15 votes

Why do we need perfect numbers?

I think it turns out that "perfect" numbers do not interact much with other parts of number theory. Some of these very old, elementary, very ad-hoc definitions of special classes of integers have ...
user avatar
  • 13.4k
14 votes
Accepted

How to arrive at infinitude of primes proof?

It might not be possible to get your brother to arrive at the proof himself, no matter how much you scaffold it. ("You can lead a horse to water" and all that.) If he's into maths and appreciates a ...
user avatar
14 votes
Accepted

Good way to explain fundamental theorem of arithmetic?

The way I have explained the fundamental theorem of arithmetic in the past is by establishing what it means to be prime (has exactly two positive divisors) and then having students construct factor ...
user avatar
14 votes

How to explain what is wrong in this "proof" that $\sqrt N$ must be irrational?

This is indeed tricky, and it seems to me the most effective way (in far more general, similar situations) is to show them the problem would be to have them apply their method to another, close ...
user avatar
13 votes

How to arrive at infinitude of primes proof?

I don't really answer the question but: why do you want your brother to come to the answer right now? Now, your brother understands that, to prove that the set of prime numbers is infinite, you can ...
user avatar
  • 1,341
11 votes

Good way to explain fundamental theorem of arithmetic?

To really understand why the integers $\mathbb{Z}$ have unique prime factorization, it helps to understand how unique prime factorization can fail in other settings. For example, $$(2 + \sqrt{10}) \...
user avatar
  • 4,755
11 votes

How to explain what is wrong in this "proof" that $\sqrt N$ must be irrational?

What's wrong is that most of the justification is missing. Why can $N = p^2/q^2$ only happen if $q^2 = 1$? This can be justified using the fundamental theorem of arithmetic (which states that integers ...
user avatar
  • 4,755
11 votes

Number theory for self-study students: books and computer languages

I would recommend Python combined with SageMath, as already recommended by Joseph O'Rourke, or rather SageMath and Python comes naturally. Python is a modern, and widely used, interpreted language (...
user avatar
  • 7,572
11 votes

Greatest common divisor applications

Another classic is the following: A rectangular floor measures $300 \text{ cm} \times 195 \text{ cm}$. What is the largest square tiles that can be used to cover the floor exactly?
user avatar
9 votes

When two equivalent algebraic statements have two "different" meanings

The two statements aren't literally "the same", because as the student observed, they say different things. However, they are logically equivalent: each implies the other. (Similarly, "4/2" isn't ...
user avatar
  • 4,755
9 votes

Greatest common divisor applications

Going back to Euclid, I have found questions such as `Given a large supply of rods of length $15$ and $21$, what lengths can be measured?' can appeal to students. This also motivates the result $\gcd(...
user avatar
9 votes

Could students learn a lot more from school if they're only taught number theory until way later?

The way I understand the question is: If students are not taught fractions, but instead formal deductive proofs of properties of natural numbers, would they learn mathematics better? I find it ...
user avatar
  • 4,544
9 votes

Analogy for multiplying modulo N

Maybe the issue is that if the values you're multiplying have units, then the result of multiplying will have the product of those units. Therefore, your result can't really be equivalent to a value ...
user avatar
  • 7,686
9 votes
Accepted

Geometrical interpretation of the identity $\operatorname{lcm}(a,b) \operatorname{gcf}(a,b) = ab$

I don't see anything wrong with what you were trying to say. In fact, I think that is a good way of saying it. Let's draw some pictures with some concrete numbers. Let's take $a=6$ and $b=9$. Then we ...
user avatar
8 votes

Differences between Hardy&Wright and Ireland&Rosen for number theory course

Ireland & Rosen is a graduate textbook which starts with general number theory and moves to algebraic number theory. Abstract Algebra is not strictly required, but students would benefit from a ...
user avatar
  • 181
8 votes
Accepted

Differences between Hardy&Wright and Ireland&Rosen for number theory course

Some caveats: I own both books, and have taken number theory courses up to graduate level. Also, your questions are clearly somewhat subjective: what is difficult, relevant or complete for one person, ...
user avatar
  • 466
8 votes

Why do we need perfect numbers?

Mathematically, even perfect numbers give a good number theory example to the general idea of classification, i.e. all even perfects have a specific form. I use perfect numbers in my number theory ...
user avatar
  • 6,350
8 votes
Accepted

Should Euclid's algorithm be taught as rigid or flexible?

There is some value beyond the algorithm to insist on the fact that quotient and remainder in an Euclidean division are uniquely determined as soon as one settles on some convention on the remainder....
user avatar
  • 7,572
8 votes

Greatest common divisor applications

Some nice geometrical applications arise in the analysis of periodical curves such as Roulettes (Spirograph curves), Star Polygons, etc. Concrete experience with implementations in toys like ...
user avatar
  • 1,009
8 votes

What is most motivating way to introduce Fermat's Little Theorem

I wrote an article for NCTM's grades 8-14 journal, The Mathematics Teacher, which is about building towards ${\rm F}\ell{\rm T}$ by thinking through ways in which the following statement can be ...
user avatar
8 votes
Accepted

What is the name of this discipline in mathematics education?

What is the name of this subdiscipline in math education? In one of the comments, Dave L Renfro has a reference to Piaget, whose work was primarily done with younger schoolchildren. With respect to ...
user avatar
8 votes
Accepted

Source material to study number theory?

I've heard very good reviews of the 2017 book, "An Illustrated Theory of Numbers" by MH Weissman. The book's main site is here; a write-up, along with some reviews, by the American ...
user avatar
8 votes

Is there a numerical base that is in any way “better” for simple mathematical calculations than others?

Clearly there is no historical data that addresses this question I want to know if there are any numerical bases that are notably well-suited for humans to learn and use at an elementary or ...
user avatar
7 votes

Should Euclid's algorithm be taught as rigid or flexible?

(Summary: I would suggest exploring it flexibly, but ensuring students also know the "rigid" version.) Rather than directly addressing Euclid's algorithm for the $\gcd$ of two whole numbers, I ...
user avatar
7 votes

Number theory for self-study students: books and computer languages

What computer languages might one recommend for, say, investigations in number theory? I find Mathematica ideal, e.g.: "Mod sequences that seem to become constant; and the number 316" "Does 53 ...
user avatar
7 votes

How to introduce Wilson's Theorem?

Wilson's Theorem is very powerful for proving things related to quadratic residues, and I think also in general a good theoretical tool. Motivating it by just saying "hey let's multiply everything ...
user avatar
  • 5,752
7 votes

Introductory book or other resource on $p$-adic numbers/number theory/analysis

A few recommendations: Fernando Q. Gouvêa's $p$-adic numbers: An Introduction. This text gives a good introduction to the $p$-adic number system and the properties of the space of $p$-adic numbers, ...
user avatar
6 votes

How to arrive at infinitude of primes proof?

I think using actual small primes actually detracts from finding the solution. If you are thinking about finding a number that is not divisible by 2,3, or 5, it is easy to come up with one (11) ...
user avatar
6 votes
Accepted

Intuition behind $\zeta(2) = \frac{\pi^2}{6}$

You probably already know that $1/\zeta(2)$ is the probability of two "random" integers being relatively prime. Since this constant is related to number theory, you should expect any proof of the ...
user avatar
6 votes

Greatest common divisor applications

Here's a word problem for the greatest common divisor: 12 boys and 15 girls are to march in a parade. The organizer wants them to march in rows, with each row having the same number of children, ...
user avatar

Only top scored, non community-wiki answers of a minimum length are eligible