# Number Theory Problem Watch

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Given that a, b and n are positive integers, and

(a^2 + b^2)/(ab - 1) = n

Prove that n = 5

First one who proves it gets rep.

Galois.

(a^2 + b^2)/(ab - 1) = n

Prove that n = 5

First one who proves it gets rep.

Galois.

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#2

Are you sure it's n=5? For some reason I got n=4...

I gotta go, so I'll check my steps to see if I added wrong or something when I get back.

I gotta go, so I'll check my steps to see if I added wrong or something when I get back.

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(Original post by

Are you sure it's n=5? For some reason I got n=4...

I gotta go, so I'll check my steps to see if I added wrong or something when I get back.

**dvs**)Are you sure it's n=5? For some reason I got n=4...

I gotta go, so I'll check my steps to see if I added wrong or something when I get back.

What values of a and b did you use?

Galois.

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#4

n = 5 is OK

a = 1 and b = 2 or vice versa

the value of the LHS decreases as a and b increase, so the above is the only solution, but I have no idea about a proof!

Aitch

a = 1 and b = 2 or vice versa

the value of the LHS decreases as a and b increase, so the above is the only solution, but I have no idea about a proof!

Aitch

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#5

(Original post by

n = 5 is OK

a = 1 and b = 2 or vice versa

the value of the LHS decreases as a and b increase, so the above is the only solution, but I have no idea about a proof!

Aitch

**Aitch**)n = 5 is OK

a = 1 and b = 2 or vice versa

the value of the LHS decreases as a and b increase, so the above is the only solution, but I have no idea about a proof!

Aitch

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#6

(Original post by

Hmm a=1 and b = 3

**RichE**)Hmm a=1 and b = 3

The other half of what I wrote is wrong too!

Aitch

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#7

(Original post by

You're right!

The other half of what I wrote is wrong too!

Aitch

**Aitch**)You're right!

The other half of what I wrote is wrong too!

Aitch

Aitch

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a=2 and b=1 (and vice versa)

a=3 and b=1 ""

a=9 and b=2 ""

a=14 and b=3 ""

a=43 and b=9 ""

a=67 and b=14 ""

There are no more pairs with a and b both less than 100. There are an infinite number of pairs a and b, but the question isn't find a and b, the question is prove n = 5.

Any one?

Galois.

a=3 and b=1 ""

a=9 and b=2 ""

a=14 and b=3 ""

a=43 and b=9 ""

a=67 and b=14 ""

There are no more pairs with a and b both less than 100. There are an infinite number of pairs a and b, but the question isn't find a and b, the question is prove n = 5.

Any one?

Galois.

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#9

I've not done it, but I got another one for number theory.

a>b>c>d are positive integers

ac+bd can divide by (a+b+d-c).

Prove that ab+cd is not a prime number.

a>b>c>d are positive integers

ac+bd can divide by (a+b+d-c).

Prove that ab+cd is not a prime number.

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#10

Suppose that it is prime, then a+b+d-c must be that same prime and equal to ab+cd;

a+b+d-c = ab + cd

a(b-1) + d(c-1) = b-c

c > b so the RHS is negative, whereas the LHS must be positive. So we have a contradiction.

a+b+d-c = ab + cd

a(b-1) + d(c-1) = b-c

c > b so the RHS is negative, whereas the LHS must be positive. So we have a contradiction.

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(Original post by

Suppose that it is prime, then a+b+d-c must be that same prime and equal to ab+cd;

a+b+d-c = ab + cd

a(b-1) + d(c-1) = b-c

c > b so the RHS is negative, whereas the LHS must be positive. So we have a contradiction.

**JamesF**)Suppose that it is prime, then a+b+d-c must be that same prime and equal to ab+cd;

a+b+d-c = ab + cd

a(b-1) + d(c-1) = b-c

c > b so the RHS is negative, whereas the LHS must be positive. So we have a contradiction.

Galois.

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#12

(Original post by

Who said c was greater than b? The RHS is also positive, because b > c.

Galois.

**Galois**)Who said c was greater than b? The RHS is also positive, because b > c.

Galois.

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Can any one do the first question?

I am calling for the best number theorist here.

Galois.

I am calling for the best number theorist here.

Galois.

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#15

Ahrr, it made my brain exploding. Dang! I have alot of this type of questions and never end up with successful answers. I'll post it after someone do Galois's question.

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(Original post by

God, this is so frustrating. I must do it.

**J.F.N**)God, this is so frustrating. I must do it.

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(Original post by

I will do it. By tonight. Don't give the answer. I must do it.

**J.F.N**)I will do it. By tonight. Don't give the answer. I must do it.

Galois.

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#20

(Original post by

OK. Remember solution in white please.

Galois.

**Galois**)OK. Remember solution in white please.

Galois.

It looks interesting, I've done P4, but can find no way into this.

Have tried a few approaches, but end up with too many variables.

When you put up the answer, and put the contenders out of their misery, can you put up a reading list, or similar?

Thanks.

Aitch

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