September 25th, 2013, 09:11 AM  #1 
Senior Member Joined: Sep 2010 Posts: 221 Thanks: 20  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? 
September 25th, 2013, 12:58 PM  #2 
Senior Member Joined: Feb 2012 Posts: 628 Thanks: 1  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 .

September 26th, 2013, 05:33 PM  #3  
Senior Member Joined: Sep 2010 Posts: 221 Thanks: 20  Re: Feature of binomials Quote:
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  
September 26th, 2013, 08:16 PM  #4 
Senior Member Joined: May 2013 From: EspaÃ±a Posts: 151 Thanks: 4  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: 1º) Prime numbers, power of the form: Example: 5, 11, 17, 23, 29, 41 2º) Prime numbers, power of the form: Example: 7, 13, 19, 31, 37, 43 Equally: 1º) Prime numbers, power of the form: 2º) Prime numbers, power of the form: Regards. 
September 27th, 2013, 08:03 AM  #5  
Senior Member Joined: Sep 2010 Posts: 221 Thanks: 20  Re: Feature of binomials Quote:
But I'm asking about proof.  

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