Beal’s Conjecture vs. “Positive Zero”, Fight

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.

What Is It?

Beal’s Conjecture is a mathematical hypothesis proposed by Andrew Beal, an American banker and mathematician, in 1993. It is a generalization of Fermat’s Last Theorem, which states that there are no integer solutions to the equation
xn + yn = zn for any integer values of x, y, and z when n > 2.

Beal’s Conjecture posits that if A, B, and C are positive integers, and if Ax + By = Cz where x, y, and z are positive integers greater than 2, then A, B, and C must have a common prime factor. In other words, if there are no common prime factors between A, B, and C, then there are no solutions to the equation with positive integer values of x, y, and z.

Despite extensive efforts by mathematicians, Beal’s Conjecture remains unproven. It has garnered significant attention due to its similarity to Fermat’s Last Theorem and the potential applications in number theory. Additionally, Beal offered a substantial monetary prize for a proof or counterexample to his conjecture, further stimulating interest in the problem.

Pack of Tools

1. The Undefeated Champion (Beal’s Conjecture): Ax + By = Cz

Where ABCxy, and are positive integers with xy> 2, then AB, and have a common prime factor.

2. Now for the Counterexample of Doom. Let the Games Begin:

Pre-Fight: Beal’s Conjecture is never true when (Ax = 1) + By = Cz. 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 one-sided limit, which may be denoted by → 0− → 0−, or → ↑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 (Ax = 1) + By = Cz 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: 13 + (+0)4= 15(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

  1. 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.
  2. Have each group create a list of all the reasons why each fighter should or should not be declared the champion.
  3. Once the groups have reached a final decision, tell each of them to present their findings to the rest of the class.
  4. If there are conflicting viewpoints, carryout a debate between the opposing sides.
  5. 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?

About the Author: Angela Moore graduated from Fairfield University in 2012 and has served as a mathematics tutor for the Connecticut Pre-Engineering 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.