A problem of Ulam

V. V. Ermakov

Matematičeskie zametki, Tome 12 (1972) no. 2, pp. 155-156.

Résumé

Let $R$ be a set of positive integers with usual operations of addition and multiplication $$ a+b=s(a,b);\quad a\cdot b=m(a,b);\quad a,b\in R. $$ A correspondence is set up between each one-to-one (Peano) mapping $p$ of the space $R\times R$ onto the whole of $R$ and the two functions $$ \begin{aligned} \sigma(c)=\sigma[p(a,b)]=s(a,b);\\ \mu(c)=\mu[p(a,b)]=m(a,b). \end{aligned} $$ It is proved in this note that there can be no Peano mapping for which $\sigma(\mu(c))=\mu(\sigma(c))$ for all $c$ in $R$.