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
