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Browse Classroom Capsules and Notes

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Displaying 1131 - 1140 of 1209

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...

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

The number of black squares in a crossword puzzle, and the number of marks needed to bound the black squares allows one to determine the number of words the puzzle contains.

In this note an elementary proof of Stirling's asymptotic formula for \(n!\) is given.

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 checked...

The authors explain certain inequalities visually.

A Fibonacci identity is depicted visually.

The authors study the upper bound for the expected differences between order statistics of a distribution.

Using just very basic knowledge of finite fields and the inclusion-exclusion formula, the authors show how one can see the shape of Gauss` formula for the number of irreducible polynomials of...

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