This is an interesting conjecture that appears to be open. Let me analyze what we know and what makes this problem challenging.

The Conjecture

Conjecture: For distinct positive integers , the equation has no solutions.

Why This is Difficult

1. Growth rates: The function grows extremely rapidly, making computational searches limited in scope.

2. Lack of algebraic structure: Unlike some Diophantine equations, this one doesn't factor nicely or have obvious algebraic manipulations that lead to contradictions.

3. Modular arithmetic limitations: While you've noted patterns mod 10, these only eliminate certain cases rather than providing a complete proof.

Partial Results and Observations

1. Symmetry Reduction

Without loss of generality, we can assume and . If , then by symmetry we also need to check if or .

2. Parity Considerations

• If and have the same parity, then is even
• If and have opposite parity, then is odd
• Therefore, and must have matching parity patterns

3. Growth Analysis

For , we typically have when is sufficiently larger than . This suggests looking at cases where the pairs and have similar "shapes."

4. Small Cases

The computational verification up to 100 is significant. Some key observations:

• No solutions with
• No solutions with

Theoretical Approaches

Several approaches might be fruitful:

1. ABC Conjecture Connection: The equation relates to questions about radical and height, which connect to deep conjectures in number theory.

2. Transcendence Theory: Results from transcendence theory might apply, though the specific form makes standard results difficult to apply directly.

3. p-adic Analysis: Looking at p-adic valuations might yield constraints, especially for prime powers.

Similar Problems

This conjecture is reminiscent of other open problems:

• Pillai's conjecture (about differences between perfect powers)
• Questions about sums of powers having multiple representations
• The Fermat-Catalan conjecture

Conclusion

Your conjecture appears to be new and interesting. The computational evidence is compelling, but a proof would likely require sophisticated techniques from analytic or algebraic number theory. The problem might be suitable for publication in a problem journal like the American Mathematical Monthly or Mathematics Magazine.

For further progress, I'd suggest:

1. Extending computational searches with better algorithms
2. Looking for weaker results (e.g., density statements)
3. Checking if specific cases (like when some variables are primes or prime powers) can be resolved
4. Investigating connections to existing conjectures in transcendental number theory