# Browse Classroom Capsules and Notes

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Displaying 61 - 70 of 133

Convergence of a complex continued fraction can be analyzed using analysis, algebra, number theory, topology or complex analysis.

The proof of the irrationality of the $n$-th root of $k$ uses congruences.

Which numbers can be written as a sum of consecutive positive integers, such as 12 = 3 + 4 + 5? The answer: all positive integers except the powers of 2. The authors prove this result and...

Can one find three distinct right triangles, each having integral sides, all having the same area? Carroll conjectured an answer of yes; he was right.

How many numbers in the integer mod $n$ ring will be perfect squares?

The authors find those arithmetic progressions such that the sums of the squares of their terms can be represented by a binomial coefficient.

Two different approaches are given to determine which regular polygons, inscribed within or circumscribed about a unit circle, possess rational area or rational perimeter.

The authors obtain a recurrence relation that yields the $n$th digit in the binary expansion of any real number.

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

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