Final Practice Problems, Math 313
1 Show that if x, y, z are integers with
x3 + y 3 = z 3 ,
then 3 divides xyz.
2 Consider the congruence
x2 ≡ n mod 1357.
(∗)
Noting that 1357 = 23·59, compute the necessary Legendre symbols or
argue otherwise to determine the number of solutions to the congruence
(∗) modulo 1357 for n = 77. Repeat this for n = 26.
3 Define two sequences of positive integers xk and yk by
√
√ k
xk + 2yk = 3 + 2 2 , k = 1, 2, . . .
Show that x2k − 2yk2 = 1. Use this to prove that there exist infinitely
many Pythagorean triples (x, y, z) with x = y + 1. Here, y is even.
4 The Diophantine equation
x3 + 2y 3 + 4z 3 = 9w3
has solutions with one or more of x, y, z, w equal to 0. Show that it
has no integral solutions with x, y, z, w all nonzero. Hint : consider the
equation modulo N , for suitable N .
5 Use the law of quadratic reciprocity to show that if p is a prime
with p > 5, then 5 is a quadratic residue modulo p if and only if the
last decimal digit of p is 1 or 9.
6 Determine whether each of the following integers is expressible as
the sum of two squares of integers :
637, 6!, 10!, 20022002 .
7 Find the largest positive integer k such that k! can be written as
the sum of two integer squares.
8 Let p be an odd prime. Show that the Diophantine equation
x2 + py + a = 0, gcd(a, p) = 1
has an integral solution if and only if the Legendre symbol (−a/p) = 1.
9 Use Continued Fraction Factoring to factor 91 and 85 .
1
2
10 If p > 3 is prime, prove that
−3
1
=
−1
p
if p ≡ 1
if p ≡ 5
mod 6
mod 6
11 Show that if n ≡ 5 mod 12 and n is an Euler pseudoprime to the
base 3, then n is a strong pseudoprime to the base 3.
12 Find, from n an √
integer with n ≥ 2, a formula for the continued
fraction expansion to n2 − 2.