Math 437/537 Homework #3
due Friday, October 17, 2003 at the beginning of class
Hint: look for short solutions to many of these problems.
I. Determine the number of solutions to the congruence 22x2 + 10x + 35 ≡ 0 (mod p)
for each of the primes p = 103, p = 149, and p = 227. Show your calculations for
this problem.
II. Find a single polynomial f ( x) with integer coefficients such that the congruence
f ( x) ≡ 0 (mod 5) has exactly two solutions, the congruence f ( x) ≡ 0 (mod 7) has
exactly one solution, and the congruence f ( x) ≡ 0 (mod 11) has no solutions.
III. Niven, Zuckerman, and Montgomery, Section 3.2, p. 141, #15-20. (Hint: The results
in Niven, Zuckerman, and Montgomery, Section 3.1, p. 137, #20 might be quite
useful were you to prove them.)
IV. Suppose that p and q are twin primes, i.e., p and q are both primes and q = p + 2.
Prove that there is an integer m such that p | (m2 − 2) if and only if there is an
integer n such that q | (n2 + 2).
V. Let p be an odd prime, and write p − 1 = 2k q with q odd and k ≥ 1. Let a be an
integer such that ( pa ) = −1. Set b = aq . Prove that b has order exactly 2k modulo p.
j
Determine the order of b2 (mod p) for every 0 ≤ j ≤ k.
VI. Given any integer a, define a function f a from the odd integers to {±1} by the
Jacobi symbol f a ( x) = ( xa ). Prove that f a is periodic with period 4a. (Note that 4a
might not be the least period.)
VII. Let p be an odd prime and a any integer. Analyze the complexity of using each
of the following methods to calculate ( pa ), in particular estimating the number of
steps each method takes and determining whether it runs in polynomial time.
(a) calculating a( p−1)/2 (mod p), which equals ( pa ) by Euler’s criterion
(b) using quadratic reciprocity and the properties of the Legendre symbol to calculate ( pa )
(c) using quadratic reciprocity and the properties of the Jacobi symbol to calculate
( pa )
VIII. Given integers a1 , . . . , a j and nonzero integers m1 , . . . , m j , describe a polynomialtime algorithm that computes the solution to the system of simultaneous congruences
x ≡ a1 (mod m1 ),
x ≡ a2 (mod m2 ),
...
x ≡ a j (mod m j ).
Notice that you are not allowed to assume that the moduli mi are pairwise relatively prime. By “describe the algorithm”, I do not mean every single line of code
you would write, but rather a medium-level description of the important choices
and computations that are made. Justify why your algorithm is polynomial-time.