Eli5: Why are there only 88 narcissistic numbers in base 10?

Question

Eli5: Why are there only 88 narcissistic numbers in base 10?

In: 2

Answer 1

I’m not sure that there’s an ELI5 answer, but here’s a go.

A number (let’s call it the ‘input’) is narcissistic when it equals the sum of each of its digits, each raised to the power of its length (let’s call that the ‘output’).

On a graph, the line of available numbers, the input, is straight – x=1, y=1 and so on.

The second line, plotting the output is wiggly – all graphs of polynomial outputs are. It is also slanted upwards. Every time the output line crosses the input line, we have a narcissistic number.

However, there comes a point where the output line is above the input line, and *never comes back below it* – ie, above a certain point, the output is always bigger than the input. From that point on, there are no more narcissistic numbers.

Edit: sum not product. As u/chromotron points out, this would be very easy if it was the product!