# Browse Classroom Capsules and Notes

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Displaying 21 - 30 of 133

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

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...

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.

Describing Pythagorean Triples with one square side and a triangular side. Their number is infinite.

Characterize positive integers which can be written as a sum of consecutive integers

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...

The author presents a simple proof of a special case of Dirichlet`s celebrated theorem on primes in arithmetic progressions.

Solving Pell Equations using Fibonacci-like sequences