Derive the continued fraction of √7. Find the value of β, given its continued fraction expression β = [1, ̄7], i.e., a0 = 1 and ai = 7 for all i ∈ {1, 2, . . .}.

Introduction to number theory

MA2011/21

Section A

1. (a) Use Fermat’s Method to factorize 1147. [5]

(b) Euclid’s algorithm applied to two numbers a and b computed the quotients

q1 = 3, q2 = 3, q3 = 3, q4 = 3, q5 = 3 (in this order) and their greatest common divisor gcd(a, b) = 3.

Compute a and b. [5]

2. Using the extended version of Euclid’s algorithm, find a solution to the following Diophantine linear equation.

77x + 91y + 143z = 2 [10]

3. Find all solutions for the following pair of simultaneous congruences.

262x ≡ 3 mod 807  3x ≡ 2 mod 5 [10]

4. Show that the equation

2x3 + 7y3 = 4 has no solution in integers. [10]

5. (a) Derive the continued fraction of √7. [5]

(b) Find the value of β, given its continued fraction expression β = [1, ̄7], i.e., a0 = 1 and ai = 7 for all i ∈ {1, 2, . . .}. [5]

MA2011/21

Section B

6. The Euler’s function φ(m) counts the number of integers a with 0 ≤ a < m and gcd(a, m) = 1.

(a) Let m = pa where p is prime. Show that φ(m) = m(1 − 1/p). [5]

(b) Let m = pa and n = qb where p and q are distinct primes. Show that

φ(mn) = φ(m)φ(n). [5]
(c) Compute φ(17) and φ(77). [4]
(d) Show that if gcd(a, m) = 1, then
aφ(m) ≡ 1 mod m. [11]

7. (a) Show that there exists a constant c > 0 such that  |b√7 − a| ≥ 1 cb , for all natural numbers a, b (b 6 = 0). [7]

(b) Compute two distinct, positive integer solutions to x2 − 17y2 = 1. [7]

(c) Let α > 0 be a real number and let pn/qn denote the corresponding n-th convergent.