## 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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