# Browse Classroom Capsules and Notes

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Displaying 71 - 80 of 133

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

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

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

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 author studies and derives properties of abundancy -- the ratio of the sum of a number's divisors to the number itself.
The authors determine the positive integers $n$ and $m$ for which $x^n=x$ holds for all integers $x \mod m$.
The author shows that two famous problems, one for triangles and one for parallelepipeds, are equivalent.
A sum is visually presented.

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

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