Chapter 2.5, Problem 32E

Show that when you substitute ${\left(3n+1\right)}^2$ for each occurrence of n and ${\left(3m+1\right)}^2$ for each occurrence of m in the right-hand side of the formula for the function $f\left(m,n\right)$ in Exercise 31, you obtain a one-to-one polynomial function $\text{Z}\times \text{Z}\to \text{Z}$ . It is an open question whether there is a one-to-one polynomial function $\text{Q}\times \text{Q}\to \text{Q}$ .

Show that ${\text{Z}}^{+}\times {\text{Z}}^{+}$ t is countable by showing that the polynomial function $f:{\text{Z}}^{+}\times {\text{Z}}^{+}\to {\text{Z}}^{+}$ with $f\left(m,n\right)=\left(m+n-2\right)\left(m+n-1\right)/2+m$ is one-to-one and onto.