Find all ordered triples of primes $(a, b, c)$ such that $a \mid {^bc} + 1, b \mid {^ca} + 1, c \mid {^ab} + 1$.

by Lê Thành Đạt   Last Updated August 14, 2019 06:20 AM

Find all ordered triples of primes $(a, b, c)$ such that $$\large a \mid {^bc} + 1, b \mid {^ca} + 1, c \mid {^ab} + 1$$

Notation: $$\large {^xy} \text{ or } x^{\underline y}$$ are nonations for tetration. ${^xy}$ simply is $y$ copies of $x$ combined by exponentiation, right-to-left. This is more of an extension of problems such that

Find all ordered triples of primes $(x, y, z)$ such that $$ \large x \mid yz + 1, y \mid zx + 1, z \mid xy + 1$$

The solution to this problem is $(2, 3, 7)$ and all of the permutations of $(2, 3, 7)$.

Find all ordered triples of primes $(m, n, p)$ such that $$ \large m \mid n^p + 1, n \mid p^m + 1, p \mid m^n + 1$$

The solution to this problem is $(2, 3, 5)$ and all of the permutations of $(2, 3, 5)$.

If I had to make a guess, the solutions might be "$(2, 5, 7)$ and all of the permutations of $(2, 5, 7)$" or "there aren't any at all".

I am fully aware of the fact much, much computing will be involved in the solution. But if anyone can come up with a solution, no matter how it is figured out, I would appreciate your work.



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