MATH 115B PROBLEM SET I
APRIL 5, 2007
(1) Prove that the quadratic congruence
6x2 + 5x + 1 0
mod p
has a solution modulo every prime p, even though the equation 6x2 + 5x + 1 = 0 has no
solution in the integers.
(2) Show that 3 is a quadratic residue modul
MATH 115B PROBLEM SET II
APRIL 17, 2007
(1) Find the value of the following Legendre symbols:
(a) (71/73)
(b) (219/383)
(c) (3658/12703) [Hint: 3658 = 2 31 59.]
(2) Prove that 3 is a quadratic non-residue of all primes of the form 22n + 1, and all primes
MATH 115B PROBLEM SET III
APRIL 24, 2007
(1) Establish each of the following assertions:
(a) Each of the integers 2n where n = 1, 2, . . ., is a sum of two squares.
(b) If n 3 or 6 (mod 9), then n cannot be represented as a sum of two squares.
n
(c) Every
MATH 115B PROBLEM SET IV
MAY 10, 2007
(1) Express each of the rational numbers below as nite simple continued fractions:
(a) 19/51
(b) 187/57
(c) 71/55
(2) Determine the rational numbers represented by the following simple continued fractions:
(a) [2; 2,
MATH 115B PROBLEM SET V
MAY 17, 2007
(1)
Compute the continued fraction expansion of:
(a) 7;
(b) (1 + 13)/2.
(2) Evaluate the following innite simple continued fractions:
(a) [2, 3];
(b) [0; 1, 2, 3].
(3)(a) For any positive integer n, show that n2 + 1
MATH 115B PROBLEM SET VI
JUNE 5, 2007
(1) Show that there are innitely many even integers n with the property that both n + 1
and (n/2) + 1 are perfect squares. Exhibit two such integers.
(2) Find the fundamental solutions of the following equations:
(i)
MATH 115B SOLUTION SET I
APRIL 17, 2007
(1) Prove that the quadratic congruence
6x2 + 5x + 1 0
mod p
has a solution modulo every prime p, even though the equation 6x2 + 5x + 1 = 0 has no
solution in the integers.
Solution:
First observe that, for p = 2, t
MATH 115B SOLUTION SET VI
JUNE 11, 2007
(1) Show that there are innitely many even integers n with the property that both n + 1
and (n/2) + 1 are perfect squares. Exhibit two such integers.
Solution:
For an integer n, suppose that n + 1 = u2 and (n/2) + 1