## Multiplicative polynomials and Fermat’s little theorem for non-primes.(English)Zbl 0885.11025

Fermat’s little theorem says that for each prime $$p$$ the polynomial $$f_p(x)= (x^p- x)/p$$ is an integer valued polynomial (it maps $$\mathbb{N}_0$$ to $$\mathbb{N}_0$$). Can this be extended to non-primes in a meaningful way? The authors suggest the following. Define a sequence of polynomials $$(g_n)_{n\in\mathbb{N}}$$ recursively by $g_1(x)= x\quad\text{and} \quad \sum_{d|n} (-1)^{n/d} g_d(x)= (-x)^n.$ Then by giving an explicit formula for the $$g_n$$, the authors show that for primes $$p$$, $$g_p(x)= x^p- x$$, and that for $$n\in\mathbb{N}$$, $$f_n(x):= g_n(x)/n$$ is integer valued. Moreover, the sequence $$\{g_n\}$$ is multiplicative with respect to the product $$\left(\sum_i a_ix^i\right)\cdot\left( \sum_j b_jx^j\right)= \sum_{i,j} a_ib_j x^{ij}$$.
The authors were led to consider the polynomials $$g_n$$ by some ideas in topological dynamics: Define an Abelian group operation $$\otimes$$ on $$\mathbb{Z}^\infty$$ by $(x_1,x_2,x_3,\dots)\otimes (y_1,y_2,y_3,\dots)= (x_1+ y_1,x_2+ x_1y_1+ y_2, x_3+ x_2y_1+ x_1y_2+ y_3,\dots),$ let $$\varphi$$ be the “natural” isomorphism from $$(\mathbb{Z}^\infty,\otimes)$$ onto the direct product group $$(\mathbb{Z}^\infty,+)$$, and define a sequence of polynomials $$(P_n)_{n\in\mathbb{N}}$$ by $$(P_n):= x\mapsto\varphi^{- 1}(x, 0,0,0,\dots)$$. Then numerical evidence suggests that $$P_n= g_n/n$$, but that appears to be an open question.

### MSC:

 11C08 Polynomials in number theory 11A07 Congruences; primitive roots; residue systems
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