An awful problem about integers in base four. (English) Zbl 0636.10003

Loxton, J. H.; van der Poorten, A. J.

Acta Arith. 49, No. 2, 193-203 (1987).

It is shown that every odd integer is a quotient of elements in the set L of integers whose expansion to the base 4 does not contain 2. The subset S of integers whose expansion only contains 0 or 1 plays an important role. For \(S-S=L\) and \(S=\cup^{\infty}_{n=1}S_ n\), where \(S_ n\) is the set of integers in S with at most n digits. The key result is that for odd k (or more generally for \(k\equiv 4^ a (mod 4^ b)\), \(a<b)\) and sufficiently large n, the number of points in \(S_ n+kS_ n\) is strictly less than \(4^ n\), so that there exist distinct pairs \((s_ 1,s_ 1')\), \((s_ 2,s_ 2')\) with \(s_ 1+ks_ 1'=s_ 2+ks_ 2',\) whence \(k=(s_ 2-s_ 1)/(s_ 1'-s_ 2').\)

Reviewer: M.M.Dodson

MSC:

11A05	Multiplicative structure; Euclidean algorithm; greatest common divisors
11A63	Radix representation; digital problems
11B83	Special sequences and polynomials

Keywords:

integers in base four; representation of odd integers