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 \(k\equiv 4^a \pmod {4^b}\), \(a<b)\) 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 (Heslington)

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