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Kurskod: TATA 54 Provkod: TEN 1 NUMBER THEORY

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Kurskod: TATA 54 Provkod: TEN 1 NUMBER THEORY, Talteori 6 hp

March 19, 2015, 14–18.

Matematiska institutionen, Link¨opings universitet.

Examinator: Leif Melkersson

Inga hj¨alpmedel ¨ar till˚atna! (For example books or pocket calculators are not allowed!)

You may write in Swedish, if you do this consistently.

You are rewarded at most 3 points for each of the 6 problems.

To get grade 3, 4 or 5, you need respectively 7, 11 and 14 points.

(1) Find the remainder when 78253? is divided by 25.

(2) (a) Can the number 1845 be written as the sum of two squares of integers?

(b) The same question for the number 3510.

(c) What is the number of ordered pairs (x, y) ∈ Z × Z of integers, such that 11 700 000 = x2+ y2?

(3) Does the congruence x2 ≡ 17 (mod 77) have a solution?

(4) (a) Compute the simple continued fraction of√ 80.

(b) Find the two smallest positive solutions x, y of the dio- phantine equation x2− 80y2 = 1

(5) (a) Show that 6 is a primitive root modulo 41.

(b) Find a primitive root modulo 82.

(6) (a) What is the largest order of an integer modulo 77.

(b) Find an integer which has the largest possible order modulo 77.

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