The number 56.96124843226

For a long time I have had an interest infunctions like the factorial that yield very large numbers for fairly small inputs.

As a student in year 11 mathematics (where my most famous memory is learning that A times B ≠ B times A) is learnt that the function nⁿ obtained by multiplying an integer by itself n times grew more prapidly than n factorial: indeed the series of n!/nⁿ I knew to converge by experiment on my calculator.

However, whereas the largest factorial below a given power of 10¹⁰ is easily given by the following sequence, finding the largest number for which nⁿ can be calculated with an ordinary hand calculator is quite hard.

Whilst a simple test will show it to be between 56 and 57, it becomes very hard when one wants a precise value of the integer n for which nⁿ = 10¹⁰⁰. Working with my Casio fx-82MS, it gave errors after about eight decimal places, and turning to a computer calculator which can go above 10¹⁰⁰, I found that the best approximation is
n = 56.96124843226

This would be a great curio for someone interested in number theory like myself. I never see nⁿ discussed serious despite its rapid growth, so I wonder what people think of the simple piece of work I did recently?!