Math Is Fun Forum

tts14414

Member

Registered: 2012-02-18

Posts: 3

a²+4ab+b²

Let A be a set of integers that can be expressed as a²+4ab+b², where a,b are integers.

If x,y are in A, then prove that xy is also in A.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,624

Re: a²+4ab+b²

hi tts14414

You could proceed like this.

write

and

Then multiply these expressions together.

You can re-arrange the result into this form:

for some P and Q.

It is finding P and Q that is proving more tricky than I was expecting.

I'm going to spreadsheet some integers and see if that will help.

I'll make a new post if I get anywhere.

Bob

Last edited by Bob (2012-02-19 01:44:22)

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

tts14414

Member

Registered: 2012-02-18

Posts: 3

Re: a²+4ab+b²

I appreciate your help, and in fact I have tried expanding and factorizing that for hours, but I just wasn't successful. I tried using (ac+bd)² + 4(ac+bd)(ad+bc) + (ad+bc)², but there's 12abcd missing, as you expand you can see, and that was quite close indeed.

Also I tried:
1. Trigonometry. Substitute sin and cos, sin² + cos² is 1, so maybe this make the expansion simpler? But then I realize that the 'a' and 'b' wouldn't be integers if it's sin/cos, this method is not possible.
2. Matrix. (a) (1 2) is actually, and magically, a²+4ab+b². But then it would be quadratic form and I don't know how to carry on with this as I haven't learnt it.
3. Number theory and some logic. I thought, since x, y are in the set A, so x,y must also be integers. Then I tried substituting x,y as a,b, and performed so transformation, but there's too little for me to carry on with.

This is a piece of work that I needed to hand in before the end of this Friday. Any help or ideas to solve this problem would be much appreciated.
                b   2 1

Bob

Administrator

Registered: 2010-06-20

Posts: 10,624

Re: a²+4ab+b²

hi tts14414

I tried using (ac+bd)² + 4(ac+bd)(ad+bc) + (ad+bc)², but there's 12abcd missing,

That's exactly what I did this morning when I first posted, full of confidence that I would have this sorted in a few lines.

But that 12abcd has me stumped too.

I've made a spreadsheet with formulas to compute

for all values of a and b from -10 to + 10 and then picked pairs to act as x and y.  So far they have all worked out ok (that is to say, I've always found a P and Q ) but I cannot see any connection between a, b, c, d and the resulting P, Q values that give xy.

With d = 0 you get P = ac and Q = bc, so we must be on the right lines.  Yet, half a day later, a solution still evades me.

Shan't give up though.  Maybe, I'll come up with something after a sleep.

Bob

Last edited by Bob (2012-02-19 08:58:55)

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Sylvia104

Banned

Registered: 2011-09-19

Posts: 29

Re: a²+4ab+b²

Write

as

by putting

and

. Conversely any number of the form

can be written as

by putting

and

. Hence

So instead of taking

and

, take

and

. Then

Last edited by Sylvia104 (2012-02-19 14:19:21)

tts14414

Member

Registered: 2012-02-18

Posts: 3

Re: a²+4ab+b²

That's simply fastastix and brilliant. Thank you for a million times. Your help is really too grateful. Thank you.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,624

Re: a²+4ab+b²

Thanks Sylvia104,

I can get some sleep now!   

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob