I need a proof for this binomial property.

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Discussion Overview

The discussion revolves around proving a specific property of Fermat numbers, particularly the relationship between Fermat numbers and a sequence defined by binomial coefficients. Participants explore various approaches and resources to assist in the proof, which is intended for further mathematical exploration.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Tony seeks a proof for the property that if \( F_n \) is prime, then \( F_n \) divides \( A_{k_n} \), where \( A_{k_n} \) is defined using binomial coefficients and powers of 2.
  • Some participants suggest using a PARI/gp program to check the formula and compute values for specific \( n \).
  • One participant provides a link to a sequence that may relate to the Fermat numbers, questioning whether the goal is to prove that only the first five Fermat numbers are prime.
  • Another participant references the binomial theorem and suggests manipulating sequences to eliminate odd terms, though they express uncertainty about its effectiveness.
  • Tony mentions a connection between Fermat numbers and Pell numbers, indicating that he believes there may be a faster proof of primality using a Lucas sequence.
  • There is a discussion about the implications of the binomial theorem and its relation to the problem, including the potential involvement of square roots and number fields.

Areas of Agreement / Disagreement

Participants express various ideas and approaches without reaching a consensus on the proof or the best method to pursue. Multiple competing views and uncertainties remain regarding the connections between the sequences and the Fermat numbers.

Contextual Notes

Some participants note the complexity of the relationships involved and the potential need for further study on the period of Pell numbers modulo Fermat numbers of primes. There are also references to unresolved mathematical steps and dependencies on definitions.

T.Rex
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Hi,
I've spent dozen of hours searching by my-self and dozen of hours searching on the Web. Now I need help.
Who could provide a proof for this binomial property ? I need it for another proof.
Thanks
Tony

Let: F_n=2^{2^n}+1 , n \geq 2 .

Prove: F_n \text{ prime } \Longrightarrow<br /> F_n \mid A_{k_n} , \text{ where } k_n=2^{3 \times 2^{n-2}-1} \text{<br /> and } A_{k_n} = \sum_{i=0}^{k_n/2}{k_n \choose 2i}<br /> 2^i
Examples:
n=2 , F_2=17 , k_2=4 , A_{k_2}=17
n=3 , F_3=257 , k_3=32 , A_{k_3}=257*1409*2448769
n=4 , F_4=65537 , k_4=2048 , A_{k_4}=\text{very big} \equiv 0 \<br /> (\text{mod} F_4)
n=5 , F_5=4294967297 , k_5=8388608 , A_{k_5}=\text{VERY big} \neq 0 \ (\text{mod} F_5)
 
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The formula can be checked with this PARI/gp program:
binomialmod(a,b,p)= B=1; while(a!=0 && b!=0,a_=a%p;b_=b%p;a=(a-a_)/p;b=(b-b_)/p; B=(B*(binomial(a_,b_)%p)%p)); return(B)

l(n)=S=0;F=2^2^n+1;k=2^(3*2^(n-2)-1);for(i=0,k/2, B=(binomialmod(k,2*i,F)*((2^i)%F))%F;S=(B+S)%F); print(n," ",F," ",k," ",S)

l(2)
 
Well, you do know that

<br /> \sum_{i=0}^n {n \choose i} x^n = (1+x)^n<br />

right?

Maybe it would help to see if you can do sequence manipulations to eliminate the odd terms from this sequence...

(then again, it might not)
 
Fermat numbers and Pell Sequence

Hi Gerald, Hurkyl,

Thanks trying to help me.
The formula I give in the first post of this thread gives Pell numbers.
While studying a certain Lucas Sequence, I found that it seemed that this Lucas Sequence could give a fastest proof of Fermat numbers primality than Pepin's test. Then I wrote a paper (no proof, only computing facts plus historical facts from E. Lucas) explaining that. The formula appeared when I tried to find other ways for computing the Pell Sequence.

I talked about this in another thread in this forum:
https://www.physicsforums.com/showthread.php?t=49778
and here is my paper: http://tony.reix.free.fr/Mersenne/PrimalityTest4FermatNumbers.pdf

I looked at the A000 sequences you talked about, but found no help.

About the (1+x)^n formula, yes I thought about using it, but I failed finding any way for finding some other formula where the Fermat number appears clearly.

If you have time, read my paper. I think I missed many things. But, what's interesting is that Edouard Lucas himself provided a theorem using the Pell Sequence that is very close to the one I talk about.
But, all of that seems to complex for me.

If someone could make a study of the period of Pell numbers modulo Fermat numbers of primes, that would help.

Regards,
Tony
 
Well, note that:

<br /> (1+x)^n = \sum_{i=0}^n {n \choose i} x^i<br />
<br /> (1-x)^n = \sum_{i=0}^n {n \choose i} (-x)^i<br />
<br /> (1+x)^n + (1-x)^n = \sum_{i=0}^n {n \choose i} (x^i + (-x)^i)<br />
<br /> (1+x)^n + (1-x)^n = 2 \sum_{i=0}^{n/2} {n \choose 2i} x^{2i}<br />
<br /> \frac{1}{2} (1+\sqrt{2})^{k_n} + (1-\sqrt{2})^{k_n}<br /> = \sum_{i=0}^{k_n/2} {k_n \choose 2i} 2^i<br />

Ick, so it involves square roots. :-p Maybe you can do the problem in a number field? I.E. use \mathbb{Z}[\sqrt{2}] instead of \mathbb{Z}. (Or are you supposed to use \mathbb{Z}[(1 + \sqrt{2})/2]? I can't remember)

Does 2 have any square roots in Z mod F_n?
 
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