Recreational Number Theory, Unsolved Problem

• MHB
• Tamas
In summary, the conversation discusses the search for a perfect power k^m > 1 where k, m, and k^m do not contain 2 in their decimal digits and do not share any decimal digit. The speaker has tried using an online calculator and manual checking, but has not found a solution. They are interested in finding a solution or proof of its impossibility, but do not have the capability to do so. The conversation also mentions that such powers exist with all other decimal digits except 2, and provides examples for each digit. The possibility of a brute force search or a proof is mentioned, but it is currently unknown if such a solution exists.
Tamas
Find a perfect power k^m > 1 where k, m, k^m do not contain 2 in their decimal digits, nor do share any decimal digit, no matter if k^m might possibly be expressed in more than one way for some value, e.g. 8^2 = 4^3. I do not know if such an integer exists at all, or how many and how large they are if they do. What did I do to try finding a solution to this problem? I cannot compute or program, so I tried an online big integer calculator with manual input and checking. This was, though methodical, but slow. I got to very large numbers without success, and the more digits appeared, the less likelihood remained for finding a match. Since I am not a mathematician, let alone a number theorist, I cannot prove or disprove the existence of such integer. Finding one can be a proof, but it is beyond my capabilities. Still, this interesting problem fascinates me and I hope others will like it too.

The more interesting it is, because powers exist with all the other individual decimal digits d missing from the otherwise also not decimal digit sharing k, m, and k^m.
So, d = 2 seems to be elusive, or, is indeed the exception?
Easily found examples for each d not equal 2 as follows:
For d = 0 -> 2^3 = 8; for d = 1 -> 3^2 = 9; for d = 3 -> 67^2 = 4489; for d = 4 -> 33^2 = 1089; for d = 5 -> 2^4 = 4^2 = 16; for d = 6 -> 7^2 = 49;
for d = 7 -> 44^2 = 1936; for d = 8 -> 34^2 = 1156; and for d = 9 -> 38^2 = 1444.
I believe a brute force search may bring up perhaps an example for d = 2, or an insightful proof is found for its impossibility and therefore non-existence.
Without these, we don't know.

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1. What is recreational number theory?

Recreational number theory is the branch of mathematics that deals with the study of numbers and patterns for fun and entertainment, rather than for practical applications.

2. What are some examples of recreational number theory problems?

Some examples of recreational number theory problems include finding patterns in prime numbers, exploring the properties of perfect numbers, and solving mathematical puzzles involving numbers.

3. What is an unsolved problem in recreational number theory?

One unsolved problem in recreational number theory is the Collatz conjecture, which states that for any positive integer, if it is even, divide it by 2, and if it is odd, multiply it by 3 and add 1. Repeating this process will eventually lead to the number 1.

4. How is recreational number theory different from other branches of mathematics?

Recreational number theory differs from other branches of mathematics in that it is not focused on solving real-world problems or developing practical applications. Instead, it is primarily concerned with exploring patterns and relationships between numbers for the sake of curiosity and enjoyment.

5. How can recreational number theory be applied in everyday life?

While recreational number theory may not have direct practical applications, it can help improve critical thinking and problem-solving skills, as well as foster a deeper understanding and appreciation for the beauty and complexity of mathematics.

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