Number of pairs with product less than a given number.

Question

How can we find out the number of positive integer pairs with product less than a specific value. Can someone help me with this.

Answer 1

I know one way. Let $a$ and $b$ be two positive numbers. Now let $max(a,b)=b$ so $ab<b^2$. Let $b^2=c$. Now $ab<c$.

Suppose we got a positive number $A$ and we wish two find two positive numbers $\alpha$ and $\beta$ s.t. $\alpha \beta < A$. Then from the first paragraph it is clear that the one sufficient but not necessary condition is that max($\alpha$,$\beta$)<$\sqrt{A}$.

Let me again clarify that max($\alpha$,$\beta$)<$\sqrt{A}$ is not a necessary condition, but it is a sufficient condition.