MATH0034 Exam 2021
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MATH0034 Exam 2021
Section A
1. (a) Find all solutions to the congruence x3 + x + 5 = 0 mod 35.
(b) Let p e N, p = 1 mod 40. Explain why 2 and 5 cannot be primitive roots mod p.
(c) Let ζ22 = e2πi/22 . Compute the product
(2 - ζ2(a)2 ).
1_a<22, hcf(a,22)=1
2. (a) (i) Compute the Legendre Symbol / 
、. State clearly all the properties pre- sented in the lecture notes that you are using.
(ii) Show that if x, y e Z satisfy the congruence 26x2 = y2 mod 97, then 26x2 = y2 mod 972 .
(b) For which primes p e N does the congruence x2 = 15 mod p have solutions? Your answer should be given in terms of congruences mod 60.
3. (a) Let f (x) = x7 + 200x4 + x3 + 9x + 32. Remark that f (1) = 0 mod 3. Find a solution to the congruence
f (x) = 0 mod 36 .
(b) (i) Using the fact that log(1 + 3x) converges 3-adically for all x e Z(3) , solve the congruence
28x = 55 mod 37 .
(ii) Solve the congruence
x28 = 55 mod 37 .
4. (a) (i) Determine the value of the continued fraction [5; 5, 10]. (ii) Find two solutions (x, y) in positive integers to Pell’s equation
x2 - 27y2 = 1.
(b) Find a solution x, y e Z to the diophantine equation
x2 + yx + y2 = 403.
Section B
1. (a) Let P (x) e Z[x] be a monic polynomial of degree at least 1.
(i) Let M e Z such that P (M) = A 
0. Show that Q(x) = A>1P (M + Ax) e Z[x].
(ii) Use the remark in the previous part to show that there are infinitely many primes dividing the integers
P (1), P (2), . . . , P (m), . . . .
(b) Let a e Z, d e N.
(i) Consider the dth cyclotomic polynomial, Φd (x). Show that if p is an odd prime dividing Φd (a) then either pld or p = 1 mod d.
(ii) Use part (a) to show that there are infinitely many primes congruent to 1 mod d in Z.
2. (a) Let
αn = 2 + 
+ 
+ .... + 
.
Show that the 2-adic valuation of αn , v2 (αn ), satisfies the inequality
v2 (αn ) > t
1 (t - v2 (t)).
(b) Show that the value of the convergent series
1
23k
k=1
is an irrational number.
2022-08-04