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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

James Moriarty
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    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." – Conifold Dec 16 '19 at 10:40
  • @Conifold I have already read this paper, thanks for mentioning – James Moriarty Dec 16 '19 at 10:50
  • A proof of this result can be obtained from Baker's Theorem about lower bounds of linear form in logarithms. – Fabio Lucchini Dec 18 '19 at 09:58
  • Wasn't this in Star Trek? – Daniel Donnelly Nov 02 '23 at 23:43
  • Kobayashi's theorem seems to be equivalent to a theorem of Pólya, which is quoted in On a problem in the elementary theory of numbers, a paper of Erdos and Turan, which states that "Their proof depends on a theorem of Mr. Pólya asserting that if we denote by $q_1<q_2< … <q_n<q_{n+1}<…$，the numbers composed of the primes $p_1, p_2, … , p_k$ ,then $q_{n+1}-q_n$ tends to infinity. But the proof of Pólya's theorem is not elementary ; it seems therefore desirable to show the above result in an elementary way." See this pdf. https://www.renyi.hu/~p_erdos/1934-03.pdf – Tong Lingling May 03 '25 at 10:55
  • Thanks to Thomas Bloom and Desmond Weisenberg, the theorem mentioned above originally came from Pólya's 1918 paper "Zur arithmetischen Untersuchung der Polynome" (Math. Z. 1 (1918), no. 2-3, 143–148). Pólya proved a stronger result, and noted the corollary above in his paper. See this MO post https://mathoverflow.net/questions/492243 – Tong Lingling May 07 '25 at 15:06

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