# Browse Classroom Capsules and Notes

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Displaying 51 - 60 of 133

The authors investigate how to partition the integers into three arithmetic progressions.

The authors consider expansions of rationals to any base $b$, with special attention to inverses of Wieferich primes.

The author discusses the solutions to the two Diophantine equations $x^2 + y^2 = z^2 + 1$ and their relations.

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

The author gives an elementary justification for the representation of $\pi$ involving Catalan numbers.

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 provides a new way of finding all rational solutions of $x^y = y^x$ as well as a historical survey of this problem.
The author finds a pair of positive integers that generates a generalized Fibonacci sequence containing no prime numbers.