There is a theorem in Number Theory due to Hiroshi Kobayashi (possibly less famous). The statement of this theorem is quite simple-looking. The original proof of Kobayashi relies on Siegel's Theorem in Diophantine Geometry, which is a deep theorem in Geometry of Numbers.

Statement of Kobayashi's Theorem -

Let $M$ be an infinite set of positive integers such that the set of prime divisors of the numbers in $M$ is finite. Then the set of primes dividing the numbers in the set $M+a:= \{ m + a \: | \; m \in M \}$ is infinite, where $a$ is a fixed non-zero integer.

Does there exist any elementary proof of this result(elementary in the sense that the proof must not include any application of geometry of numbers or Diophantine geometry)?

The original paper of Hiroshi Kobayashi can be found here

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    $\begingroup$ In Musings on the Prime Divisors of Arithmetic Sequences Morton gives an elementary proof of a weaker result and states:"It can also be deduced from a striking theorem of H. Kobayashi... Can Kobayashi's theorem be proved in an elementary way? I do not know." $\endgroup$
    – Conifold
    Dec 16, 2019 at 10:40
  • $\begingroup$ @Conifold I have already read this paper, thanks for mentioning $\endgroup$
    – ShBh
    Dec 16, 2019 at 10:50
  • $\begingroup$ A proof of this result can be obtained from Baker's Theorem about lower bounds of linear form in logarithms. $\endgroup$ Dec 18, 2019 at 9:58
  • $\begingroup$ Wasn't this in Star Trek? $\endgroup$ Nov 2, 2023 at 23:43


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