The unique decomposition of natural numbers using prime numbers by the gamma function.

0.参考文献

[1]Gakken Group, "The definition and properties of the Gamma function (a generalization of factorial) | Beautiful Stories of High School Mathematics"<https://manabitimes.jp/math/960> 2024.7.5.23:02 Accessed.

1.What is the Gamma function[1]

The Gamma function is a generalization of the factorial. For a natural number nn, it becomes

Thoughts

If the uniqueness of prime factorization holds, then for the Gamma function, there should exist some prime numbers such that for any natural number nn, Γ(n)Γ(n) can be uniquely expressed in terms of Γ(p)Γ(p) (excluding permutations).

Let's examine this in detail.

We will not provide a proof, but it appears that this is likely to hold.