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 \(u_1, v_1, \ldots, u_8, v_8 \ll p^{3/2}\log^2 p\).
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 \(u_1, v_1, \ldots, u_8, v_8 \ll p^{3/2}\log^2 p\).
Reviewer: Krassimir Atanassov (Sofia)
MSC:
| 11B39 | Fibonacci and Lucas numbers and polynomials and generalizations |
| 11B50 | Sequences (mod \(m\)) |
References:
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