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Questions tagged [number-theory]

For questions related to the teaching of number theory, the part of mathematics concerned with properties of the positive integers.

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176 views

Analogy for multiplying modulo N

Sometimes I want to explain to laymen/new students/laywomen how addition modulo N works. There are some instructive analogies: Addition on the clock (12), Addition on weekdays (7). They illustrate the ...
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3answers
210 views

Could students learn a lot more from school if they're only taught number theory until way later?

According to https://www.inc.com/bill-murphy-jr/science-says-were-sending-our-kids-to-school-much-too-early-and-that-can-hurt-th.html, when students get taught a concept when they're so young, they're ...
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1answer
134 views

Duodecimal by Stealth

It is widely recognised that the Duodecimal number system is superior to the decimal system. However, it is plainly obvious that trying to introduce such a system would be difficult, especially in a ...
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1answer
87 views

Making modular arithmetic interesting for school kids

This is a pattern even school kids could discover (when gently pointed to). I never did conciously, and cannot remember to have been pointed to explicitly, neither at school nor later: $$\color{red}{\...
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0answers
155 views

Succinct description of situations where naively obvious is correct, but for far more complicated reasons?

What is the name for a situation where the obvious thing turns out to be true, but the reasoning is more complicated? In abstract algebra we can say the rational numbers - the fractions, $\mathbb{Q}...
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1answer
108 views

Pythagorean triples

What is the most motivating way to introduct Pythagorean triples to undergraduate students? I am looking for an approach that will have an impact. Good interesting or real life examples will help. Is ...
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2answers
90 views

Subject advice in Number Theory [closed]

At my University, we have the optional feature to write a project like a Bachelor Thesis. This semester have finished and I would like to work in the summer in project like this. So, I'm searching for ...
4
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2answers
134 views

Source material to study number theory?

I don't know if this is the correct site to ask this question, but I felt it was off-topic on the Mathematics forum. I really like Number Theory and would like study some on my own. Which books should ...
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1answer
83 views

Does studying elementary number theory improve one's proof skills and ability to understand algebra and analysis? [closed]

I'm taking a number theory course and don't know whether it's worth it. I currently can't understand algebra and real analysis and decided to take # theory to see whether this would help me prove and ...
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2answers
139 views

What can we learn as we reduce the fraction at the end of the quadratic formula process? [closed]

In the final throes of the quadratic formula, you reduce a fraction. Consider the following two examples. $y = 6x^2 + 11x + 3$; the quadratic formula reveals the roots $x = -4/12$ or $x = -18/12$. ...
3
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1answer
123 views

“Seeing” GCD and LCM in Word Problems

Last year, I taught GCD and LCM and then gave my students word problem relating to these concepts ("Two runners with given different speeds; when will they meet again?", "Having three kinds of flowers ...
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288 views

Using number theory instead geometry to introduce proof in Basic School?

It seems there is an overall agreement that Geometry is the right place to introduce proof in Basic School. However, number theory (arithmetic) looks like to be a more simple environment (consider, ...
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2answers
116 views

Interesting math lesson on integers, Euclid's Elements, polyhedra, prime numbers, non-Euclidean geometry, arithmetic functions or graphs

I have to deliver a lecture for secondary school, about one of these topics: integers, Euclid's Elements, polyhedra, prime numbers, non-Euclidean geometry, arithmetic functions or graphs. It should ...
7
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1answer
208 views

What is the name of this discipline in mathematics education?

I am struggling with my students who can think only in concrete terms, they can compute with concrete numbers but are not able to think in terms of e.g. functions on natural numbers and come up with ...
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2answers
167 views

How to introduce Wilson's Theorem?

What is the most motivating way to introduce Wilson’s Theorem? Why is Wilson’s theorem useful? With Fermat’s little Theorem we can say that working with residue 1 modulo prime p makes life easier ...
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4answers
578 views

What is most motivating way to introduce Fermat's Little Theorem

What is the best way to introduce Fermat’s Little Theorem (F$l$T) to students? What can I use as an opening paragraph which will motivate and have an impact on why students should learn this theorem ...
8
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1answer
124 views

What specifically should I include to my self study notes?

I am a high school student who will be entering college in one and a half year (It is so long though). For me, Mathematics primarily means Number Theory. My interest in Number Theory always motivates ...
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6answers
1k views

Greatest common divisor applications

What are some real-life applications of gcd? I am looking for a motivating way of introducing this topic in an elementary number theory course.
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4answers
394 views

Number theory for self study students, books and computer languages

Sometimes students will contact me, as my email is visible. This time, an undergraduate in Sri Lanka has no number theory courses available and is self studying. My own experience is that it helps ...
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0answers
204 views

How do i deal with students who make these mistakes? [closed]

I came across some interesting mistakes in many area of mathematics with my students and do not let me also to tell you for university students level, I would like to know How do i deal with ...
15
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4answers
490 views

How to explain what is wrong in this “proof” that $\sqrt N$ must be irrational?

Here is the problem that I asked undergraduate students of an introductory number theory course to prove: Prove that if $N$ is a nonsquare natural number, then $\sqrt N $ is irrational. Many ...
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3answers
238 views

Planning high school workshop on Goldbach Conjecture

So I'm doing a mathematics education extension for my current undergraduate maths course, and for one bit of the final assessment we're asked to create a detailed lesson plan on the (strong) Goldbach ...
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3answers
401 views

When two equivalent algebraic statements have two “different” meanings

Suppose I want to prove $\sqrt{7}$ is not a rational number. I suppose it is and it brings me to a contradiction. Here how it goes line be line: First line. $\sqrt{7}=\frac{m}{n}$ Second line (...
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2answers
706 views

Good way to explain fundamental theorem of arithmetic?

Some students understand how this works, like they know what the theorem means, but, say, imagine some student asks why, not how. Not really proving the theorem, rather why does it exist or why is it ...
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2answers
237 views

Explaining difference between natural numbers, integers, rationals, reals, complex numbers, Gaussian integers

I am teaching an introduction to number theory for high schoolers right now, and there seems to be quite a bit of confusion on what the difference between the natural numbers, the integers, the ...
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3answers
912 views

Should Euclid's algorithm be taught as rigid or flexible?

Euclid's algorithm is a way to find the greatest common divisor of two natural numbers $a$ and $b$. In the usual version of the algorithm one tries to find $p,q\in\mathbb N$ so that $a=pb+q$ and $0\...
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1answer
153 views

Resources for Pell's equation

What is the best way to introduce Pell’s equation on a first elementary number theory course? Are there any practical applications of Pell’s equation? What are the really interesting questions about ...
5
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1answer
176 views

Self Teaching Theory for Olympiad. Need advice

(Cross-posted in MSE 1301476.) I want to start to do Olympiad type questions but have absolutely no knowledge on how to solve these apart from my school curriculum. I'm 16 but know maths up to the 18 ...
6
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2answers
176 views

Differences between Hardy&Wright and Ireland&Rosen for number theory course

My professor advised us to get either Hardy&Wright or Ireland&Rosen for our number theory course. I would like to ask what are the differences between these textbooks in terms of pedagogical ...
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3answers
255 views

Teaching number theory: geometric approach

Are there any books that are substantially based on a geometric approach to explain topics in number theory (elementary and more advanced)? If so, is such approach -- judging from your teaching (or ...
6
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4answers
360 views

Intuition behind $\zeta(2) = \frac{\pi^2}{6}$

The result $$\zeta(2) = \frac{\pi^2}{6},$$ tends to amaze young students because of its beauty. However, although in literature there are many proofs of this result, I find that none is suitable for ...
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4answers
751 views

How to arrive at infinitude of primes proof?

I know Euclid's proof of there being infinite number of primes. I want to let my brother (age 15) arrive at that proof by himself. He knows definition of a prime number (number divisible only by 1 and ...
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5answers
1k views

Why do we need perfect numbers?

Why are perfect numbers important? What is the best way of introducing these numbers to a first course on number theory? I could not find any application apart from the relation to Mersenne primes. ...