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,
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
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 p
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 allowe
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 quad
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 w
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
Solut
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 quad
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,
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.