# Browse Classroom Capsules and Notes

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Displaying 41 - 50 of 133

The author shows that two famous problems, one for triangles and one for parallelepipeds, are equivalent.
The authors determine the positive integers $n$ and $m$ for which $x^n=x$ holds for all integers $x \mod m$.
The author studies and derives properties of abundancy -- the ratio of the sum of a number's divisors to the number itself.

The authors state and prove a theorem on the number of partitions of an integer into consecutive parts.

The authors provide a proof, accessible to beginning number theory students, of Chebychevs upper bound on the number of primes no greater than n.

After a brief historical discussion of the aliquot parts of an integer the authors use elementary arguments to prove a theorem about the sums of its principal divisors.

The author discusses an alternate way of developing Binets formula. See also a follow-up letter from Art Benjamin in Mathematics Magazine, Volume 78, No. 2, page 97.

A 1-1 correspondence between Pythagorean triplets and the factorization of certain even squares is exhibited.

Certain subsets of the ring of integers mod $n$ with hidden group structure are discussed.

An identity between certain squares and sums of fourth powers is exhibited, which can be used to establish a Fibonacci identity.