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August 30th, 2012, 08:09 AM   #1
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squares

If any base greater than 6 is used, 14,641 is a perfect square, more precisely prove that 14641 is the fourth power of an integer in any base greater than 6.

How do you prove this?
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August 30th, 2012, 10:35 AM   #2
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Re: squares

In base 10, 14,641 is 1 * 10^4 + 4 * 10^3 + 6 * 10^2 + 4 * 10^1 + 1 * 10^0. What is it in base b? Can you factor that?
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August 30th, 2012, 11:15 AM   #3
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Re: squares

Here is another hint:

This is the first few rows of Pascal's Triangle:

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1

The number 121 is a perfect square in any base greater than 2, and the number 1331 is a perfect cube in any base greater than 3.
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August 30th, 2012, 03:53 PM   #4
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Re: squares

Ooh... nice hint, but probably harder to get from that to the answer than from the answer to that!
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September 1st, 2012, 03:39 AM   #5
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Re: squares

I'd like to see the generalized Pascal thing proven. Sounds interesting.
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September 1st, 2012, 04:49 AM   #6
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Re: squares

Sorry, I don't get this at all...... can someone give me a solution to this?... I'm really bad at this.
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September 1st, 2012, 04:56 AM   #7
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Re: squares

Let be the base of the number system. Then the integer 14641 is represented by:

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