7

Part of 2020 South East Mathematical Olympiad

Problems(2)

Sylvester-Schur revised

Source: 2020 China Southeast 10.7

Given any prime

p \ge 3

. Show that for all sufficient large positive integer

x

, at least one of

x+1,x+2,\cdots,x+\frac{p+3}{2}

has a prime divisor greater than

p

Square-free numbers with certain gap

Source: 2020 China Southeast 11.7

Arrange all square-free positive integers in ascending order

a_1,a_2,a_3,\ldots,a_n,\ldots

. Prove that there are infinitely many positive integers

n

a_{n+1}-a_n=2020