# Tag Info

### 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 ...
• 13.4k
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 ...
• 304
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 ...
• 18.1k

### 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 ...
• 8,713

### 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 ...
• 1,341

### 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}) \...
• 4,755

### 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 ...
• 4,755

### 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 (...
• 7,572

### 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?
• 2,420

### 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 ...
• 4,755