4

Part of 2018 China Second Round Olympiad

Problems(2)

Coprime to sum of terms before it

Source: China second round (A) 2018 Q4

Define sequence

\{a_n\}

a_1

is any positive integer, and for any positive integer

n\ge 1

a_{n+1}

is the smallest positive integer coprime to

\sum_{i=1}^{n} a_i

and not equal to

a_1,\ldots, a_n

. Prove that every positive integer appears in the sequence

\{a_n\}

number theorySequence

a^k+n are all composite numbers

Source: China second round 2018 (B) Q4

Prove that for any integer

a \ge 2

and positive integer

n,

there exist positive integer

k

a^k+1,a^k+2,\ldots,a^k+n

are all composite numbers.