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

March 8, 2012, 14–18.

Matematiska instituitonen, Link¨opings universitet.

Examinator: Leif Melkersson

Inga hj¨alpmedel ¨ar till˚atna! (E.g. no pocket calculators are allowed!) You can 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) Show that n8 ≡ n2 (mod 126) for all integers n.

(2) (a) Find a primitive root of 25.

(b) Solve the congruence x7 ≡ 7 (mod 25).

(3) Is it possible to write the number 1729 as the sum of (a) two squares of integers

(b) four squares of integers (c) three squares of integers

(4) Show that 121 is an Euler pseudoprime to the base 3.

(5) (a) Find the continued fraction expansion of √ 95.

(b) Find the least solution in positive integers of the Pell equa- tion x2− 95y2 = 1.

(6) How many ordered pairs (x, y) ∈ Z × Z of integers are there such that x2+ y2 = 63700

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