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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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