Proofs and puzzles for beginning mathematicians

[Imaginer1]

I am a freshman in High school, however I've been working quite a lot in the field of number theory for quite some time. However, I've been beginning to feel slightly bad that I haven't actually proven anything. It's not like I want to make a brand new theorem, no; but I would like to start to prove statements. However, all the questions I've come up with have been out of my reach of understanding. So, any mathematical statements to prove at an intermediate level for practice? Dump them here. I'm sure everyone will be entertained by them, and I sure hope I will be, too.

 

[micromass]

1) Prove that the sum of odd numbers is always a perfect square. For example, 1+3+5+7 is a perfect square.

2) Find a five digit number such that the third power of the first two digits is the entire number. So find abcde such that (ab)³=abcde. Try to reason through it, don't use trial and error.

3) Classify all natura numbers x,y such that $x^2-y^2=17$. Generalize.

4) Not number theory and not easy, but very nice: Take a perfect n-gon in the unit circle. Take a special point x of the n-gon. Connect all other points of the n-gon with x. Prove that the length of all line segments is n.

 

[Imaginer1]

Yes.
Those seem exceedingly entertaining.
I SHALL BEGIN.

 

[checkitagain]

http://www.mathschallenge.net


There are four difficulty levels of exercises with hidden solutions.

Many are number theory type with other math subject area problems.

 

[morphism]

4) Not number theory and not easy, but very nice: Take a perfect n-gon in the unit circle. Take a special point x of the n-gon. Connect all other points of the n-gon with x. Prove that the length of all line segments is n.

I suppose what you meant to say here is: Take a perfect n-gon with its vertices on the unit circle. Take a special vertex x. Connect all other vertices to x. Prove that the product of the lengths of the all line segments is n.

Right?


re Imaginer1: Have you thought about getting a textbook on basic number theory? There are plenty of good ones that have lots of fun exercises.

 

[micromass]

I suppose what you meant to say here is: Take a perfect n-gon with its vertices on the unit circle. Take a special vertex x. Connect all other vertices to x. Prove that the product of the lengths of the all line segments is n.

Yes, of course, thank you!

 

[Imaginer1]

re Imaginer1: Have you thought about getting a textbook on basic number theory? There are plenty of good ones that have lots of fun exercises.

I've thought about it, but haven't. Don't assume, however, that implies my knowledge on number theory doesn't stack up- actually, I'm currently working with the Mertens function growth rate, but that's a bit beyond the point.

 

[Frogeyedpeas]

You might like the AoPS book-line just google that acronym and check out the website

 

[Frogeyedpeas]

And once you're done with that I guess just buy other books/read the wikipedia articles if you have the patience

 

[20Tauri]

If you like number theory and you are interested in learning formal proof, Number Theory: A Lively Introduction by Pommersheim, Marks, and Flapan might be a good choice for you. It's an undergrad text, but I think it'd be accessible to you. It is full of puns, which may be a plus or a minus depending on your sense of humor. My understanding of it is that while it's not the most rigorous text, it does treat its math seriously and it's a good, non-threatening introduction for students who maybe haven't done a lot of upper-level math yet. It also introduces things like proof by induction. We used it in the number theory course I took in the fall.

A selection of things to try proving, cribbed from my memory of number theory assignments:

Prove that for natural numbers x, y, z and prime p, x^p +y^p =z^p implies p divides x+y-z.

Prove that the greatest common divisor of two consecutive Fibonacci numbers is always 1.

Prove that the product of an even number and an odd number is always even.

For integers a and b and prime p, show that if a and b are both quadratic residues modulo p, then the product ab is also a quadratic residue modulo p.