Kobayashi's theorem explained

In number theory, Kobayashi's theorem is a result concerning the distribution of prime factors in shifted sequences of integers. The theorem, proved by Hiroshi Kobayashi, demonstrates that shifting a sequence of integers with finitely many prime factors necessarily introduces infinitely many new prime factors.^[1]

Statement

Kobayashi's theorem: Let M be an infinite set of positive integers such that the set of prime divisors of all numbers in M is finite. For any non-zero integer a, define the shifted set M + a as

M + a = \{m + a | m \in M\} . Kobayashi's theorem states that the set of prime numbers that divide at least one element of M + a is infinite.

Proof

The original proof by Kobayashi uses Siegel's theorem on integral points, but a more succinct proof exists using Thue's theorem.

Kobayashi's theorem is also a trivial case of the S-unit equation.

Example

Problem (IMO Shortlist N4): Let

n > 1 be an integer. Prove that there are infinitely many integers k \ge 1 such that \lfloor \tfrac{n ^{k}{k}} \rfloor is odd.

See also

• Thue's theorem
• Siegel's theorem on integral points
• Olympiad mathematics

Notes and References

1. Kobayashi. Hiroshi. On Existence of Infinitely Many Prime Divisors in a Given Set. Tokyo Journal of Mathematics. 4. 2. 379–380. 1981-12-01. 10.3836/tjm/1270215162.