Dear godelproof,
It was proved that there are only 88 daffodil numbers (commonly known Narcissistic Numbers) which are in base 10. Please refer, Narcissistic Number -- from Wolfram MathWorld.
I googled it and found none... can somebody give a proof that there are infinite many of them? thanks
BTW, a daffodil number is an n-digit number whose value is equal to the sum of n-th power of each digit, like
Dear godelproof,
It was proved that there are only 88 daffodil numbers (commonly known Narcissistic Numbers) which are in base 10. Please refer, Narcissistic Number -- from Wolfram MathWorld.
Dear godelproof,
You are welcome.Wow, 88 is a lovely number... I'd thought there were infinitely many of them! thank you~!
These are called "Perfect digital invariants". It is still not known whether Perfect digital invariants are finite or infinite for a given base. Please refer Narcissistic number - Wikipedia, the free encyclopediaWell... the proof is almost trivial though
so let's relax the problem a little...
Define a rose number to be an n-digit number whose value is equal to the sum of m-th power of each digit~ like this one given below
And i can think again ...
Thank you! You are being so helpful!
But there doesn't seem to be any articles about PDI in wiki! I can only find some theorems here Digital Invariants: Observations & Theorems
" It is still not known whether Perfect digital invariants are finite or infinite for a given base" where does this conclusion come from?
It is stated in the Wikipedia link with a citation. Read, Finite or Infinite?
Interesting... They can prove that there are infinitely many bases in which there are infinitely many PDIs, but they can NOT prove there whether there are infinitely many PDI in the 10-base case. Hmmm... seems so classic hard number theorical problem: the devil hides in specifics