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MTH5130: Number Theory
创建日期：2024-04-15 16:03:18
所属分类：数学mathematics
标签： MTH5130
Number Theory

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MTH5130: Number Theory
You should attempt ALL questions. Marks available are shown next to the ques-
tions.
In completing this assessment, you may use books, notes, and the Internet. You
may use calculators and computers, but you must show your work for any
calculations you do. You must not seek or obtain help from anyone else.
At the start of your work, please copy out and sign the following declaration:
I declare that my submission is entirely my own, and I have not sought
or obtained help from anyone else.
All work should be handwritten and should include your student number.
The exam is available for a period of 24 hours. Upon accessing the exam, you will
have 3 hours in which to complete and submit this assessment.
When you have finished your work:
? scan your work, convert it to a single PDF file, and submit this file on QMPlus;

? with your e-mail, include a photograph of the first page of your work together
with either yourself or your student ID card.
You are expected to spend about 2 hours to complete the assessment, plus the time
taken to scan and upload your work. Please try to upload your work well before the
end of the submission window, in case you experience computer problems. Only one
attempt is allowed – once you have submitted your work, it is final.


MTH5130 (2021) Page 2
Question 1 [17 marks].
(a) State Fermat’s Little Theorem, and use it to prove that 15 is a composite
number. [6]
(b) Let φ be Euler’s totient function. What is the parity of φ(15841)? Justify your
answer. State clearly any results you use from the lecture material without
proofs. [3]
(c) Find all the primitive roots mod 11 in {1, 2, . . . , 10}. Justify your answer. [8]
Question 2 [10 marks]. Using the Chinese Remainder Theorem, solve the
following simultaneous congruence equations in x. Show all your working.
9x ≡ 3 mod 15,
5x ≡ 7 mod 21,
7x ≡ 4 mod 13. [10]
Question 3 [30 marks].
(a) Assume that 3083 and 3911 are prime numbers. Using properties of Legendre
symbols, compute the Legendre symbol
(
3083
3911
)
. Justify your answer. [8]
(b) Which of the following congruences are soluble? If soluble, find a positive
integer solution less than 79; if insoluble, explain why.
(i) x2 ≡ 41 mod 79. [4]
(ii) 41x2 ≡ 43 mod 79. [8]
(c) Using Hensel’s Lemma, find an integer 1 ≤ z ≤ 125 satisfying z3 ≡ 2 mod 125.
Show all your working. [10]
? Queen Mary University of London (2021) Continue to next page
MTH5130 (2021) Page 3
Question 4 [20 marks].
(a) Compute the continued fraction expression of
√
11. [4]
(b) Compute the convergents
s0
t0
,
s1
t1
,
s2
t2
and
s3
t3
to
√
11. [4]
(c) Find the smallest and the fourth smallest positive integer solutions to the
equation
x2 ? 11y2 = ±1. [6]
(d) Compute the convergent
s7
t7
. [6]
Question 5 [10 marks]. Use 672 ≡ ?1 mod 449 and Hermites’ algorithm to find a
pair of positive integers s and t such that
s2 + t2 = 449. [10]
Question 6 [13 marks].
(a) What is the definition of a unit in a ring R? [3]
(b) How many units are there in the following rings? If finitely many, list them all;
if infinitely many, describe them all.
(i) Z[
√?1]. [4]
(ii) Z[
√
11]. [6]

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