1.1

Part of 2022 Saudi Arabia BMO + EGMO TST

Problems(2)

product of first k primes + 1= perfect power

Source: 2022 Saudi Arabia November Camp Test 1.1 BMO + EGMO TST

Find all positive integers

k

such that the product of the first

k

primes increased by

1

is a power of an integer (with an exponent greater than

1

number theoryPerfect PowersPerfect power

M_{P(x)} = x\in [0,2021] \max}{|P(x)|.

Source: 2022 Saudi Arabia January Camp Test 1.1 BMO + EGMO TST

For each non-constant integer polynomial

P(x)

, let’s define

M_{P(x)} = \underset{x\in [0,2021]}{\max} |P(x)|.

1. Find the minimum value of

M_{P(x)}

P(x) = 1

.
2. Suppose that

P(x) \in Z[x]

P(x) = n

2 \le n \le 2022

M_{P(x)} \ge 1011

algebrapolynomial