1
$\begingroup$

A PDDI (perfect digit-to-digit invariant) is a number which is also the sum of each of it's digits raised to them self.

My main problem with the number 438579088 being a PDDI is the 7th number - 0. Breaking 438579088 down to check if it's a PDDI or not:

  • 4^4 = 256
  • 3^3 = 27
  • 8^8 = 16777216
  • 5^5 = 3125
  • 7^7 = 823543
  • 9^9 = 387420489
  • 0^0 = 1
  • 8^8 = 16777216
  • 8^8 = 16777216

I had a program to find the sum of these numbers and I've checked it numerous times with different approaches, and each time I receive the number 438579089, 1 number off 438579088. My hypothesis is that the 0 is being miscalculated - I do seem to remember any number to the power of 0 is 1, however.

In my code, (where num is 438579088) total would indeed be 438579088:

 for i in str(num):
     if int(i) != 0:
         total += pow(int(i), int(i))

but, when I stopped checking if i was 0,

for i in str(num):
     total += pow(int(i), int(i))

I received 438579089. Why does this happen? This isn't so much a code problem but a mathematical problem. I can't understand why many sources say this number is a PDDI where also saying that 0^0 = 1.

$\endgroup$
2
$\begingroup$

I've never come across this before. But one can define concepts however one likes, and a quick check around indicates that a PDDI comes with the convention that $0^0 = 0$. See for instance wikipedia or magic-squares.net.

You might find it interesting to study Meunchhausen Numbers. It is not all obvious to me that there are exactly four Meunchhausen numbers. This gives a base-level interest to using the convention that $0^0 = 0$, since

You might be interested in studiying PDDI numbers using the more standard convention that $0^0 = 1$. I encourage you - go for it! This is part of the attraction of recreational mathematics.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.