54

"Because we said so" is a bit of a conversation closer, I agree. But "Because some people agreed a long time ago to define it that way so we could have conversations where we all understood each other. Does that seem like it would be a good idea?" is both more inclusive and more correct. I don't even think it's that hairy to talk through the FTA with ...


35

A really nice example is factorization of the Hilbert numbers - that is the numbers $$1,\,5,\,9,\,13,\,17,\,\ldots,4n+1,\,\ldots.$$ Now, we can talk about factoring in this domain too - for instance, $45=5\times 9$ is a factorization of $45$. Moreover, $5$ and $9$ are both "Hilbert Primes" - they are not the product of two other Hilbert numbers - meaning ...


34

There was a multiplication table posted on the wall. Like this \begin{alignat}4 1 &\quad 2 &\quad 3 &\quad 4 &\quad\cdots\\ 2 &\quad 4 &\quad 6 &\quad 8 &\quad\cdots\\ 3&\quad 6 &\quad 9 &\quad 12 &\quad\cdots\\ 4&\quad 8 &\quad 12 &\quad 16 &\quad\cdots\\ \vdots&\quad \vdots &\quad \...


22

Integer factorization itself isn't unique: for example, $6 \cdot 2 = 4 \cdot 3$. The remarkable theorem is that there is only one factorization (up to reordering and sign) in which all the factors are prime. Or, you could give an example where unique factorization into irreducibles genuinely fails. If we look at numbers of the form $m + n \sqrt{10}$, where $...


18

You can restrict to subsets of the integers to get nice examples of structures that do not have unique factorization. For example, take the set of all natural numbers, or integers it does not really matter, that are congruent to $1$ modulo $4$. This is a semigroup with identity under multiplication, and every element is the product of irreducible elements. ...


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 ...


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

You can "factor" every positive integer into a sum of distinct Fibonacci numbers. This is not unique. However, it is unique if you add the condition that no two summands are adjacent Fibonacci numbers.


10

FYI: here's some pro and con: http://primefan.tripod.com/Prime1ProCon.html One was originally considered prime. It is prime with the most convenient ("natural") definition. It got excluded from prime-ness because many other higher theorems would be complicated by leaving it as prime. Essentially "prime" -> "prime*". The definition of primeness was ...


9

Your student may be interested in solving the Project Euler series of problems. These are programming exercises to solve mathematical challenges, many of which deal with number theory. All of the exercises are designed such that the running time on a normal personal computer should be not much more than a minute. However, achieving reasonable running ...


8

How should one talk about the question of 1 or 0 being prime ... with primary or middle school children? Depending on what you did before you will have an easy or a hard task: If the children were told: A prime number is a natural number which cannot be divided by other numbers than by 1 and by itself. ... you will have problems explaining why 1 is not ...


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 ...


5

This one is really very simple. First, tell them what a prime number is: A prime number has exactly two different factors. (If they don't know what factors are, and they ask about primes, the correct answer is "well, first you have to know about factors...") With that definition, it is very easy to figure out 0 and 1. Is 1 a prime? No, because it only ...


5

A good way to lead to the uniqueness of prime factorization and the convention that $1$ is not a prime is to build factor trees (that's common in elementary school these days in fourth grade, sometimes third grade). 24 24 24 8 3 6 4 2 12 2 4 3 2 2 2 3 ...


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 know this might be perhaps be too advanced, but perhaps you can look at other integer-like structures, (non unique factorization domains) like e.g. the classical counterexample $\mathbb Z[\sqrt{-5}]$ where $6= 2 \cdot 3= (1+\sqrt{-5})(1-\sqrt{-5})$ (which are all primes). Perhaps someone can come up with another example, but I fear this is the simplest one....


5

side note: If he is not using iPython, he might give it a go. There are free cloud based iPython notebooks and it is an excellent environment for investigations, with easy graphing etc. A couple of my students of similar age use it. Test for primality rather than finding all primes First I would show him how to extend his research by tweaking his current ...


5

Ask them how much they would like to discover on their own and reinvent, and how much they would like to do research and verify results as a stepping stone to becoming an independent researcher. Ideally, the student will find resources on his own, such as the Prime Pages by Chris Caldwell, Ribenboim's books on primes and Diophantine equations, Crandall and ...


4

The problems you describe are the tip of an iceberg call "Computation Number Theory" and a google search should turn up a variety of resources at various levels. One possible route is to have the student learn about polynomial time vs. non-polynomial time algorithms and then try to implement the polynomial time primality testing algorthim (called the AKS ...


4

If you build each number n using n square blocks in rectangular configurations, there are multiple configurations for each composite number. (4 is 4 by 1 or 2 by 2.) The primes are the ones that can only be built as a 1 by n rectangle. It seems clear that 0 would be neither prime nor composite, when looked at this way. The easiest way to understand why we ...


4

I'm kind of restating what other answers have said, but I wanted to practice expressing it in the clearest, most concise way I could think of. (Coincidentally, this came up with my partner tonight, so I got a test-run with it, and got an entirely satisfying result). Consider only natural numbers (i.e, positive integers). It seems like the number of ...


3

You are correct in that the coprimality of $a$ and $b$ is not used in its full strength. It is adequate to merely assume that they are not both even. But since people are so used to reducing a rational number to lowest terms, making this assumption improves the readability of the proof. The unnecessary stronger assumption of coprimality is not used, but it ...


3

We don't need the full FTA upfront if we limit our discussion for the moment to obvious examples of what's necessary for a factorization to be unique. The FTA provides analogous sufficiency conditions they'll probably guess on their own, even if they don't know how it's proven. You can say, $1$ isn't considered prime because then there wouldn't be unique ...


2

I would start by showing them, on paper, what they already know - that in the context of multiplication the number 1 is useless. It is the identity function. It simply reflects the original number. It is a mirror. "And just like your reflection in a mirror is not a real person, neither is 1 a real number when multiplying (not to be confused with a Real ...


1

The problem here is defining primes in their own right rather than defining them in terms of factorization. Start with a number like 30. Writing 30 = 2 x 15 tells us something new; writing 30 = 1 × 30 doesn't. Writing 30 = 2 x 3 x 5 tells us something new again, whereas writing 30 = -2 x -15 doesn't. Once we get to 30 = 2 x 3 x 5, we can't break down any ...


1

\begin{align} & \begin{array}{cccccccccccccccccccc} & & & & & & & & & 840 \\[12pt] = {} & & & & & 28 & & & & \times & & & & & 30 \\[12pt] = {} & & & & 4 & \times & 7 & & & \times & & & & 5 & \times & 6 \\...


1

When I was at school, after being taught about integer division, I was told: "A natural number is prime if it has exactly 4 integer divisors" Then 2 is prime, as it can be divided by -2, -1, 1, and 2. Then one can be divided by -1 and 1, and those are only 2 divisors. Hence not prime. Zero can be divided by anything but itself, yielding zero, hence ...


1

(If your context does not include negative numbers, turn all the negatives below positive. This almost won't change the discussion.) Everything divides zero, so zero can't be prime. $0 \cdot 7 = 0$ means $0$ and $7$ divide $0$. $0 \cdot -8 = 0$ means minus eight also divides zero. Can we see that everything divides zero, so zero is very far from being ...


1

Too top down and too proofy/hard in places. Do something more experiential and experimental. "Explain the conjecture, provide some simple examples to begin" Nope, run an exercise and have students find if they can figure out the sums (small numbers). Even let group pick a couple small numbers (maybe two digit ones) and let group find sum that ...


1

Continuing with the prime theme there is a natural progression to prime polynomials. Then there's primitive polynomials.


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