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The University of British Columbia Final Examinations – April 2005 Mathematics 313 Instructor: V. Vatsal Time: 2.5 hours
Name: Student Number:
Signature: Section Number:
Special instructions:
- No calculators, books, notes, or other aids allowed.
- Answer all 8 questions. All questions are worth 5 marks.
- Give your answer in the space provided. If you need extra space, use the back of the page.
- Show enough of your work to justify your answer. Show ALL steps.
Problem 1: Find all primes p for which the congruence X^2 + 3X + 1 ≡ 0 (mod p) has a solution.
Problem 3: Find the first 5 convergents of the continued fraction expansion for e = [2, 1 , 2 , 1 , 1 , 4 , 1... ].
Problem 4: If d > 1, show that the continued fraction expansion of
d^2 − 1 is given by [d − 1 , 1 , 2 d − 2 , 1 , 2 d − 2... ] (the string 1, 2 d − 2 repeats).
Problem 6: Find positive integers x and y such that x^2 + y^2 = 34255 = 5 · 13 · 17 · 29.
Problem 7: Show that the equation x^2 − 5 y^2 = 3z^2 has no solutions with x, y, z nonzero integers.
