
Beal's Conjecture vs. "Positive Zero", Fight
by Angela
Moore
ABSTRACT.
This article seeks to spark debates amongst today’s youth
regarding a possible solution to Beal’s Conjecture. It breaks
down one of the world’s most difficult math problems into
layman’s terms and forces students to question some of the most
fundamental rules of mathematics. More specifically; it
reinforces basic algebra/critical thinking skills, makes use of
properties attributed to the number one and reanalyzes the
definition of a positive integer in order to provide a potential
counterexample to Beal's Conjecture.
1. The
Undefeated Champion (Beal’s Conjecture): A^x + B^y = C^z
Where A,
B, C, x, y, and
z are positive integers with x,
y, z > 2, then A, B, and C
have a
common prime factor.
2. Now for the
Counterexample of Doom. Let the Games Begin:
PreFight: Beal’s
Conjecture is never true when (A^x= 1) + B^y = C^z. This is
because 1 has no prime factors.
Final Match: There are
instances when positive 0 is not the same as zero. “Signed zero
is zero with an associated sign. In ordinary arithmetic, −0 = +0
= 0. However, in computing, some number representations allow
for the existence of two zeros, often denoted by −0 (negative
zero) and +0 (positive zero)”^{i}.
Furthermore, the same website proved that signed 0 sometimes
produces different results than 0. “…the concept of signed zero
runs contrary to the general assumption made in most
mathematical fields (and in most mathematics courses) that
negative zero is the same thing as zero. Representations that
allow negative zero can be a source of errors in programs, as
software developers do not realize (or may forget) that, while
the two zero representations behave as equal under numeric
comparisons, they are different bit patterns and yield different
results in some operations.” Additionally, the site confirmed
that signed zero can be used to represent different concepts
“…signed zero echoes the mathematical analysis concept of
approaching 0 from below as a onesided limit, which may be
denoted by x → 0^{− }, x → 0−, or x
→ ↑0. The notation "−0" may be used informally to denote a small
negative number that has been rounded to zero. The concept of
negative zero also has some theoretical applications in
statistical mechanics and other disciplines.” Since 0 is an
integer and it is possible for positive 0 to not be the same as
0; that means there are some rare instances when positive 0
could technically be considered a positive integer, due to the
fact that it is both positive and an integer.
FATALITY: If the
formula (A^x= 1) + B^y = C^z is used when the existence of
positive 0 that can technically be considered a positive integer
is allowed, then the following statement disproves Beal’s
Conjecture: 1^3 + (+0)^4= 1^5
(When reduced this is equivalent to
1+0=1).
3. Moves Performed During
the Fatality (values used): A=1 B=+0
C=1 x=3 y=4 z=5
4A. Can you choose a
winner?: Some people may say that
Beal’s Conjecture won this fight due to the fact that zero is
NEVER included in “the positive integers.” Furthermore, some
people may say that it is highly implied that the numbers used
in the final answer must be greater than zero. Others may argue
that Positive Zero won the match due to the fact that it is
never specifically stated in the question that a positive
integer has to be greater than zero or part of the “official
positive integers.” The question only states that the integer
must be positive and that ANY counterexample to the question
posed is acceptable. Based on what you have read in this story
and what you have learned in previous math classes, who do you
think won the final match?
4B. Activity
 Now that your students
have finished reading the story, split them into groups of 2
or 3 and ask them to discuss who they think the true winner
of this match is.
 Have each group create a
list of all the reasons why each fighter should or should
not be declared the champion.
 Once the groups have
reached a final decision, tell each of them to present their
findings to the rest of the class.
 If there are conflicting
viewpoints, carryout a debate between the opposing sides.
 Declare a winner once
the debate has concluded.
Questions to consider:
Are there ever exceptions to rules in mathematics?
Are there certain rules that
always remain true in mathematics?
Should a question be
judged on what is written or what is implied?
When is it acceptable to
alter a mathematics rule?
If the definition of a
positive integer changes, should Beal be allowed to alter
his question?
Is positive zero really different from zero? Does allowing
the use of positive zero alter the original question? Can
you think of more?

References
About the Author: Angela Moore
graduated from Fairfield University in 2012 and has served as a
mathematics tutor for the Connecticut PreEngineering Program.
Additionally, she is the sole illustrator and author of, Truth,
30% off, a social development comic book series. In 2013, she
began working for Yale University in the Human Research Protection
Program. 

