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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.