# Math Help - Prove there are infinitely many daffodil numbers

1. ## Prove there are infinitely many daffodil numbers

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

2. ## Re: Prove there are infinitely many daffodil numbers

Originally Posted by godelproof
I googled it and found none... can somebody give a proof that there are infinite many of them? thanks

BTW, a daffodil number is a 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.

3. ## Re: Prove there are infinitely many daffodil numbers

Originally Posted by Sudharaka
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.
Wow, 88 is a lovely number... I'd thought there were infinitely many of them! thank you~!

4. ## Re: Prove there are infinitely many daffodil numbers

Originally Posted by Sudharaka
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.
Well... 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 ...

5. ## Re: Prove there are infinitely many daffodil numbers

Dear godelproof,

Wow, 88 is a lovely number... I'd thought there were infinitely many of them! thank you~!
You are welcome.

Well... 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 ...
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 encyclopedia

6. ## Re: Prove there are infinitely many daffodil numbers

Originally Posted by Sudharaka
It is still not known whether Perfect digital invariants are finite or infinite for a given base. Please refer Narcissistic number - Wikipedia, the free encyclopedia
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?

7. ## Re: Prove there are infinitely many daffodil numbers

Originally Posted by godelproof
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?

8. ## Re: Prove there are infinitely many daffodil numbers

Originally Posted by Sudharaka
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