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Number of odd quadratic residues mod n

Source: Serbia MO 2018 P2

April 2, 2018
number theoryQuadratic ResiduesSequence

Problem Statement

Let n>1n>1 be an integer. Call a number beautiful if its square leaves an odd remainder upon divison by nn. Prove that the number of consecutive beautiful numbers is less or equal to 1+⌊3n⌋1+\lfloor \sqrt{3n} \rfloor.
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