# Browse Classroom Capsules and Notes

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Displaying 11 - 20 of 1135

Given a positive integer $$m$$, the authors exhibit a group with the probability $$1/m$$ that two randomly selected elements commute.

The number $$2$$ is a quadratic residue mod $$p$$ if $$p = 8k + 1$$ or $$p = 8k + 7$$, but not if $$p = 8k + 3$$ or $$p = 8k + 5$$. This is proved by a simple counting argument, assuming the...
The author presents three solutions to a problem concerning the terms of a certain linear recurrence modulo prime numbers.
How do sequences of the form $$(1+x/n)^{n + \alpha}$$ with $$x > 0$$ approach their limits?
A combinatorial proof of a formula for the sum in the title is provided.
The author uses Newton`s impact method to compute the centripetal force exerted on a particle moving uniformly on a circumference of a noneuclidean circle.
The author gives a mapping of the ordered pairs of positive integers onto the positive integers which is immediately recognizable as a bijection.

By exploiting the geometry of the cobweb plot, the author provides a simple and elementary derivation of the parameter for the period-three cycle of the logistic map.

The result in the title is demonstrated graphically.

This note uses the Gauss sums to evaluate certain sums of trigonometric functions. It includes a short history of Gauss sums.