Challenge Level

A googol is the number $10^{100}$. What is the smallest whole number $n$ for which

$$

n^4-6n^2> \mbox{googol}?

$$

Can you carefully write out the number on the left-hand side of this inequality for this value of $n$ in base 10?

Did you know ... ?

Although computers are very useful in checking calculations in number theory, it is very difficult to use them to perform calculations involving very, very large numbers. A googol is a big number, but a googolplex, defined to be $10^{\mbox{googol}}$ is unimaginably larger. Such gigantic numbers do make appearances in mathematics from time to time and require the power of pure thought and mathematics to yield to analysis. Rather amusingly, the Googol Corporation call their headquarters 'the googolplex'.

Although computers are very useful in checking calculations in number theory, it is very difficult to use them to perform calculations involving very, very large numbers. A googol is a big number, but a googolplex, defined to be $10^{\mbox{googol}}$ is unimaginably larger. Such gigantic numbers do make appearances in mathematics from time to time and require the power of pure thought and mathematics to yield to analysis. Rather amusingly, the Googol Corporation call their headquarters 'the googolplex'.