54 votes

Explaining why (or whether) zero and one are prime, composite or neither to younger children

"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 ...
  • 5,569
35 votes

A Non-Unique Factorization of Integers!

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, $...
34 votes

Explaining why (or whether) zero and one are prime, composite or neither to younger children

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\\ ...
  • 6,552
22 votes

A Non-Unique Factorization of Integers!

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 ...
  • 4,813
18 votes

A Non-Unique Factorization of Integers!

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 ...
  • 7,612
11 votes

A Non-Unique Factorization of Integers!

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....
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10 votes

Explaining why (or whether) zero and one are prime, composite or neither to younger children

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 ...
  • 109
9 votes

Student Project about Prime Numbers: How to Continue?

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 ...
8 votes

Explaining why (or whether) zero and one are prime, composite or neither to younger children

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 ...
5 votes

Planning high school workshop on Goldbach Conjecture

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) ...
5 votes

Explaining why (or whether) zero and one are prime, composite or neither to younger children

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, ...
5 votes

A Non-Unique Factorization of Integers!

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 $\mathbb Z[\sqrt{-5}]$...
  • 379
5 votes

Student Project about Prime Numbers: How to Continue?

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 ...
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5 votes

Student Project about Prime Numbers: How to Continue?

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. ...
5 votes

Explaining why (or whether) zero and one are prime, composite or neither to younger children

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 ...
  • 51
4 votes

Explaining why (or whether) zero and one are prime, composite or neither to younger children

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 ...
4 votes

Student Project about Prime Numbers: How to Continue?

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 ...
  • 6,740
4 votes

Explaining why (or whether) zero and one are prime, composite or neither to younger children

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 ...
  • 18.7k
3 votes

Explaining why (or whether) zero and one are prime, composite or neither to younger children

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 ...
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3 votes

Missing Step in Most Proofs of the Irrationality of $\sqrt{2}$

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 ...
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2 votes

Explaining why (or whether) zero and one are prime, composite or neither to younger children

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 ...
  • 129
1 vote

Explaining why (or whether) zero and one are prime, composite or neither to younger children

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 × ...
1 vote

Explaining why (or whether) zero and one are prime, composite or neither to younger children

\begin{align} & \begin{array}{cccccccccccccccccccc} & & & & & & & & & 840 \\[12pt] = {} & & & & & 28 & & & & \times & &...
1 vote

Explaining why (or whether) zero and one are prime, composite or neither to younger children

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, ...
  • 339
1 vote

Explaining why (or whether) zero and one are prime, composite or neither to younger children

(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. $0 \cdot 7 = 0$ ...
1 vote

Planning high school workshop on Goldbach Conjecture

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 ...
  • 1,774
1 vote

Student Project about Prime Numbers: How to Continue?

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

A Non-Unique Factorization of Integers!

In addition to the $\sqrt{-5}$ example, there is also the $R=\mathbb Z[\sqrt{-3}]$ example, which fails to be unique factorization for the "secret" reason that it is not integrally closed: we should ...
  • 13.8k

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