## A note on Fibonomial coefficients.(English)Zbl 1459.11039

Let $$F_0, F_1, \dots$$ be the Fibonacci numbers. For $$n \geq 1$$ it is put $[0]_F = 1, [n]_F = \prod_{u=1}^n F_k.$
For $$n \geq k \geq 0$$, the Fibonomial coefficient is given by $\binom{n}{k}_F = \frac{[n]_F}{[k]_F [n-k]_F} = \frac{F_{n-k+1} \cdots F_n}{F_1 \cdots F_k}.$
In the paper it is proved that for almost primes $$p$$, each residue class $$\lambda$$ modulo $$p$$ can be written as $\binom{ u_1}{v_1}_F + \ldots + \binom{ u_8}{v_8}_F \equiv \lambda \pmod p,$ for positive integers $$u_1, v_1, \ldots, u_8, v_8 \ll p^{3/2}\log^2 p$$.

### MSC:

 11B39 Fibonacci and Lucas numbers and polynomials and generalizations 11B50 Sequences (mod $$m$$)

### Keywords:

Fibonacci numbers; sum-product phenomenon
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### References:

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