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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17 votes
4 answers
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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 ...
Amir Asghari's user avatar
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15 votes
3 answers
1k 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\...
Joonas Ilmavirta's user avatar
11 votes
4 answers
829 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 ...
user13107's user avatar
  • 307
11 votes
5 answers
7k 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. ...
user92877's user avatar
  • 211
11 votes
5 answers
897 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 to ...
Will Jagy's user avatar
  • 415
11 votes
4 answers
696 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 ...
matqkks's user avatar
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11 votes
1 answer
434 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, ...
Humberto José Bortolossi's user avatar
10 votes
6 answers
4k 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.
matqkks's user avatar
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10 votes
3 answers
335 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 ...
Adrian Hindes's user avatar
9 votes
4 answers
552 views

Geometrical interpretation of the identity $\operatorname{lcm}(a,b) \operatorname{gcf}(a,b) = ab$

Does anyone know a good geometrical representation of the fact that $\DeclareMathOperator{lcm}{lcm}\DeclareMathOperator{gcf}{gcf}\lcm(a,b) \gcf(a,b) = ab$? Because $\lcm$ and $\gcf$ are abstract ...
Jonas Gomes's user avatar
9 votes
2 answers
332 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 introductory number theory course. I would like to ask what are the differences between these textbooks in terms of ...
Dal's user avatar
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9 votes
2 answers
449 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 ...
user166854's user avatar
8 votes
3 answers
638 views

Is there a numerical base that is in any way “better” for simple mathematical calculations than others?

I want to know if there are any numerical bases that are notably well-suited for humans to learn and use at an elementary or grade-school level. I know that different numerical bases (i.e. decimal/...
Lawton's user avatar
  • 189
8 votes
1 answer
283 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 ...
Gergely's user avatar
  • 183
7 votes
2 answers
154 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 $x = -18/12$. ...
Chaim's user avatar
  • 655
7 votes
1 answer
160 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 ...
Vidyanshu Mishra's user avatar
6 votes
4 answers
420 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 ...
Dal's user avatar
  • 1,111
6 votes
4 answers
553 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 ...
Francisco José Letterio's user avatar
6 votes
2 answers
228 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 ...
matqkks's user avatar
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6 votes
2 answers
445 views

Introductory book or other resource on $p$-adic numbers/number theory/analysis

I am having problems understanding $p$-adic numbers/$p$-adic number theory/$p$-adic analysis. I have tried some notes on the internet, but these notes were not helpful. Can anyone suggest a book, ...
Consider Non-Trivial Cases's user avatar
6 votes
1 answer
227 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 ...
matqkks's user avatar
  • 1,243
5 votes
2 answers
922 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 ...
Buffer Over Read's user avatar
5 votes
4 answers
508 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 ...
Dal's user avatar
  • 1,111
5 votes
3 answers
463 views

Why do we write numbers with decreasing place values?

This question came up while teaching ~16 year olds binary numbers. Why do place values increase to the left and not the other way round?
Jasper's user avatar
  • 2,699
5 votes
2 answers
499 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 ...
MKu's user avatar
  • 151
5 votes
1 answer
190 views

Are there any mathematics based game apps which require students (between 10 - 16 years) to apply their maths knowledge to play the game

So, what we essentially mean is students will apply their knowledge on divisibility, factorization, prime numbers, lcm, gcf, decimals, fractions, etc to play the game. A somewhat different approach to ...
GanitCharcha's user avatar
4 votes
3 answers
236 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 ...
user12061's user avatar
4 votes
1 answer
152 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}{\...
Hans-Peter Stricker's user avatar
3 votes
2 answers
307 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}...
john mangual's user avatar
3 votes
1 answer
431 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 ...
Behzad's user avatar
  • 2,363
3 votes
0 answers
124 views

congruency: how widely used?

Today I was made aware of the term "congruency" as a word related to congruence in the same way that equality is related to equation. I have never seen the term "congruency" used ...
KCd's user avatar
  • 3,486
3 votes
0 answers
223 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 ...
zeraoulia rafik's user avatar
2 votes
5 answers
672 views

How can I explain construction of the Bézout's identity to my kid?

My kid is soon 7 years old, he could understand fractions, linear equation and modulo operation. I've just taught him Chinese remainder theorem, looking to introduce some more basic number theory ...
athos's user avatar
  • 737
2 votes
1 answer
212 views

What is the terminology for integers with the same oddness or evenness?

If two integers are either both negative or both positive, we can say they have the same sign. How about two integers that are either both odd or both even? Is there any term for them?
D G's user avatar
  • 131
2 votes
1 answer
118 views

Reference request: an introduction to triangular, square, and other figurate numbers

There are dozens (maybe thousands) of websites that explain what triangular numbers, square numbers, etc. are. I'm searching for a printed book that includes this material, preferably at a level that ...
mweiss's user avatar
  • 17.4k
1 vote
3 answers
181 views

Whole numbers as sets vs abstracted properties of sets

I recently landed on a book written for elementary school teachers which introduced the concept of whole numbers in the following manner: We have a set $\{\alpha, \beta, \gamma\}$. There are other ...
Harshit Rajput's user avatar
1 vote
1 answer
185 views

What should I say about elementary number theory?

I need to give an option talk (a 10 min talk given to students who are selecting their options for sophomore mathematics) about an elementary number theory module. The students will have completed a ...
matqkks's user avatar
  • 1,243
1 vote
2 answers
125 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 ...
Chris's user avatar
  • 113
1 vote
1 answer
106 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 ...
user avatar
1 vote
2 answers
153 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 ...
xyzt's user avatar
  • 113
1 vote
1 answer
128 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 ...
matqkks's user avatar
  • 1,243
1 vote
0 answers
124 views

Number theory in an introductory course on discrete dynamical systems

Benjamin Hutz, in Chapter 10 of his An Experimental Introduction to Number Theory, allows for the optional inclusion of discrete dynamical systems with a number-theoretic flavor in an undergraduate ...
J W's user avatar
  • 4,663
0 votes
3 answers
457 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 (...
Amir Asghari's user avatar
  • 4,428
0 votes
1 answer
265 views

Limitations of applying the factor theorem

Here are three situations in which students might try to apply the factor theorem. Proving that $x + 1$ is a factor of the polynomial $x^3 + x + 2$ can be done using the factor theorem by showing ...
Janaka Rodrigo's user avatar
0 votes
1 answer
155 views

Which academic subjects examine what the advantages and disadvantages of the various number bases are?

Which academic subjects examine what the advantages and disadvantages of the various number bases are, e.g. besides base ten: base twelve, base sixteen, base eight, base two and the ways that they can ...
Matthew Christopher Bartsh's user avatar
-1 votes
3 answers
643 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 ...
Timothy's user avatar
  • 499