Number Theory. Final Exam from Spring 2013. Solutions
1.
(a) (5 pts) Let d be a positive integer which is not a perfect square.
Prove that Pells equation x2 dy 2 = 1 has a solution (x, y) with
x > 0, y > 0 and y even.
(b) (7 pts) Find a solution (x, y) to
Number Theory, Spring 2014. Midterm #2.
Due Wednesday, April 16th in class
Directions: Provide complete arguments (do not skip steps). State clearly
and FULLY any result you are referring to. Partial credit for incorrect solutions, containing steps in the
Number Theory, Spring 2014.
Solutions to the second midterm
1. Let p be an odd prime, and let x Z be a primitive root mod p.
(a) (2 pts) Prove that x is a primitive root mod p2 xp1 1 mod p2 .
(b) (4 pts) Let i [1, p 1]. Use (a) and the lifting theorem to
Math 5653. Number Theory. Spring 2014. First Midterm.
Wednesday, February 26th, 2-3:20pm
Directions: No books, notes, calculators, laptops, PDAs, cellphones, web
appliances, or similar aids are allowed. All work must be your individual
eorts.
Show all yo
Homework #10. Solutions to selected problems.
1. Let R, and assume that the continued fraction for is innite
periodic. Prove that is a quadratic irrational, that is, Q, but is a
root of a nonzero quadratic polynomial with integer coecients. Hint: Start
wi
Homework #9. Solutions to selected problems.
1. Let p be a prime of the form 4k + 3.
(a) Prove that if p a or p b, then p (a2 + b2 ).
(b) Use (a) to prove that ordp (a2 + b2 ) is even for any a, b Z with a = 0
or b = 0.
Solution: (a) If p divides one of t
Solutions to homework #8
1. Let p > 3 be a prime. Prove that
3
p
1
1
=
if p 1 or 11 mod 12
if p 5 or 7 mod 12
in two dierent ways:
(i) using quadratic reciprocity
(ii) directly using Gauss lemma
Solution: (i) Using quadratic reciprocity: If p 1 mod 4, the
Homework #7. Solutions to selected problems.
p1
a=1
1. Let p be an odd prime. Prove that
a
p
= 0.
Solution: By Lemma 7.3 from the book, precisely half of the integers in
the interval [1, p 1] are quadratic residues (while the other half are nonresidues).
Number Theory. Final Exam from Spring 2013.
Directions: Provide complete arguments (do not skip steps). State clearly
and FULLY any result you are referring to. Partial credit for incorrect
solutions, containing steps in the right direction, may be given.
Homework #6. Solutions to selected problems.
1.
(a) Let G1 , . . . , Gk be nite groups. Prove that
exp(G1 . . . Gk ) = lcm(exp(G1 ), . . . , exp(Gk ),
where as usual exp(G) denotes the exponent of G.
(b) Give an example showing that if G is nite, but non-