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Substract primes

Source: St Petersburg Olympiad 2010, Grade 11, P4

September 14, 2017
number theory

Problem Statement

Natural number NN is given. Let pNp_N - biggest prime, that ≤N \leq N. On every move we replace NN by N−pNN-p_N. We repeat this until we get 00 or 11. If we get 11 then NN is called as good, else is bad. For example, 9595 is good because we get 95→6→195 \to 6 \to 1. Prove that among numbers from 11 to 10000001000000 there are between one quarter and half good numbers
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