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Sequence with Primitive Prime Factor

Source: China 2016 TST Day 2 Q4

March 16, 2016
number theorySequenceprime numbers

Problem Statement

Let c,d≥2c,d \geq 2 be naturals. Let {an}\{a_n\} be the sequence satisfying a1=c,an+1=and+ca_1 = c, a_{n+1} = a_n^d + c for n=1,2,⋯n = 1,2,\cdots. Prove that for any n≥2n \geq 2, there exists a prime number pp such that p∣anp|a_n and p∤aip \not | a_i for i=1,2,⋯n−1i = 1,2,\cdots n-1.
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