Definition: A Rashee number is an integer that can form a palindrome through the iterative process of repeatedly reversing its digits and subtracting the resulting numbers.
To check if the number $3226$ is a Rashee number or not:
$3226 \to (3226-6223) = -2997 \to (-2997+7992)=4995 \to (4995-5994)=-999$
So yes, $3226$ is a Rashee number.
1) give one example of non-Rashee number in base-$10$?
2) show that there are infinitely many Rashee numbers in base-$10$?
3) show that there are infinitely many non-Rashee numbers in base-$10$?
Does the idea of this Rashee number already exist? Is it as hard as proving Lychrel number?
Best Answer
1) $2178$ is NOT a Rashee number: $$2178 \to (2178-8712)=-6534 \to (-6534+4356)=-2178\\ \to (-2178+8712)=6534 \to (6534-4356)=2178$$ and we have a cycle. Another one is, of course, $6534$.
2) Any palindromic number is a Rashee number so they are infinite. Also $2\underbrace{99\dots 9}_{\text{$n$ times}}1$ is a Rashee number: $$2\underbrace{99\dots 9}_{\text{$n$ times}}1 \to (2\underbrace{99\dots 9}_{\text{$n$ times}}1-1\underbrace{99\dots 9}_{\text{$n$ times}}2)=\underbrace{99\dots 9}_{\text{$n+1$ times}}.$$
3) It is easy to verify that $21\underbrace{99\dots 9}_{\text{$n$ times}}78$ is NOT a Rashee number and therefore there are infinitely many non-Rashee numbers.