Kobayashi's theorem

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.

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.

Proof by Thue's theorem:

Suppose for the sake of contradiction that the set of prime divisors of M+a is finite. Enumerate $M=\{m_{n}\}_{n=1}^{\infty }$ and $M+a=\{m_{n}+a\}_{n=1}^{\infty }$, and write each element as m_n = mx³ and m_n + a = ny³ for m and n cube-free integers. If the prime divisors of M and M+a are finite, then there is only a finitely many possible values of m and n; hence, there is a finite number of equations of the form ny³ - mx³ = a. Since the left-hand side is irreducible over the rational numbers, by Thue's theorem, each equation has a finite number of solutions in integers x and y, which is not possible because the set M is unbounded. Thus our original assumption was incorrect, and the set of prime divisors of M+a is infinite.

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 $k\geq 1$ such that $\lfloor {\tfrac {n^{k}}{k}}\rfloor$ is odd.

See also

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