A Real Life Story

I majored in physics engineering for five years before I realized that I didn't like it. I had always wanted to know what made the world tick and thought science would tell me. But after repeatedly being told that the laws I learned the previous quarter were not really true and what I was to learn the current quarter was the real skinny, I gave up in disgust and searched desperately for another major.

Since I had always done well in mathematics, I searched my soul and devoured books on philosophy to find some justification for studying mathematics. At the time I saw mathematics as a tool of science, not an end in itself. Gradually, like a cult leader, I convinced myself that mathematics is indeed a worthy subject. Here was my argument.

Science invents a theory to explain physical phenomena and then designs experiments which either disprove the theory or add to its plausibility. The musings of G. H. Hardy notwithstanding, science proves nothing. On the other hand, proof is everything in mathematics. A theorem proved in freshman calculus is still true after graduation. What a comfort!

Armed with the zeal of a Moonie, I switched majors to mathemaics and hounded everyone I could to give me mathematical problems to solve. You know first hand how irritating that can be. One buddy, to shut me up, gave me the following problem.

For integers n > 1, n⁴ + 4 is not a prime.

I worked on this problem for days, applying in vain all the number theory I knew or could learn. Finally, sufficiently humbled now, I asked the buddy how to solve this problem. He said "Dummy, the polynomial factors!". And indeed it does. It is the difference of two squares in disguise:

n⁴+ 4 = n⁴+ 4n²+ 4 - 4n² = (n² + 2)² - (2n)² = (n²+ 2 + 2n)(n²+ 2 - 2n)

My "irrational exuberance" was quelled and I learned my first painful lesson in humility.