[seqfan] Re: A property of n-bonacci constant

[Vladimir Shevelev]

In my previous message, I wrote:
 Denote by S_i the i-th power sum for roots of equation (1): x_1^i +...+x_n^i.
Calculating in a usual way S_i, we find S_1=1, S_2=3, S_3=7,...,S_p=2^p-1.
However, it is true for i<= n, while I used "sufficiently large p".  So, sorry, I
 gave a wrong proof. Below I give a right proof. Consider sequence defined
by recursion:
a_k=a_(k-1)+a_(k-2)+...+a_(k-n) + n-1.        (1)
Since characteristic equation of  the the corresponding homogeneous recursion 
(a_k=a_(k-1)+a_(k-2)+...+a_(k-n) ) is 
x^n-x^(n-1)-...-x-1=0,                       (2)
and a private solution of (1) is, trivially, a_k=-1, then  a  general solution of (1)
is a_k=C_1*x_1^k + C_2*x_2^k+...+C_n*x_n^k -1. (3)
>From Theorem 3.3 in our  with  S.  Litsyn paper “Irrational Factors Satisfying the Little Fermat Theorem” (International Journal of Number Theory, vol.1, no.4 ( 2005), 499-512) for x=1, it follows that
1) for every prime k=p, a_p==0 (mod p);
2) If the inintial conditions  for (1) are a_i=2*sum{1<=j<=i/2}Binomial (i,2*j), i=1,...,n, then all coefficients
in (3)  equal 1. Thus we have
a_p=x_1^p + x_2^p+...+x_n^p -1==0 (mod p), where x_1>1 is n-bonacci constant. 
Now ,for sufficiently large p, we have round(x_1^p)==1 (mod p).
It follows from our conjecture that all absolute values of other roots of (2) are less than 1.

Best regards,
Vladimir


________________________________________
From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Vladimir Shevelev [shevelev at exchange.bgu.ac.il]
Sent: 23 March 2014 13:16
To: seqfan at list.seqfan.eu
Subject: [seqfan] A property of n-bonacci constant

Dear Seqfans,

One can  prove that, for n>=2, the equation
x^n-x^(n-1)-...-x-1=0      (1)
has an unique real  root x_1>1. Besides, I conjecture  that, for n>=3, the absolute values of
 complex  roots <1 (probably, it is known).
Number x_1=x_1(n) is called "n-bonacci constant". So, for n=2 we have fibonacci
constant phi or golden ratio (A001622); for n=3 we have tribonacci constant (A058265);
for n=4 we have tetranacci constant (A086088), for n=5 (A103814), for n=6 (A118427), etc.
 Denote by S_i t he i-th power sum for roots of equation (1): x_1^i +...+x_n^i.
Calculating in a usual way S_i, we find S_1=1, S_2=3, S_3=7,...,S_p=2^p-1.
If p is odd prime, then, by little Fermat theorem  we have S_p==1 (mod 2*p).
On the other hand, |x_2^p+...x_n^p|<=(n-1)*|x*|^p, where x* is a root with
 the maximal absolute value among roots x_2,...,x_n. Since |x*|<1, then for
sufficiently large p, we have (n-1)*|x*|^p<0.5. Hence round(x_1^p)=S_p==1(mod 2*p).
 Therefore, the sequence {(round(x_1^prime(k))-1)/prime(k)} has integer and even
terms for sufficiently large k. For example, see our with Peter  sequences
A239502, A239544, A239564-A239566.
Although in our  with  S.  Litsyn paper “Irrational Factors Satisfying the Little Fermat Theorem” (International Journal of Number Theory, vol.1, no.4 ( 2005), 499-512)
 the cases n>=4 of n-bonacci constants are not considered, it contains many
 other  interesting statements on close topic.

Best regards,
Vladimir







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