Category Archives: elementary number theory

Generalized Binomial Coefficients from Multiplicative and Divisible Functions

Given a function f from the natural numbers to the natural numbers, one way to generalize the binomial coefficient is via . The usual binomial coefficient of course has f as the identity function . Question: What kinds of functions f guarantee that … Continue reading

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Some Divisibility Properties of the Binomial Coefficients

Recently I read in Koshy’s book [1] on Catalan numbers some divisibility properties of the binomial coefficients I had not seen before.  Koshy credits them to Hermite.   They are particularly interesting to me because (as Koshy notes) some of the most famous … Continue reading

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Some Stirling Number Congruences

Here is Problem 51, parts a and b, in Chapter 6 of the classic text Concrete Mathematics: 51.  Let p be a prime number. a.  Prove that , for . b.  Prove that , for . In this post I will show that the converses … Continue reading

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Checking Multiplication via Digit Sums

Last week a friend who is a fourth grade teacher came to me with a math problem.  The father of one of his students had showed him a trick for checking the result of a three-digit multiplication problem.  The father … Continue reading

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A Combinatorial Proof of a Formula Involving Euler’s Totient Function

Several years ago I found a proof of the following identity, which I had not seen before (and still haven’t seen anywhere else): where is Euler’s totient function, the number of positive integers less than or equal to and relatively … Continue reading

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Perfect numbers and Pythagorean triples

For my first entry, let’s talk about a problem I posed in Mathematics Magazine several years ago.  (Plus it gives me a chance to mention Fermat, which I really should.) Two of the more famous topics in elementary number theory … Continue reading

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