Category Archives: math

Anything mathematical.

Art vs. science in mathematical discovery

Is mathematics science or art? Mathematics resembles science. In math, data and examples are collected and hypotheses made. There is a difference in the hypotheses of math versus science: in the former they can be proved, but that could be considered a small point. But I have met mathematicians who don't consider themselves scientists, and […]

A little intro to the Jacobi symbol: Part 3

This is the final post on the Jacobi symbol. Recall that the Jacobi symbol $(m/n)$ for relatively prime integers $m$ and $n$ is defined to be the sign of the permutation $x\mapsto mx$ on the ring $\Z/n$. In the introductory post we saw this definition, some examples, and basic properties for calculation purposes. In Part […]

Ideal in a union of ideals

Suppose $I$ is an ideal in a ring $R$ and $J,K$ are ideals such that $I\subseteq J\cup K$. Then either $I\subseteq J$ or $I\subseteq K$. Indeed, suppose that there is some $x\in I$ such that $x\not\in J$. If $y\in I$ is arbitrary and $y\not\in K$ then $x + y$ is in neither $J$ nor $K$. […]

Sum of squares a square?

The sum of squares of two integers can be an integer. For example, $3^2 + 4^2 = 25^2$. Those are also the legs of a right-angle triangle. This example is special in that the numbers $3$ and $4$ are consecutive integers! Can you find a sum of squares of three consecutive integers that make a […]