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LANCASTER UNIVERSITY
2008 EXAMINATIONS
PART II (Third or Fourth Year)
MATHEMATICS & STATISTICS 2 hours
Math 328: Number Theory
You should answer ALL Section A questions and TWO Section B questions.
In Section A there are questions worth a total of 50 marks, but the maximum mark that you can gain there is capped at 40.
The notation (a, b) is used for the greatest common divisor of a and b.
SECTION A
A1. What are the possible values mod 8 of (i) squares, (ii) numbers of the form a^2 + 2b^2? [5] A2. Show that (284, 821) = 1 and find an integer x (between 1 and 821) such that 284 x ≡ 1 mod 821. [5] A3. Show that 13n^ + 5^2 n−^1 is a multiple of 6 for all n ≥ 1. [3] A4. Let a, n be positive integers with (a, n) = 1. Prove the following statements. (i) If n|ab, then n|b. (ii) There exists x such that ax ≡ 1 mod n, and if y is any other number such that ay ≡ 1 mod n, then y ≡ x mod n. [7] A5. Factorise 18 419 by Fermat’s method. [4] A6. Prove that there are infinitely many prime numbers. [5] A7. An outdoor pursuits course, which limits its numbers to 800, forms groups of eight for rowing, groups of nine for hiking and groups of eleven for football. Four participants are left over in the case of rowing, three in the case of hiking and six in the case of football. How many participants are there? [8] A8. Define Euler’s function φ and show that φ(n) is even for n ≥ 3. [6] please turn over
SECTION A continued
A9. Let uS (n) =
1 if n is a square, 0 otherwise. Show that the function uS is multiplicative. [7]
SECTION B
B1. Let τ (n) denote the number of divisors of n, and σ(n) the sum of the divisors of n.
(i) Show that τ (n) is odd if and only if n is a square. Deduce that if n is odd, then σ(n) is odd if and only if n is a square. [9] (ii) Derive an expression for τ (n) in terms of the prime factorization ∏mj=1 pk j jof n. Deduce that τ is multiplicative. [6] (iii) Prove that τ (mn) ≤ τ (m)τ (n) for any positive integers m, n. [7] (iv) Describe, in terms of the possible patterns of prime factors, the numbers n with τ (n) = 16, and find the smallest such number. [8]
B2. (i) Let^ p^ be prime. Show that if^ r^2 ≡^ s^2 mod^ p, then^ r^ is congruent to either^ s^ or^ −s^ mod p. [3] (ii) Let p ≥ 3 be prime, and write p = 2q + 1. By considering pairs of congruence classes of the form {r,ˆ rˆ−^1 }, or otherwise, show that −1 is a quadratic residue mod p if p ≡ 1 mod 4, but not if p ≡ −1 mod 4. You may assume that there are exactly q distinct quadratic residues mod p. [10] (iii) Let n be an integer, not a square. Suppose that there exists u such that u^2 ≡ −1 mod n. By considering numbers of the form r − su for a suitable range of r and s (or otherwise), prove that n is the sum of two squares. [9] (iv) Suppose that n can be expressed as x^2 + y^2 , with (x, y) = 1. Prove that there exists u such that u^2 ≡ −1 mod n, and deduce that all odd prime factors of n are congruent to 1 mod 4. [8]
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