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Part of 2021 Saudi Arabia Training Tests

Problems(1)

each integer k satisfies both |x - k^2| > 10^6 and |x - k^3| > 10^6

Source: 2021 Saudi Arabia Training Lists p33 https://artofproblemsolving.com/community/c2758131_2021_saudi_arabia_training_tests

Call a positive integer

x

to be remote from squares and cubes if each integer

k

|x - k^2| > 10^6

|x - k^3| > 10^6

. Prove that there exist infinitely many positive integer

n

2^n

is remote from squares and cubes.

number theoryinequalities