Permalink Submitted by Chris G on December 12, 2017

I think I found an interesting fact about e independently, which of course doesn't make it a new discovery.

I was looking at self roots, that is numbers that are the nth root of n. 1 root 1 = 1, 2 root 2 = 1.4142..., 3 root 3 = 1.4422. 4 root 4 = 1.4142..., 5 root 5 = 1.3797..., 6 root 6 = 1.3480, 100 root 100 = 1.0471..., 1000 root 1000 = 1.0069...

At first the values in the list increase from 1 to a maximum with 3 root 3, then appear to converge on 1 again. But while 3 may have the biggest self root of any integer at 1.4422, a smaller number, 2.7, has an even bigger one at 1.444655705... But e root e rises to 1.444667861..., while 2.718981828 root 2.71898828 (which is just 0.0007 bigger than e) declines again to 1.444667843...

So e has the biggest self root of any number.

(Another interesting fact to emerge is that 2 root 2 = 4 root 4, are they the only two equal self roots?)

Please tell me where I can find this in the literature.

## The self root of e

I think I found an interesting fact about e independently, which of course doesn't make it a new discovery.

I was looking at self roots, that is numbers that are the nth root of n. 1 root 1 = 1, 2 root 2 = 1.4142..., 3 root 3 = 1.4422. 4 root 4 = 1.4142..., 5 root 5 = 1.3797..., 6 root 6 = 1.3480, 100 root 100 = 1.0471..., 1000 root 1000 = 1.0069...

At first the values in the list increase from 1 to a maximum with 3 root 3, then appear to converge on 1 again. But while 3 may have the biggest self root of any integer at 1.4422, a smaller number, 2.7, has an even bigger one at 1.444655705... But e root e rises to 1.444667861..., while 2.718981828 root 2.71898828 (which is just 0.0007 bigger than e) declines again to 1.444667843...

So e has the biggest self root of any number.

(Another interesting fact to emerge is that 2 root 2 = 4 root 4, are they the only two equal self roots?)

Please tell me where I can find this in the literature.