Why was I thinking about exponential Diophantine equations at seven in the morning while lying in bed, half asleep?

摘要

Why was I thinking about exponential Diophantine equations at seven in the morning while lying in bed, half asleep? First of all, not only was I thinking about the equation, I constructed a way to generate infinitely many integer solutions to the equation a~x + b~y = c~z, where a, b, c, x, y, and, z are integers greater than 1 and where a, b, and c are relatively prime (i.e. their greatest common factor is 1). I wasn't thinking about them because I had an exam, I wasn't writing a paper on them, and I didn't have to teach them to my students. I was thinking, in those early morning hours, voluntarily, and the thought was a pleasant reverie. I wasn't thinking about them because I am a mathematical savant who always thinks about maths. I am a maths education professor with a Bachelor's degree in maths and many years of maths teaching experience, but maths has never been extremely easy for me, particularly not pure mathematics like number theory or Diophantine equations. I had success in mathematics because I did enough algebra-type manipulation problems (like factorizing, simplifying rational expressions, and solving linear equations) that I had no problem mastering high school maths and calculus. I struggled, however, with proofs and anything that had to do with deep, unconventional, reasoning about numbers and mathematical objects like Diophantine equations.