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Browse Classroom Capsules and Notes

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Displaying 1 - 10 of 133

Two Irrational Numbers from the Last Nonzero Digits of \(n!\) and \(n^n\)

The author considers two infinite decimals, where the \(n\)th digit is the last non-zero digit of \(n!\), creating the number \(F\), and using \(n^n\), creating the number \(P\). The author shows...

Proof Without Words: How did Archimedes Sum Squares in the Sand?

The geometry of Archimedes` proof of a formula for a sum of squares is depicted visually.

Fibonacci Identities via the Determinant Sum Property

This capsule uses the determinants of matrices to study Fibonacci numbers. Specifically, the sum property of the determinant is used to derive identities between Fibonacci numbers.

Using Random Tilings to Derive Fibonacci Congruence

This capsule follows the technique of random tilings used in the proof of the closed form for Fibonacci Numbers. By relaxing the condition of probability, the authors are able to obtain...

Parity and Primality of Catalan Numbers

Based on a closed form and recurrence relation of Catalan numbers, the authors proves their parity and primality.

Searching for Möbius

This capsule introduces a way of producing the Möbius function in Number Theory through the algebra of formal series. The author hopes that this exposition can lobby more mathematicians...

No Arthmetic Cyclic Quadrilaterals

Based on the notion of "arithmetic triangles," arithmetic quadrilaterals are defined. It was proved by using an elliptic curve argument that no such quadrilateral can be inscribed on...

Streaks and Generalized Fibonacci Sequences

This capsule discusses an alternative way of examining the Fibonacci sequence. As a result, a class of generalized  Fibonacci sequences of numbers can be defined.

Summing Cubes by Counting Rectangles

Starting with a homework problem on combinations, the capsule applies the "checkerboard" logic to derive identities involving summing squares and cubes.

Writing Numbers in Base \(3\), the Hard Way

Every integer can be expressed in base \(2\) using the set \(\{-1, 0, 1\}\) as coefficients. Does one need to use this set, or might another set of numbers do as well? The author investigates...

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