The Beal Conjecture

I was led to the Beal Conjecture by following the answers to this question in Quora:

What are three distinct positive integers a, b and c such that \(a^3+b^3=c^4\)

Of course, we know from Fermat's Last Theorem that there are no solutions to \(a^3+b^3=c^3\) but what if c is raised to the fourth power as it is here in this question? It turns out that there are infinitely many solutions and one nice, easy solution generator was proposed in one of the answers and that's shown below:

It's clear that all solutions will involve numbers that have a common factor because \(a = kc \) and \( b = lc \). No pair of numbers can be coprime. Another answer however, referred me to the Beal Conjecture. Below is an excerpt from Wikipedia:

The article goes on to say that:

Fermat's Last Theorem established that \( A^n + B^n = C^n \) has no solutions for n > 2 for positive integers A, B, and C. If any solutions had existed to Fermat's Last Theorem, then by dividing out every common factor, there would also exist solutions with A, B, and C coprime. Hence, Fermat's Last Theorem can be seen as a special case of the Beal conjecture restricted to x = y = z.

I should say a little about Andy Beal whose name adheres to the conjecture:

(In 1993) Andy Beal was working on Fermat's Last Theorem when he began to look at similar equations with independent exponents. He constructed several algorithms to generate solution sets but the very nature of the algorithms he was able to construct required a common factor in the bases. He began to suspect that co-prime bases might be impossible and set out to test his hypothesis by computer. Andy Beal and a colleague programmed 15 computers and after thousands of cumulative hours of operation had checked all variable values through 99. Many solutions were found: all had a common factor in the bases. While certainly not conclusive, Andy Beal now had sufficient reason to share his discovery with the world. In the fall of 1994, Andy Beal wrote letters about his work to approximately 50 scholarly mathematics periodicals and number theorists. By offering a cash prize for the proof or disproof of this important number theory relationship, Andy Beal hopes to inspire young minds to think about the equation, think about winning the offered prize, and in the process become more interested in the wonderful study of mathematics. Information regarding the $1,000,000 cash prize that is held in trust by the American Mathematics Society can be obtained at the University of North Texas web site: http://www.math.unt.edu/~mauldin/beal.html.

Daniel Andrew "Andy" Beal is an American banker, businessman, investor, poker player, and amateur mathematician. He is a Dallas-based businessman who accumulated wealth in real estate and banking. Wikipedia

Born: November 29, 1952 (age 64), Lansing, Michigan, United States

Net worth: 10.5 billion USD (2017) Forbes

Education: Baylor University, Michigan State University

Organizations founded: Beal Bank, Beal Aerospace