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Guessing a number from prime tests

Source: 2023 Grosman MO, P6

September 8, 2023
number theorycombinatoricsInfo

Problem Statement

Adam has a secret natural number xx which Eve is trying to discover. At each stage Eve may only ask questions of the form "is x+nx+n a prime number?" for some natural number nn of her choice.
Prove that Eve may discover xx using finitely many questions.
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