关键词: Algebraic number, Bases, Digit

Abstract: Let a,b and n be integers larger than 1 and let α,β be integers satisfying 0 ≤α<a, 0 ≤β<b. Denote L_α,a ( n ) ,L_β,b ( n ) the numbers of digits in the canonical expansion of n in base a and b which are different from α and β, respectively. Define L_α,a,β,b (n)=L_α,a (n)+L_β,b (n). In this short note, the lower bound of L_α,a,β,b ( n ) was considered, i.e., if loga/logb is irrational, the estimate L_α,a,β,b (n)>loglogn/(logloglogn+C)−1, holds for C ?logab and n>25, which deepens the result of Stewart.

Key words: Algebraic number, Bases, Digit