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Category Archives: number theory
Some Corrections to My Paper “A Combinatorial View of Sums of Powers”
A few months ago Mathematics Magazine published a paper of mine, “A Combinatorial View of Sums of Powers.” In it I give a combinatorial interpretation for the power sum , together with combinatorial proofs of two formulas for this power … Continue reading
Posted in combinatorics, number theory, publishing, Stirling numbers
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A Connection Between the Riemann Zeta Function and the Power Sum
The Riemann zeta function can be expressed as , for complex numbers s whose real part is greater than 1. By analytic continuation, can be extended to all complex numbers except where . The power sum is given by . … Continue reading
Posted in Bernoulli numbers, number theory
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The Sum of Cubes is the Square of the Sum
It’s fairly well-known, 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 well-known … 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
Posted in number theory
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A Lesson on Converting Between Different Bases
We’re in the time of COVID-19, 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 11-year-old 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
Posted in number theory
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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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A Narrow Margin 