[seqfan] A property of n-bonacci constant

Vladimir Shevelev shevelev at bgu.ac.il
Sun Mar 23 12:16:28 CET 2014

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,
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