Track / Stage 6 / 7 of 400 #1007 of 1964

We say that a positive integer is lazy if it has no prime divisor greater than $3$. Prove that there are at most two lazy numbers strictly between two consecutive square numbers.

This one wants a proof. Work it on paper, then check yourself against the publisher's own solution, linked below. Be honest about it: the record is only any use to you if it is.

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Source: KöMaL, licensed Rights held by KöMaL and the MATFUND Foundation. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project. Solutions are the publisher's, linked not copied.