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September 20th, 2011, 05:38 AM   #1
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Exponents - conditions

Hi!

As I was solving a math problem (an equation with exponents) I found myself unable to go any further without having determined what conditions must be met by integers so that

.

I can post the original quetion, if needed.

Thanks,

Arnold
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September 20th, 2011, 06:01 AM   #2
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Re: Exponents - conditions




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September 20th, 2011, 06:48 AM   #3
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Re: Exponents - conditions

Quote:
Originally Posted by arnold
Hi!

As I was solving a math problem (an equation with exponents) I found myself unable to go any further without having determined what conditions must be met by integers so that

.
m = 1, n = 0.

It's clear that m > n, else the left side will not be positive. Further, n = 0 else the left side will be a multiple of 5. So the problem becomes


But then you have two consecutive powers which can only happen if a base or exponent is 0 or 1, or else if the numbers are 8 and 9, by Catalan's conjecture (now a theorem).

Cannon, meet fly.
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September 20th, 2011, 08:10 AM   #4
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Re: Exponents - conditions

Thanks a lot!
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September 20th, 2011, 08:38 AM   #5
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Re: Exponents - conditions

Could you explain it to me why n=0? How do I prove it?
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September 20th, 2011, 08:46 AM   #6
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Re: Exponents - conditions

I mean, I understand why n=0. I just want to know if it can be proven somehow.
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September 20th, 2011, 08:56 AM   #7
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Re: Exponents - conditions

Quote:
Originally Posted by arnold
I mean, I understand why n=0. I just want to know if it can be proven somehow.
If n > 0, then either the left is negative or a multiple of 5. Powers of 2 are never negative and are never multiples of 5.
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September 20th, 2011, 08:59 AM   #8
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Re: Exponents - conditions

Ok, thanks. So I don't need to apply modulo, etc, do I?
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September 20th, 2011, 10:40 AM   #9
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Re: Exponents - conditions

Quote:
Originally Posted by arnold
Ok, thanks. So I don't need to apply modulo, etc, do I?
Depending on how you formalize it you might use the modulus operator to express "divisible by 5".
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