# Chapter 12

#### *Exercise 1.* This software finds all
solutions to the equation *x*^{2} + y^{2} = 0
in *Z*_{p}.
Run your program for all odd primes up to 37. Make a conjecture about the
the number of solutions in *Z*_{p} (where* p*
is a prime) and
the form of *p*.

*Exercise 4*.
This software determines the order of the group of units in the
ring of 2 by 2 matrices over
*Z*_{n} (that is, the group
*GL*(2,*Z*_{n})) and
the subgroup *SL*(2,*Z*_{n}).
Run the program for *n* = 2, 3, 5, 7, 11, and 13. What
relationship do
you see between the order of *GL*(2,*Z*_{n}) and the order
of *SL*(2,*Z*_{n}) in these cases? Run the program
for *n
= 16, 27, 25,* and *49*. Make a conjecture about the
relationship between
the
order of *GL*(2,*Z*_{n}) and the order of
*SL*(2,*Z*_{n})
when *n* is a power of a prime.
Run the program for *n =
32*. (Notice that when you run the program for *n = 32* the
table shows the orders for all divisors of 32 greater than 1.) How
do the orders the two groups change each time you increase the
power of 2 by 1?
Run the program for *n =
27*. How do the orders the two groups change each time you increase the
power of 3 by 1?
Run the program for *n =
25*. How do the orders the two groups change when you increase the
power of 5 by 1?
Make a conjecture about the relationship between
*|SL(2,Z*_{pi})| and
*|SL(2,Z*_{pi+1})|.
Make a conjecture about the relationship between
*|GL(2,Z*_{pi})| and
*|GL(2,Z*_{pi+1})|.
Run the program for *n* = 12, 15,
20, 21, and 30. Make a conjecture about the order of
*GL*(2,*Z*_{n})
in terms of the orders of *GL*(2,*Z*_{s}) and *GL*(2,*Z*_{t})
where *n = st* and *s* and *t *are relatively prime.
(Notice that when you run the program for *st* the table
shows the values for * st, s* and * t *.)
For each
value of *n* is the order of *SL*(2,*Z*_{n}) divisible
by *n*? Is it divisible by *n* + 1? Is it divisible by *n*
- 1?

*Exercise 5.* In the ring *Z*_{n}
this software finds the number of solutions to the equation *x*^{2}
= -1.
Run the program for all primes between 3 and 29. How does the answer depend
on the prime? Make a conjecture about the number of solutions when *n*
is a prime greater than 2.
Run the program for the squares of all primes between 3 and 29. Make a conjecture
about the number of solutions when *n* is the square of a prime greater
than 2.
Run the program for the cubes of primes between 3 and 29. Make a conjecture
about the number of solutions when *n* is any power of an odd prime.
Run the program for *n* = 2, 4, 8, 16, and 32. Make a conjecture
about
the number of solutions when *n* is a power of 2.
Run the program for *n* = 12, 20, 24, 28, and 36. Make a
conjecture about
the number of solutions when *n* is a multiple of 4.
Run the program for various cases where *n* = *pq* and *n*
= 2*pq* where *p* and *q* are odd primes. Make a conjecture
about the number of solutions when* n* = *pq* or *n* = 2*pq*
where *p* and *q* are odd primes. What relationship do you see between
the number of solutions for *n = p* and *n = q* and
*n = pq*?
Run the program for various cases where *n* =* pqr* and *n*
= 2*pqr *where *p*, *q* and *r* are odd primes. Make a
conjecture about the number of solutions when *n* = *pqr* or *n*
= 2*pqr *where *p*,* q* and *r* are odd primes. What relationship
do you see between the number of solutions when *n* = *p*,
*n = q*
and *n = r *and the case that *n = pqr*?

####

*Exercise 6.* This software determines
the number of solutions to the equation *X*^{2} = -I where *X
*is a *2 *x* 2 *matrix with entries from *Z*_{n}
and *I * is the identity. Run the program for *n = 32*. Make a conjecture
about the number of solutions when *n = 2*^{k} where *k
> 1*. Run
the program for* n = 3, 11, 19, 23,* and * 31 *. Make a
conjecture
about the
number of solutions when *n *is a prime of the form *4q + 3*. Run
the program for *n = 27* and * 49*. Make a conjecture about
the number of
solutions when *n* has the form *p*^{i} where *p* is
a prime of the form *4q + 3*. Run the program for *n = 5, 13, 17,
29,*
and* 37*. Make a conjecture about the number of solutions when
*n*
is a prime of the form *4q + 1*. Run the program for *n = 6, 10, 14,
22; 15, 21, 33, 39; 30, 42. * What seems to be the relationship between
the number of solutions for a given *n *and the number of solutions for
the prime power factors of* n* ?

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Making of the project / Project Home Page / UMD Undergraduate Math Program
}