2019/09/24 by Ulas, Maciej
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1909.10844
Let Bn(t) be the nth Stern polynomial, i.e., the nth term of the sequence defined recursively as B0(t)=0, B1(t)=1 and B2n(t)=tBn(t), B2n+1(t)=Bn(t)+Bn-1(t) for n∈\N. It is well know that ith coefficient in the polynomial Bn(t) counts the number of hyperbinary representations of n-1 containing exactly i digits 1. In this note we investigate the existence of odd solutions of the congruence Bn(t)≡ 1+rt\fracte(n)-1t-1\pmodm, where m∈\N≥ 2 and r∈\0,…,m-1\ are fixed and e(n)=\opdegBn(t). We prove that for m=2 and r∈\0,1\ and for m=3 and r=0, there are infinitely many odd numbers n satisfying the above congruence. We also present results of some numerical computations.