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If P(0),P(1),...,P(2^n+1) are divides by 2^{2^n}, then all values are also

Source: Saint Petersburg olympiad 2024, 11.6

September 22, 2024

number theoryPolynomialsalgebrapolynomial

Problem Statement

Polynomial

P(x)

with integer coefficients is given. For some positive integer

n

numbers

P(0),P(1),\dots,P(2^n+1)

are all divisible by

2^{2^n}

. Prove that values of

P(x)

in all integer points are divisible by

2^{2^n}