Episode #125 of the Stack Overflow podcast is here. We talk Tilde Club and mechanical keyboards. Listen now
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

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

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 ...


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 ...


12

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 ...


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 ...


10

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 ...


10

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.) ...


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

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 (...


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

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?


8

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.


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....


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

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 ...


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

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 ...


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

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 ...


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 $\...


6

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 theory, ...


5

Euler's original heuristic may "fly" with people who don't know about calculus: just as polynomials (e.g., with all real roots) are essentially products of linear factors $x-\alpha$ where $\alpha$ runs through the roots, one might imagine that $\sin \pi z$ (using radian measure) is a products of something like $1-{z\over n}$ for $n$ non-zero integer, and ...


5

I like to ask my probability students the question: If you pick an integer between 2 and 100 uniformly at random, what's the probability that it's the average of two (not necessarily different) primes? I like that above question (easily equivalent to Goldbach) because there's no preference for even versus odd numbers as in Goldbach. It also gets ...


5

I am not sure whether this really answers your question, but I could think of the following strategy of introducing these sets of numbers. It is not based on any kind of research and just based on personal experience with first year university students where I sometimes give something similar as a "naive idea" in addition to the proper definitions. The ...


5

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 ...


5

Look at patterns in decimal expansions: what is the period of the repeating decimal of $1/n$? From numerical data, the period is at most $n-1$, and you only get equality when $n = p$ is prime (but not conversely: $1/11$ has decimal period $2$, not $10$). If you look at the period of the decimal of $1/p$ for primes $p$ besides $2$ and $5$, you find that ...


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