Sum equals product

I love investigating patterns in numbers and shapes, I particularly like finding equations describing the patterns. Few years ago, while working on a maths assignment, I wondered if the number 2 is the only number that gives the same answer when added together and multiplied together i.e 2+2=2*2=4. It seemed obvious that 2 had to be the only number that you can do this with, but was there other numbers that when added together N number of times would give the same result as being multiplied together N times. So I decided to investigate and see where it leads me. I originally called the results of these sums and products ‘perfect numbers P’ , but I later realised that perfect number was a name given to something else.
I’ll start with defining what the number P is, P satisfies the following:

Where N is the number of terms and x is the number we’re adding and multiplying together. If we equate the above two and rearrange to get:

From equations 3 and 4 we can immediately see that N cannot equal zero or one otherwise we get undefined results. I’ll show you later how N can approach 0 and 1 to give some nice results but first I want to use equations 3 and 4 to produce some values of x and P.

Dx and Dp are the difference in the values of x and P respectively. From the table we can see that Dp seems to be converging to 1 and Dx is converging towards 0. This tells us that as N goes to infinity so does P, x however is converging towards a single value of 1.
Now lets consider what happens when N=0. We already know that we can’t just enter 0 into the equation, but we can sort of cheat and say what happens when N=0+h where h can be as small as we like, or to put it another way the lim(0+h)=0 as h approaches 0. If you can remember your basic differentiation you’ll know what I have done there. So if we put this new value into equation 4 we get P=1. A far more interesting result is obtained when we apply the same method to N=1. So once again the lim(1+h) = 1 as h approaches 0. Now if you put that into equation 4 we get p=e where e is the base of the natural logarithm also known as Euler’s number. I don’t know about you but I find this result amazing. We can also do the same thing for x. When N=0+h we get x=infinity and when N=1+h we once again get the number e. One final bit of observation from the results is that P starts from 1 and goes to infinity whereas x starts from infinity and goes to 1.
I realise that this is not very mathematically rigorous but I hope you can appreciate the results. I have not worked on this any further so if anyone else finds any more interesting results please let me know.

Link:
Mathcentral