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Browse Classroom Capsules and Notes

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Displaying 21 - 30 of 133

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

A wide-ranging discussion of the question in the title of the paper is presented.

A visual proof for the sum of octagonal numbers is presented.

The authors investigate the occurrence of composite values of non-constant polynomials with integer coefficients evaluated at integer points.

Golomb's theorem on \(x/\pi(x)\) about prime number distribution is generalized and has a new proof.

Many calculations in number theory have different rules for different residue class mod an integer \(r\). The author shows how to combine these cases using a single amalgamated formula....

This article gives an explanation why the solution to a number partitioning problem would be \(3\). The answer is that \(3\) is close to \(e\).

A combinatorial proof of the sum of the cubes of the first \(n\) integers is presented, by counting edges in complete bipartite graphs.

This paper offers a visual illustration of the fact that every octagonal number is the difference of two squares.

By providing increasingly simpler test functions, this note places in context a primality test developed by Dennis P. Walsh ("A curious test for primes," this Magazine 80(4), October...