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An Introduction to the Theory of Numbers

Exercise 2.1.11 (Least positive integer $x$ such that $13 \mid (x^2 + 1)$)

Find the least positive integer $x$ such that $13 ∣ (x^{2} + 1)$ .

Answers

Proof. $13 ∣ 5^{2} + 1 = 26$ (and $13 ∤ 1^{1} + 1 = 2, 13 ∤ 2^{2} + 1 = 5, 13 ∤ 3^{2} + 1 = 10, 13 ∤ 4^{2} + 1 = 17$ ).

Thus $x = 5$ is the least positive integer such that $13 ∣ (x^{2} + 1)$ . □

2024-08-21 10:20

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