MATH0034 Number Theory 2020
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MATH0034
Number Theory
Section A (standard calculations)
Questions in this section are standard calculations.
You must show your working clearly to be awarded marks.
Answer all questions.
1. Find integers h and k such that 39h + 43k = 1.
2. Calculate 39-1 modulo 43.
3. Solve the congruence 117x = 9 mod 129.
4. Solve the simultaneous congruences
x = 5 mod 39, x = 6 mod 43.
5. Calculate ϕ(54), where ϕ is the Euler totient function.
6. Calculate 3736724 modulo 54, writing your answer as an integer between 0 and 53.
7. Calculate 54100! mod 2020, writing your answer as an integer between 0 and 2019.
8. Calculate the cyclotomic polynomial Φ 15 (X).
9. Which of the following are primitive roots modulo 101?
100, 3, 19.
Justify your answers.
10. Calculate the quadratic residue symbol ╱ 
、.
11. Which of the following congruences have solutions? Justify your answers.
x2 = 17 x2 = 17 x2 = 17 x2 = 17
mod 19,
mod 191000 ,
mod 381000 ,
mod 3801000 .
12. Using Hensel’s Lemma, find a solution to the following congruence:
x3 + 10x + 6 = 0 mod 54 .
Write your answer as an integer between 0 and 624.
13. Which of the following series converge 7-adically? Justify your answer carefully.
o 7n2 o 7n2 o 7n2
6n2 + 1 , (n!)3 , (n!)! .
For those which converge, calculate the sum modulo 74 .
14. Write 
as a finite continued fraction.
15. Write ′ 15 as an infinite continued fraction.
16. Assuming that ′70 = [8, 2, 1, 2, 1, 2, 16], find the fundamental solution to Pell’s equation x2 一 70y2 = 1.
17. Find a solution in integers to the equation x2 一 70y2 = 11 with y > 200.
1 Section B
Questions in this section are more difficult.
It is not anticipated that many student will be able to answer all the questions. Answer as many as you can.
You may use any theorem from the course without proof.
If you wish to use a theorem not included in the course, then you should include a full proof of the theorem.
1. (a) Prove that for every integer x ∈ z, the number x102 一 x2 is a multiple of 2020.
(b) Find the largest integer n such that x102 一 x2 is a multiple of n for all integers x. Justify your answer carefully.
2. (a) Let x be an integer and let d be a factor of 2x2 一 1. Prove that d2 = 1 mod 16. (b) Hence prove that there are infinitely many prime numbers p such that p2 = 1 mod 16.
3. Let z(7) be the local ring at 7. Find a 7-adically convergent series x = an with an ∈ z(7) ,
such that
x2 = 2 mod 7r for all r > 0.
(Justify your answer carefully.)
4. Assume n is a positive integer, such that the following equation has a solution in integers:
x2 一 17y2 = n. (1)
Prove that equation (1) has a solution in integers x, y satisfying the following bounds:
1 ≤ x + y ′17 < 66 and 
< x 一 y ′17 ≤ n.
(You may assume that N (33 + 8 ′ 17) = 1).
2022-04-25