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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 Chebychev`s 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 Binet`s 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.