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Congruence properties of coefficients of certain algebraic power series. (English) Zbl 0661.10019

The author studies the coefficients u(n) of the Taylor expansion of \((1+\sum^{e}_{i=1}a_ iX^ i)^{-1/e}\) where \(e\in {\mathbb{N}}\), \(e>1\) and \(a_ i\in {\mathbb{Z}}\) for all i. Letting p be a prime with \(p\equiv 1 (mod e)\), the author proves two remarkable congruences for u(n) modulo prime powers of p. The first of them is \(u(mp^ r)\equiv u(mp^{r-1}) (mod p^ r)\) for all \(m,r\in {\mathbb{N}}\). The second result gives a lower bound for the number of primes dividing u(n) as soon as p divides \(a_ i\) for some i. Let \(n=n_ 0+n_ 1p+...+n_ tp^ t\) be the expansion of n in base p. Let \(J=\{1\leq j\leq p-1:\) \(p| a_ i\}\) and \(S=\{k\in {\mathbb{N}}:\) \(n_ k\in J\}\). Then \(ord_ p u(n)\geq (| S| +1).\) The proofs are tedious but elementary.
Reviewer: F.Beukers

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11A07 Congruences; primitive roots; residue systems
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References:

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