#fermat

I had an entire plan for abstract algebra, but then:

He didn't write down his proof

Y'all started obsessing over the problem, and

Y'all missed the elegant proof I had planned for you to use :/

Fermat was a man of the 17th century who indulged in mathematics as a hobby, allegedly, and yet he's still considered as one of the greatest intellectuals in history. However his theorems were often shared through informal correspondence, he'd leave each of them unsolved in hopes that they'd be proven ages ahead of him.

As for his last theorem, considered the trickiests of the bunch, he'd written in the margin of a text book, claiming that he already knew how to solve it; the proof had departed along with him, leaving his peers and those which followed to search for the missing pieces.

Quite a fascinating individual I'd say, mathematicians are certainly in a league of their own. Wouldn't you agree?

Ah… Fermat’s Last Theorem. Yes, I am familiar with it. One could hardly walk the halls of mathematics without eventually brushing against that particular ghost.

What captivates me is not merely the theorem itself, but the audacity of the man. To encounter a truth so complete in one’s own mind, and yet leave it behind like a sealed letter addressed to the future… that requires either remarkable confidence or a very particular sense of mischief. I suspect Fermat possessed both.

As for the theorem itself… yes, I have indulged in it. Repeatedly. Not because I believed I might succeed where centuries failed, but because it forces one to confront the limits of intuition. Fermat’s claim appears deceptively simple, yet it resists every naïve approach. That contradiction is what makes it so elegant.

You see, I don’t admire Fermat because he was right. I admire him because he trusted time to vindicate him. And indeed, it did, though not in the way he could ever have imagined. The proof required tools that did not yet exist, minds shaped by generations of failure. In that sense, the theorem is less a statement about numbers, and more a demonstration of collective human progress.

Tell me, do you find the theorem fascinating because it was solved… or because, for so long, it refused to be?

Alan: Don’t worry, I got this.

Alan: *stomps his feet*

Fermat’s little theorem

26. Forgotten hero everyone should know about

This could have been a perfect occasion to talk about Claude-Antoine Prieur again, but given that I plan to devote him many future posts on my blog, I thought it would have been more appropriate to use this ask to share my knowledge about an important and unfortunately still rather unknown STEM personality, who truly inspired me when I was a young student. I'm referring to Sophie Germain.

Born in Paris in 1776, Sophie was one of the rare mathematiciennes of the 18th-19th century. She had her first approach with mathematics during the days of the storming of the Bastille, when it was too dangerous for a young 13 years old girl to go outside. To pass the time, she turned to her father's library and a book named "Histoire des mathématiques" by Jean-Étienne Montucla captured her interest. The story of Archimedes narrated in the book fascinated her deeply, eventually leading her to start studying mathematics on her own through the works by famous mathematicians like Euler, Newton, Cousin. Her interest and dedication to the discipline was so strong, that during winter, when her parents denied her warm clothes and a fire in her bedroom to prevent her from studying she kept doing it anyway despite the cold; at the time maths wasn't considered appropriate as a studying discipline for a woman.

When the Polytechnic school opened in 1794, women couldn't attend, but the policy of the school allowed to everyone, who asked for them, notes of the lectures. She requested them under the pseudonym of Antoine-Auguste Le Blanc, a former student who had dropped out. Given that, as a student of the Polytechnic school, one was expected to send written observations about the lectures - a sort of homework - Germain wrote and sent hers to Joseph-Louis Lagrange, one of the teachers and renowned mathematician. The latter was so positively impressed by her essays that requested a meeting with the brilliant student LeBlanc, who unexpectedly had improved so much. She was then forced to reveal her identity. Lagrange was pleasantly surprised to realize Monsieur Le Blanc was in reality a young and talented woman and decided to support her, becoming her mentor.

One of her most noteworthy contribution to mathematics was in number theory, where she proved a special case of the so-called Last Fermat's Theorem (1), which has remained one of the hardest mathematical theorems to prove for more than three centuries and whose final proof was actually found only in 1994 by Andrew Wiles. Other important works of hers include treatises on elastic surfaces, one of which, Recherches sur la théorie des surfaces élastiques, awarded her a prize from the Paris Academy of Science in 1816.

Although she often faced prejudice for being a woman, Germain was praised and also supported by various well-known mathematicians of the time. Some of them include the aforementioned Lagrange, Legendre, who thanks to her work on the Fermat's theorem, was able to prove it for another special case; Cousin himself, Fourier, who managed to grant her the permission to follow the sittings held at the Paris Academy of science and last, but obviously not least, the great Gauss, who after Germain's death advocated for giving her an honorary degree in mathematics.

Notes

(1) In short, the Last Fermat's Theorem asserts that for n > 2 there are no integer solutions to the following equation:

Einstein said THIS about a man who fell off a roof

When he said no three positive integers a, b, and c satisfy the equation aⁿ + bⁿ = cⁿ for any integer value of n greater than 2, I was SHOCKED