Browse Classroom Capsules and Notes

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On the Sum of Consecutive \(K\)th Powers

The author provides a direct and elementary proof of the Bernoulli formula for the sum of consecutive \(K\)th powers.

Proof without Words: \(\pi^e < e^{\pi}\)

The author presents a geometric proof of \(\pi^e < e^{\pi}\) by using a suitable function.

Special Sets of Smith Numbers

Using doubles of certain prime numbers, the author finds large Smith numbers.

False Conjecture Is True ne-Way

A plausible (but false) conjecture about the maximal product of equal summands for a given sum is modified and then proven, using basic properties of continued fractions.

The Abundancy Ratio, a Measure of Perfection

The author studies and derives properties of abundancy -- the ratio of the sum of a number's divisors to the number itself.

A Periodic Property in \(Z_m\)

The authors determine the positive integers \(n\) and \(m\) for which \(x^n=x\) holds for all integers \(x \mod m\).

Perfect Cuboids and Perfect Square Triangles

The author shows that two famous problems, one for triangles and one for parallelepipeds, are equivalent.

Proof without Words: A Triangular Sum

A sum is visually presented.

Partitions into Consecutive Parts

The authors state and prove a theorem on the number of partitions of an integer into consecutive parts.

Proof without Words: Pythagorean Triples and Factorizations of Even Squares

A 1-1 correspondence between Pythagorean triplets and the factorization of certain even squares is exhibited.