My Math Forum Feature of binomials

 Number Theory Number Theory Math Forum

 September 25th, 2013, 08:11 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 $(X+Y)^N-[X^N+Y^N ]$ is divisible by $X^2+XY+Y^2$ if sum of coefficients of middle terms i.e. $2^N-2$ is divided by 3. Starting with N=7 the polynomial is divisible by $(X^2+XY+Y^2)^2=X^4+2X^3 Y+3X^2 Y^2+2XY^3+Y^4$ if the sum $2^N-2$ is divided by 9. Can it or has it been proved or is it just a conjecture?
 September 25th, 2013, 11:58 AM #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 $3^K$ is the largest power of 3 that divides $2^N - 2$, and N is prime, then $(X + Y)^N - [X^N + Y^N]$ is divisible by $(X^2 + XY + Y^2)^K$.
September 26th, 2013, 04:33 PM   #3
Senior Member

Joined: Sep 2010

Posts: 221
Thanks: 20

Re: Feature of binomials

Quote:
 Originally Posted by icemanfan Not just a conjecture, I think. In fact, I would presume that if $3^K$ is the largest power of 3 that divides $2^N - 2$, and N is prime, then $(X + Y)^N - [X^N + Y^N]$ is divisible by $(X^2 + XY + Y^2)^K$.
If it's not a conjecture does a proof of divisibility of $(X + Y)^N - [X^N + Y^N]$ by $X^2 + XY + Y^2$ exist?
I tested the case of N=37 where $2^{36}-1$ is divided by $3^3$. And $(X+Y)^{37}-(X^{37}+Y^{37})$ appeared divisible by $(X^2 + XY + Y^2)^2$ but I failed to divide the quotient by the third $X^2+XY+Y^2$

 September 26th, 2013, 07:16 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: $6k-1$ Example: 5, 11, 17, 23, 29, 41 $x^2+xy+y^2 | (x+y)^n-(x^n+y^n)$ 2º) Prime numbers, power of the form: $6k+1$ Example: 7, 13, 19, 31, 37, 43 $(x^2+xy+y^2)^2 | (x+y)^n-(x^n+y^n)$ Equally: 1º) Prime numbers, power of the form: $6k-1$ $x^2-xy+y^2 | (x-y)^n-(x^n-y^n)$ 2º) Prime numbers, power of the form: $6k+1$ $(x^2-xy+y^2)^2 | (x-y)^n-(x^n-y^n)$ Regards.
September 27th, 2013, 07:03 AM   #5
Senior Member

Joined: Sep 2010

Posts: 221
Thanks: 20

Re: Feature of binomials

Quote:
 Originally Posted by mente oscura 1º) Prime numbers, power of the form: $6k-1$ Example: 5, 11, 17, 23, 29, 41 $x^2+xy+y^2 | (x+y)^n-(x^n+y^n)$ 2º) Prime numbers, power of the form: $6k+1$ Example: 7, 13, 19, 31, 37, 43
You are right. Obviously there may be infinite number of examples.

 Tags binomials, feature

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post PhizKid Algebra 2 May 14th, 2013 12:48 PM RocketBG Algebra 2 January 16th, 2012 12:53 PM ChristinaScience Algebra 3 February 14th, 2011 06:37 PM micahat Algebra 3 May 27th, 2009 03:34 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top