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

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
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