0

I was solving this initial problem at Project Euler website (link: https://projecteuler.net/problem=30). It asks to find the sum of all the numbers in which the sum of the digits to the fifth power is equal to the number itself. I basically searched in the space between $10$ and $10^6$. I've chosen this upper limit since I thought that, at the edge, all digits would be $9$, maximizing the sum, but $7*(9^5) < 10^6$, and, as the number grows, it get even further from that maximal possible sum, even with the number of digits still increasing. But I know this is not very formal and I would like to know how can I describe that condition in a more formal way, in order to help me think for future problems.

Thank you!

Kamal Saleh
  • 6,497
drover
  • 1

1 Answers1

1

You can say that if the number $n$ has $d$ digits, then $10^{d-1} \leq n < 10^d$, and if $s(n)$ is the sum of the fifth powers of the digits of $n$, then $s(n) \leq d \times 9^5 < 60000d$, so there can be no solutions when $10^{d-1} > 60000d$.

There is a well-known result that $a^d$ grows faster than any polynomial of $d$, so once you find a value of $d$ that causes $10^{d-1}$ to be bigger than $60000d$ you know that that's your upper bound, and you can usually find that easily enough with a bit of educated guessing.

ConMan
  • 24,300