Problem 5

Consider an infinite sequence

a_1,a_2,\ldots

of positive integers such that

100!(a_m+a_{m+1}+\cdots+a_n)

is a multiple of

a_{n-m+1}a_{n+m}

for all positive integers

m,n

with

m\le n

. Prove that the sequence is either bounded or linear. A sequence is bounded if there is a constant

N

such that

a_n<N

for every positive integer

n

, and it is linear if

a_n=n\cdot a_1

for every positive integer

n

Step 9 of 10: A large prime power persists in every sufficiently large term

Previous step Next step

p^e>K^2,\quad p^e\mid a_r\Longrightarrow a_s\ge a_r\Rightarrow p^{e'}\mid a_s,\quad p^{e'}>K>c

Detailed analysis

Suppose the normalized sequence is unbounded. Then some term $a_r$ is divisible by a prime power $p^e>K^2$ : otherwise all prime powers in all terms would be at most $K^2$ , and only finitely many primes and exponents could occur, making the sequence bounded. For every term whose value $a_s\ge a_r$ , the quotient $a_r/\gcd(a_r,a_s)$ is at most $K$ , so $a_s$ is divisible by $p^{e'}$ with $p^{e'}\ge p^e/K>K$ . In particular $e'>v_p(c)$ , because $p^{v_p(c)}\le c<K$ .