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September 25th, 2013, 09:11 AM   #1
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Feature of binomials

I recently discovered for myself the following feature.
When N is prime starting with N=5 the expanded polynomial
is divisible by if sum of coefficients of middle terms i.e. is divided by 3.
Starting with N=7 the polynomial is divisible by
if the sum is divided by 9.

Can it or has it been proved or is it just a conjecture?
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September 25th, 2013, 12:58 PM   #2
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Re: Feature of binomials

Not just a conjecture, I think. In fact, I would presume that if is the largest power of 3 that divides , and N is prime, then is divisible by .
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September 26th, 2013, 05:33 PM   #3
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Re: Feature of binomials

Quote:
Originally Posted by icemanfan
Not just a conjecture, I think. In fact, I would presume that if is the largest power of 3 that divides , and N is prime, then is divisible by .
If it's not a conjecture does a proof of divisibility of by exist?
I tested the case of N=37 where is divided by . And appeared divisible by but I failed to divide the quotient by the third
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September 26th, 2013, 08:16 PM   #4
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Re: Feature of binomials

Hello.
Do not be if I have understood well the question. If it is not like that, I ask for excuses.
In his day I verified all the prime, major numbers that 3, up to 43.
It is fulfilled:

1║) Prime numbers, power of the form:


Example: 5, 11, 17, 23, 29, 41



2║) Prime numbers, power of the form:


Example: 7, 13, 19, 31, 37, 43



Equally:

1║) Prime numbers, power of the form:




2║) Prime numbers, power of the form:




Regards.
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September 27th, 2013, 08:03 AM   #5
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Re: Feature of binomials

Quote:
Originally Posted by mente oscura

1║) Prime numbers, power of the form:


Example: 5, 11, 17, 23, 29, 41



2║) Prime numbers, power of the form:


Example: 7, 13, 19, 31, 37, 43
You are right. Obviously there may be infinite number of examples.
But I'm asking about proof.
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