# Math 3240 - Introduction to Number Theory Due in MSB

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```					Math 3240 - Introduction to Number Theory                                             Due in MSB 318
Problem Set 3                                                                         10/9/09 at 4 PM

I feel a responsibility [. . . ] to teach that doubt is not to be feared, but that it is to be
welcomed as the possibility of a new potential for human beings. If you know that you
are not sure, you have a chance to improve the situation.                  Richard Feynman

The midterm exam is on Oct. 15th in class.
Required Reading: Handout on Wolfram Alpha and either text sections 9ABC or the handouts
on Fermat’s compositeness test and Euler’s theorem.

1. (Induction practice)
The congruence x2 ≡ 2 mod 7 has the solution 3 and the congruence x2 ≡ 2 mod 49 has the
solution 10, which reduces to 3 mod 7.
a) Use Wolfram Alpha to ﬁnd a solution to x2 ≡ 2 mod 73 , x2 ≡ 2 mod 74 , and x2 ≡ 2 mod 75
such that x ≡ 3 mod 7 each time. (When I tried this the ﬁrst time I got an answer from
Wolfram Alpha that was garbage. If you meet the same problem, you’ll have to think about
other ways of feeding the task into Wolfram Alpha. If you have no patience – a pity – then
tell me what other computer package you use instead.)
b) Use induction to show for each k ≥ 1 there is an integer xk such that x2 ≡ 2 mod 7k .
k
(Hint: Starting from a solution to x2 ≡ 2 mod 7k , show there is an integer yk such that
k
(xk + 7k yk )2 ≡ 2 mod 7k+1 . Then the number xk+1 := xk + 7k yk solves the problem.)
Note. This lifting problem modulo powers of 7 resembles the exercise on the previous set
about squares ending in themselves, where you found consistency in the successive answers
5, 25, 625, 90625, . . . . This resemblance becomes apparent if you write the answers in part a
using base 7 rather than base 10: c0 + c1 · 7 + c2 · 72 + · · · where the digits ci run from 0 to 6.
2. a) Find the smallest Fermat witness for 2701.
b) Use Wolfram Alpha to factor 2701 into primes and then state Euler’s congruence (Euler’s
theorem) for this modulus.
c) Let m = 56052361. Use Wolfram Alpha to see if the numbers 2, 3, 4, 5, 6, 7, 8, 9, 10, or
11 are Fermat witnesses for m. What do you ﬁnd? Does this tell you anything with certainty
about m being prime or composite?

3. (Exploration)
a) Look at the ﬁles “Squares Modulo Primes” and “Primes and Congruence Conditions”, which
both concern primes up to 200. (Be sure to look at both ﬁles together, not just the ﬁrst one!)
Writing for an unknown square, conjecture from the ﬁles a set of congruence conditions on
all primes p which characterize those for which −1 ≡ mod p, with ﬁnitely many possible
exceptions. Your characterization should account for all p < 200 for which −1 ≡ mod p and
not include any p < 200 for which −1 ≡ mod p.
Then do the same for each of the conditions 2 ≡ mod p, −2 ≡ mod p, 3 ≡     mod p,
−3 ≡ mod p, 5 ≡ mod p, and −5 ≡ mod p. The last case, with −5, will be harder than
the rest!
(The idea of a “congruence condition” was seen already on Set 2, where x2 ending in x when
x has n digits means x2 ≡ x mod 10n and the period length of Fn mod p appears to divide
p − 1 if and only if p ≡ 1, 9 mod 10.)
b) We call a unit modulo m a generator if its powers gives all the units mod m. For example, the
units modulo 5 are {1, 2, 3, 4}, and 2 is a generator of these since 21 = 2, 22 = 4, 23 ≡ 3 mod 5,
and 24 ≡ 1 mod 5. Modulo 9, the units are {1, 2, 4, 5, 7, 8}, and again 2 is a generator:

21 = 2, 22 = 4, 23 = 8, 24 ≡ 7 mod 9, 25 ≡ 5 mod 9, 26 ≡ 1 mod 9.

Modulo 8 there is no generator for the units: the units are {1, 3, 5, 7} and every unit mod
8 squares to 1: a2 ≡ 1 mod 8 if a is odd, so the sequence of powers of a mod 8 looks like
a, 1, a, 1, a, 1, . . . . There is no unit mod 8 whose powers give all the units mod 8.

(i) For m = 11, 14, 18, and 22, ﬁnd a generator for the units mod m and explicitly show that
the powers of your generator run through all the units modulo m.
(ii) In the ﬁle “Moduli with a Generator for the Units” is a table of all moduli below 302 where
the units have a generator. Propose a general characterization of the m ≥ 2 for which
the units modulo m have a generator. Your characterization should not only include all
integers m ≤ 302 which appear in the table but also exclude all positive integers up to
302 which are not in the table. (Hint: Consider ﬁrst odd m, then even m.)

4. (More simultaneous congruences)
a) Describe the solutions to the general pair of congruences x ≡ a mod 12 and x ≡ b mod 91
for any a, b ∈ Z, as a single congruence condition on x modulo 1092. (Your answer will depend
on a and b).
b) For relatively prime integers m1 and m2 , prove that the simultaneous congruence conditions

x ≡ a1 mod m1 ,   x ≡ a2 mod m2

are equivalent to a single congruence condition on x mod m1 m2 .
5. For any non-square d ∈ Z, let
√           √
Z[ d] = {a + b d : a, b ∈ Z}.

For example,
√           √                      √        √           √
Z[ 5] = {a + b 5 : a, b ∈ Z} = {0, 6, 5, 7 + 4 5, −9 + 11 5, . . . }.
√
The elements of Z[ d] are closed under addition, subtraction, and multiplication. (The case
d = −1 is the Gaussian integers, so you can compare the√     results below to what you have
√
seen in Z[i] on the previous assignment.) For α = a + b d in Z[ d], set the norm√ α
√         √                                                      of
to be N(α) = (a + b d)(a − b d) = a2 − db2 ∈ Z. For example, the norm of 7 + 4 5 is
72 − 5 · 42 = −31, so norms can be negative.
√                                                                       √
We say α ∈ Z[ d] is a unit when it has a multiplicative inverse: αβ = 1 for some β ∈ Z[ d].
√
a) Show N(αβ) = N(α) N(β) for all α and β in Z[ d].
√
b) Prove α ∈ Z[ d] is a unit if and only if N(α) = ±1.
n
c) If√ = 1 then un v√ = 1 for any integer n, so any integral power of a unit is a unit. Show
uv                                                                √                    √
1 + 2 is a unit in Z[ 2] and then compute the ﬁrst 8 powers of 1 + 2 in the form a + b 2.
Where have you seen the coeﬃcients of some of these powers earlier in the course?
√                                                                  √
d) Two obvious units in Z[ d] are ±1. For d = 3, 5, 6, 7, 8, 10, 11, and 12, ﬁnd a unit in Z[ d]
other than ±1 and list for each unit what its inverse is. (Be sure your inverses are correct!)
√
What can you say about units in Z[ d] if d < 0 and d = −1?
√                                            √
e) A unit multiple of α ∈ Z[ d] is √ product αu, where u is a unit in Z[ d]. (For instance,
√           a      √           √                          √
one unit multiple of 5 + 2 is (5 + 2)(1 + 2) = 7 + 6 2.) For any unit u in Z[ d], show u
and αu are divisors of α. (Units and unit multiples of α are considered the trivial divisors of
α, just like ±1 and ±n are trivial divisors of an integer n.)
√
f) When α ∈ Z[ d] is not a unit (i.e., | N(α)| > 1 by part b), call α prime if its only divisors
√
in Z[ d] are units and unit multiples of α, as in part e. If N(α) = ±p for a prime number
√                                                           √
p, show α is prime in Z[ d]. Then use this to give examples of four primes in Z[ 3] with
diﬀerent norms.
√
g) Call a nonzero α ∈ Z[ d] composite if it is not a unit and not a prime. Show α is composite
if and only if it has a factorization α = βγ where | N(β)| < | N(α)| and | N(γ)| < | N(α)|. Then
√
use induction on the absolute value of the norm to prove every α ∈ Z[ d] with | N(α)| > 1 is
√
a product of primes in Z[ d].

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