Category Archives: Elementary Number Theory
An Interesting Problem Involving a Partition of $\mathbb{N}\setminus \left\lbrace 0\right\rbrace$
Suppose the positive integers are partitioned as $\left\{\left\{1\right\},\left\{2,3\right\},\left\{4,5,6\right\},\ldots\right\}.$ Call the elements of the partition $A_{n}$ where $A_{n}$ contains $n$ integers. Then, what is the sum of the integers in $A_{n}$? Call these sums $S_{n}.$ We can calculate a few sums … Continue reading
Freshman’s Dream Come True! (But only in characteristic $p$)
A well-known fallacy committed by students is the so-called “Law of Universal Linearity” (the link is to a discussion of this phenomenon on Mathematics Stack Exchange). The most famous example of this is the statement $$\left(x+y\right)^n = x^n + y^n,$$ … Continue reading
Divisibility Rules: An Introduction to Modular Arithmetic
There are many rules for determining whether a number is divisible by another. For example, we know that a number is even (divisible by 2) whenever the one’s digit is even. We also know that a number is divisible by … Continue reading