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Category Archives: number theory
The Sum of Cubes is the Square of the Sum
It’s fairly wellknown, to those who know it, that . In other words, the square of the sum of the first n positive integers equals the sum of the cubes of the first n positive integers. It’s probably less wellknown … Continue reading
Posted in number theory
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No Integer Solutions to a Mordell Equation
Equations of the form are called Mordell equations. In this post we’re going to prove that the equation has no integer solutions, using (with one exception) nothing more complicated than congruences. Theorem: There are no integer solutions to the equation … Continue reading
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A Lesson on Converting Between Different Bases
We’re in the time of COVID19, and that has meant taking far more direct responsibility for my children’s learning than I ever have before. It’s been a lot of work, but it’s also been fun. In fact, I’ve been surprised … Continue reading
Posted in number theory
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Strong Induction Wasn’t Needed After All
Lately when I’ve taught the second principle of mathematical induction – also called “strong induction” – I’ve used the following example to illustrate why we need it. Prove that you can make any amount of postage of 12 cents or … Continue reading
Posted in number theory, proof techniques
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Arguments for 0.9999… Being Equal to 1
Recently I tried to explain to my 11yearold son why 0.9999… equals 1. The standard arguments for (at least the ones I’ve seen) assume more math background than he has. So I tried another couple of arguments, and they seemed … Continue reading
Posted in arithmetic, number theory
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A Quasiperfect Number Must Be an Odd Perfect Square
Let be the sum of divisors function. For example, , as the divisors of 5 are 1 and 5. Similarly, , as the divisors of 6 are 1, 2, 3, and 6. A perfect number is one that is equal … Continue reading
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The Validity of Mathematical Induction
Suppose you have some statement . Mathematical induction says that the following is sufficient to prove that is true for all natural numbers k. is true. For any natural number k, if is true, then is true. The idea is that the first … Continue reading
Posted in number theory, proof techniques
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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
Posted in binomial coefficients, number theory
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Integrality of the Catalan Numbers via Kummer’s Theorem
Why is the nth Catalan number, , an integer? If you know one of its combinatorial interpretations, then the answer is clear, but how do you get integrality strictly from this formula? In this post I’m going to discuss how one can … Continue reading
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
Posted in binomial coefficients, number theory
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