In 1637, while studying Arithmetic books, mathematician Pierre de Fermat had an insight that led him to analyze the Pythagorean Theorem from another perspective. It states that the sum of the areas of the squares on the legs equals the area of the square on the hypotenuse, described in the equation x²+y²=z². But what if we raised it by a number greater than 2? At this point, Fermat wrote in the book’s margin: that the equation xn+yn=zn has no integer solutions for n>2. I have found a wonderful demonstration, but the margin of this book is too small for it.
There is no record of what the proof proposed by Fermat would be. However, a simple statement in the corner of the page gave rise to one of the greatest mysteries in mathematics, known as Fermat’s Last Theorem. What fascinated mathematicians over the centuries was the simplicity of the statement and the idea that there was a solution, but no one has been able to solve it. Given its complexity, most mathematicians thought such proof did not exist or was simply impossible to solve. After many unsuccessful attempts, the theorem fell by the wayside as other important questions in mathematics arose.
In 1963, Andrew Wiles was just ten years old when he found a copy of Fermat’s Last Theorem book in his town library. He was intrigued by a problem that even he could understand but that no one had been able to solve in over 300 years. “I knew at that moment that I would never give up. I had to solve it,” he said. Andrew grew up and devoted himself entirely to Mathematics. He majored in mathematics at Oxford and completed his Ph.D. at the University of Cambridge. Today, Wiles is a research professor at the Royal Society at the University of Oxford.
After more than 30 years of study, in 1995, Andrew Wiles finally proved the theorem, ending this nearly 400-year-old mystery. His curiosity guided him through all these years and, thanks to this passion for the challenge, he remained steadfast throughout the journey. Reaching this result was not only the highlight of his career, but also the culmination of a 10-year-old boy’s personal journey that began three decades earlier. The discovery won Wiles the Abel Prize in 2016, considered the Nobel Prize in Mathematics, in addition to generating one of the most complex areas of Number Theory and considerable advances in the field of Mathematics.
But the question remains: did Fermat really have a solution to this theorem? We will never know. However, it is believed not. To arrive at the result, Wiles used the ideas of dozens of other mathematicians of the 20th century, and even today, it is believed that only a tenth of the mathematicians in the world can fully understand Wiles’ solution.
Wiles could not rely on previous concepts as there was no correct answer to this situation. No one knew the answer, or even if there was one. When asked how he got the inspiration or ideas to solve this age-old problem, Wiles replied, “[I] tried to find general patterns. I did calculations that explained small math results to me, then tried to fit these calculations into my idea. Sometimes this would lead me to consult some books to see how something similar had been done. Other times, I had to make modifications and do more calculations. But I realized that these calculations had never been done before, so I had to work on something totally new.”
Just like Andrew, I’ve been always fascinated by things I don’t know. Curiosity has been a good friend, and sometimes I think it is important to just let yourself go and pursue questions made by your curiosity -even if they seem useless.
As I argued elsewhere, learning normally comes from questions we ask about the world around us. Also, with computers getting better and better at giving answers, we need people who know how to ask good questions.
The writer Pico Iyer put it differently:
“At some point, knowledge gives out. (…) [But] I don’t believe that ignorance is bliss. Science has unquestionably made our lives brighter and longer and healthier. (…) The opposite of knowledge, in other words, isn’t always ignorance. It can be wonder. Or mystery. Possibility. And in my life, I’ve found it’s the things I don’t know that have lifted me up and pushed me forward much more than the things I do know. It’s also the things I don’t know that have often brought me closer to everybody around me.”
Questions, after all, open more doors than answers ever could.