Kobayashi's theorem

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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 $M=\{m_n{\}}_{n=1}^{\mathrm{\infty }}$ and $M+a=\{m_n+a{\}}_{n=1}^{\mathrm{\infty }}$, 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

$k\ge 1$

such that

$\lfloor \frac{n^k}{k}\rfloor$

is odd.

See also

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

References

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📰 Article(s) of the same category(ies)

• Abouabdillah's theorem
• Lamé’s Theorem

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