Article
Proof and Disproof of the Beal Conjecture- How a Mathematics Conjecture Can be Proven and Disproven while A Person Could Arguably be Added as Supreme Court Justice To promote Diversity of Universities
American Journal of Math, Fordham Law Review, Illinois Journal of Math, Illinois Issues, New York Journal of Math, Lithuanian Journal of Math, Michigan Journal of Math (2016)
Abstract

Proof and Disproof  and Study of the Beal Conjecture using Case studies and Example
By
James T. Struck
Case studies and example can be used as a type of proof or disproof.
Here we use 2 examples to show a proof and disproof of the Beal Conjecture. Here is the Beal Conjecture
BEAL'S CONJECTURE: If Ax + By = Cz, where A, B, C, x, y and z are positive integers and x, y and z are all greater than 2, then A, B and C must have a common prime factor.
Proof
23 +23 = 24
8 + 8= 16
2 is the common prime factor of all these components of the equation
2 Disproofs of Conjecture
13 + 13 = 13
2 does not equal 1
2 NOT EQUAL 1 disproof of conjecture and A, B, C do not have a common prime factor as 1 is not a prime number
03 + 03 = 03
As O can be seen as a positive integer as 0 is not negative, but 0 is not a prime number disproving the conjecture
Neither Disprove nor prove but tending toward disproof
I choose numbers that are prime for A,B and C 2,3,5 and numbers greater than 2 for x, y and z
23 + 33 = 53
8 + 27 does not equal 125
2,3 and 5 do not have a prime factor, and the initial if is not equal. I do not really disprove or prove here but it tends towards disproof as we can take later conditions or then conditions like being prime or commonly prime or having a characteristic of commonly prime and disprove the IF statement.  Similarly take numbers with common prime factor of 2
23 + 24 does not equal 25
8 + 16 does not equal 32
Take prime factor numbers, but we do not get the IF Statement.
We use case studies or examples as a type of disproof and proof of the conjecture and a study of an equation or relationship of numbers. Pierre de Fermat was a human being who may have meant for his conjecture to be proved and disproved or just studied and played with like a game. He lived from 1601-1655 and his purpose may have been to show that conjectures can be proven and disproven. We present here a different type of proof where both the If statement is shown false by conditions and the then statement is shown false by conditions. The where’s are emphasized showing that under certain conditions, both if and then parts of conjectures can be disproven.
1. We can prove an equation,
2. We can disprove a THEN statement
3. We can disprove an IF statement using the THEN characteristics
4. I disprove the IF using some of the THEN characteristics like commonly prime but not common prime factor
4. Equations can be studied and analyzed rather than simply proven or disproven.
At the Same Time I could be added as a 10th Supreme Court Justice to promote knowledge of the United States and help people understand the diversity of Universities and Colleges a person can be linked to. Here are some of the colleges, universities I am linked to. As I have links to Kent College of Law for 9 years, Fordham University/U of Pittsburgh/Vanderbilt/Drew University/Washington and Lee University Presidential Scholarship winner, University of Michigan for 4 weeks working as debate judge and housing assistant, Northwestern University for 8 plus years, University of Chicago for many years with a BA in Religion and the Humanities, Loyola University for 1 year, Northeastern Illinois University for a BS in Biology, Rosary College with a Masters and Library and Information Sciences, ISU 2nd JV State,  Vermont University with a debate tournament 2nd place, 2nd speaker, Emory University 3rd place finish, Redlands University 1st place, Loyola Marymount University top seed, UIC all A's in education work, ISU 1st place, Hoover School 1st place, Harvard University 1st place, 1st Seed at St. Marks in Texas, I worked at DePaul University as a judge 3 or more times,    LSU Multidimensional Calculus, Truman College CNA, Harold Washington College Associates in Arts, work for 10 years at Roosevelt University, work for 2.5 years at  Adler Planetarium, 3 months work at Field Museum, 2.5 years work at Crain Communication,  1 year work with Lincoln Park Zoo, and arguably PHD with Imaginary Heaven University and Deer Beaver College, I have more diverse background so I should be considered as the 10th Supreme Court Justice candidate.
People involved in math can also be involved in attempts to promote diversity of educational diversity at the Supreme Court of the United States.
Keywords
• Beal Disproved,
• Prime Number,
• Supreme Court Justice Diversity Universties Colleges Proof Disproof
Disciplines
Publication Date
May 1, 2016
Citation Information
James T Struck. "Proof and Disproof of the Beal Conjecture- How a Mathematics Conjecture Can be Proven and Disproven while A Person Could Arguably be Added as Supreme Court Justice To promote Diversity of Universities" American Journal of Math, Fordham Law Review, Illinois Journal of Math, Illinois Issues, New York Journal of Math, Lithuanian Journal of Math, Michigan Journal of Math (2016)
Available at: http://works.bepress.com/james_struck/66/