# Tag Info

15

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 proven (and will prove) to interact interestingly with other ideas, but some seem not to. It's not easy for a beginner to guess the significance or subtlety of one ...

14

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 good proof, you might get the desired enthusiasm by just showing him the proof! That said, here's an idea. Forget primes for a moment, and think about ...

14

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 trees (where the prime factors at the end are circled). The prime definition avoids some of the caveat-language otherwise seen ("a number only divisible by ...

14

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 problem where the answer is actually opposite. Either they will explain why it does not apply there, and you can argue that the difference is subtle enough to warrant ...

13

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 assume it is finite. And, from this pool of prime numbers, you can construct a new one that is not in the family. Why don't you let your brother play with that ...

11

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, $$(2 + \sqrt{10}) \cdot (2 - \sqrt{10}) = -6 = -2 \cdot 3,$$ so prime factorizations in $\mathbb{Z}[\sqrt{10}] = \{a + b\sqrt{10}: a, b \in \mathbb{Z}\}$ aren't unique. On the other ...

11

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 have unique prime factorization), but that justification needs to be made explicit, and they need to make sure that the fundamental theorem of arithmetic has ...

11

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 (no compilation needed) it supports big integers via the bignum type. (But using SageMath I think this is tangential, I mention it for completeness mainly.) ...

10

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?

9

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 literally the same expression as "5 - 3", but they provably represent the same number.) If we could only prove things by repeating the same statement, we couldn't ...

9

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 $\gcd(a,b) = ra+sb$ of the (extended) Euclidean Algorithm.

9

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 unlikely. Students struggle with fractions in primary school. Students usually struggle with proofs in gymnasium or university. It seems unlikely that primary school ...

9

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 in the base set, because the units are different. Therefore, if applying this to the real world, I suggest considering the repeated-addition interpretation (...

9

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 can draw an $a\times b$ grid as shown below. Now as you said we can show the greatest common divisor $d$ is the side length of the largest square that can ...

8

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 course in it. The first half of the textbook can be used, with some care, as an advanced undergraduate textbook. Hardy & Wright is an introductory number ...

8

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, depends on their previous exposure, inclination, course content, and ability. So...in my opinion.. both books are classics, which means they have been around ...

8

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 class for two or three pedagogical reasons: with some trial and error (and the help of some computational software), I have the students essentially discover ...

8

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. What they are exactly depends on ones convention (non-negative remained, smallest absolute value, or still something else) but if one fixes a convention then ...

8

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 Spirograph also provides excellent motivation for more abstract concepts such as cyclic groups.

8

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 generalized: If you subtract a natural number from its square, then the result is even. I wrote more about this in some earlier StackExchange posts, e.g., MESE ...

8

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 extending this work into the older years of one's education, a potentially good place to look would be APOS Theory, which is due to Ed Dubinsky and collaborators....

8

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 Mathematical Society can be found here. To quote from the latter (emphasis added by me): An Illustrated Theory of Numbers gives a comprehensive introduction to number ...

8

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 grade-school level since we have ten fingers and humans have learned only decimal arithmetic for everyday use. I just finished four weekly sessions with fifth ...

7

(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 believe one witnesses similar phenomena when covering the standard algorithm for division. For example, I observe analogs with your remarks of: The flexible method ...

7

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 diverge to infinity in this Collatz-like sequence?" But: (a) there is a huge start-up learning curve, and (b) Mathematica is not free. Because of the latter, I ...

7

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 together" seems very reasonable and even playful. Why is this theoretical? It is basically saying that if you multiply all units together you get a specified ...

7

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, vector spaces over $\mathbb{Q}_p$, and the metrically completed algebraic closure of $\mathbb{Q}_p$. There is also some discussion of $p$-adic analysis near ...

6

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) without having to use a formula. Thus, I would get him thinking about primes more symbolically. Given 3 primes, $p_1, p_2, p_3$ what is their LCM? What is ...

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 Basel formula to be hard work. Probably the best proof from the point of view of transparency is obtained by evaluating \int_0^1\int_0^1 \frac{1}{1-xy}{\rm d}x{\...

6

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, and with each row composed of children with the same gender. What is the largest number of children per row that satisfies these constraints? There should be \$\...

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