes

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A Reformulation of the Goldbach Conjecture

The author gives a new formulation of the conjecture.

A Lesser-Known Goldbach Conjecture

Goldbach's conjecture on the expression of odd numbers and some examples

In the Gaussian Integers \(\alpha^4 + \beta^4 \ne \gamma^4\)

A simple proof for Last Fermat Theorem when n=4 in Gaussian Integers

Proof without words

A simple geometric proof for the formula \(1 + 3 + 5 + \ldots + (2n - 1) + (2n + 1) + (2n - 1) + \ldots + 5 + 3 + 1 = n^2 + (n + 1)^2\)

A Graph of Primes

The graph obtained by constructing an edge between each pair of primes with difference a power of 2 is not connected.

A Dynamical Systems Proof of Fermat's Little Theorem

An interesting unusual visual and rigorous proof of Fermat`s Little Theorem, with unusual generalizations

When Is the Unitary Group Cyclic? An Algebraic Approach

This article presents an algebraic proof of the primitive root theorem.

Mutual Multiples in \(Z_n\)

If two elements in a ring are multiples of each other, are they necessarily unit multiples? In general, the answer is no; it is yes for the integers mod \(n\).

Fermat's Little Theorem: A Proof by Function Iteration

The author proves Fermat's little theorem using iteration of functions.

Solutions of \(X^n + Y^n = Z^{n+1}\)

The author constructs infinitely many integer solutions to the title equation \(X^n + Y^n = Z^{n+1}\).