Math Is Fun Forum

Agnishom

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From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

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Number Theory

Prove that x+y divides x^n + y^n for all n which is odd

Last edited by Agnishom (2014-05-09 01:24:29)

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'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

Bob

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Registered: 2010-06-20

Posts: 10,643

Re: Number Theory

I can show a proof by induction.  It'll take a while to Latex it.

Bob

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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

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

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Re: Number Theory

Induction, hmm.

Thanks for the hint. I'll try it myself.

---

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: Number Theory

QED. 

Bob

Last edited by Bob (2014-05-09 05:55:42)

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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

Agnishom

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From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

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Re: Number Theory

Thanks. Can you prove this:

If a,b are positive integers:

And if a + b + ab = 2020

then a+b=88?

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'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Number Theory

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: Number Theory

Maybe like this:

Find the (only two) factors of 2021. (other than 1 and 2021)

Bob

Last edited by Bob (2014-05-09 06:27:17)

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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

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

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Re: Number Theory

Thanks, bob

What would Ramanujan do bobbym? What is the D command?
By the way, FindInstance has already given me 42 and 46

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'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Number Theory

One of Ramanujan's great answers was when he reduced a problem to 7 possibilities and tried them all! In the age before computers, that guy was doing experimental math.

That is the divisors operator. Notice that they are paired up to product to 2020 - 88. Now just find 2 that sum to 88. They are easily found.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

Website

Re: Number Theory

I've got FindInstance. Why must I work so hard?

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'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Number Theory

Because I want to show the work. Those divisors can be done by hand.

I would take Bob's idea but you are still going to have to factor a number.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

Website

Re: Number Theory

What work do you want to show?

I'll factor it with a computer

---

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Number Theory

I hid that answer because it is a last resort and much better than having no answer.

There is an amusing story of the mathematician who walked up to the board wrote this:

then he did the multiplication. Took 10 minutes to do, never said a word and just sat down. That was his proof that 2^64+1 is composite.

I thjought post #7 was good for an answer, so take that one.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

Website

Re: Number Theory

I cannot do that. I will most certainly make mistakes.

---

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Number Theory

Here is another way that uses only algebra:

Substituting for b using the 2nd equation.

That is a quadratic that is easily solved and gets a =42 and a = 46.

So we have a = 42 and b = 46 or a = 46 and b = 42.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Nehushtan

Member

Registered: 2013-03-09

Posts: 957

Re: Number Theory

Consider

This is a G.P. with common ratio

so we have

But n is odd so

.

Multiplying both sides by

gives

Since the LHS is an integer, the RHS must be an integer. QED.

Last edited by Nehushtan (2014-05-09 08:16:02)

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Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: Number Theory

I would take Bob's idea but you are still going to have to factor a number.

Earlier I number crunched it in Excel, but further thought provides immediate factors:

Factors of 2021 (assume just two to provide a unique solution)

To end in 1 the number pair must come from one of these sets

{11,21,31,41} {19,29,39} {13,23,33,43 with 17,27,37,47} [anything over 50 is too big]*

You can dismiss all of the first as 41 x 41 is only just over 1600.

You can dismiss the second as 39 x 39 < 1600.

So any factor pair must come from pairing a number ending in 3 with a number ending in 7

37 x 43 < 40 x 45 = 1800 so only 47 need be considered.

47 x 33 < 50 x 35 = 1750 so you only need to try 43 x 47 by long multiplication.

Bob

* Just realised something like 37 x 53 might work .... working on this modification

Last edited by Bob (2014-05-09 08:45:17)

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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

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: Number Theory

OK.  Above is wrong although you could still get that answer quickly without lots of number crunching. 

However for a complete analysis see the screen shot below.  I think it is self explanatory.

Bob

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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

zetafunc.

Guest

Re: Number Theory

Thanks. Can you prove this:

If a,b are positive integers:

And if a + b + ab = 2020

then a+b=88?

zetafunc.

Guest

Re: Number Theory

Thanks. Can you prove this:

If a,b are positive integers:

And if a + b + ab = 2020

then a+b=88?

You may have noticed that I ignored the cases a = 0, b = 2020 and b = 0, a = 2020. Your constraint was that a and b are positive, hence we can ignore these cases.

zetafunc.

Guest

Re: Number Theory

There are also the cases for negative a and b, but, again, these do not satisfy the constraints of your problem.

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

Website

Re: Number Theory

Here is another way that uses only algebra:

Substituting for b using the 2nd equation.

That is a quadratic that is easily solved and gets a =42 and a = 46.

So we have a = 42 and b = 46 or a = 46 and b = 42.

You are asked to prove that a+b = 88, You cannot begin by accepting that fact

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'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Number Theory

You can accept the statement and work backwards like a Reductio ad Absurdum. Then you try to work towards a contradiction which in this case does not happen. But I will ask for some other opinion on it.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

Website

Re: Number Theory

In reductio ad absurdum, you do not accept the statement. You and bob had a huge argument about this process somewhere but I do not remember

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'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Number Theory

In reductio ad absurdum, you do not accept the statement.

I think you do.

anonimnystefy vindicated the idea of working forward back then. I will ask the mathematicians what they think.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.