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Browse Classroom Capsules and Notes

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Displaying 121 - 130 of 133

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

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....

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

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

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

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

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

The author presents three solutions to a problem concerning the terms of a certain linear recurrence modulo prime numbers.

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...

A covering system is a system of \(k\) arithmetic progressions whose union includes all integers. This paper presents upper bounds on the number of consecutive integers which need to be...

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