Proving Fermat’s Little Theorem

Berkeley Connect Math students explore Fermat’s Little Theorem & its proofs

What’s simple can be deceptive, as Berkeley Connect Fellow Pablo Solis proved at a recent small group meeting. During the hour, Berkeley Connect Math students dissected proofs of Fermat’s Little Theorem, which states that for every prime number p, ap – a (a being any integer) would be an integer multiple of p. In other words, the answer to 67 – 6, for example, would be a multiple of 7. Don’t believe me (or rather Fermat)? 67 – 6 = 279930. Divide that by 7 and you get a clean 39990. Works for every prime number and integer you plug in. Simple, right? The proofs, on the other hand – not so simple.

Pablo began the hour with a magic trick with a proof of its own and challenged students to figure it out. Who said math couldn’t be fun? He also answered questions about how to talk to professors and gave tips on applying for graduate school. Then, he asked the students: “What are proofs?” The answer, Pablo noted, is simple:  they are “nothing more than good reasons to believe in something.” He went on to explain, “Proofs can also reveal a lot about the culture of mathematics – how to learn math, different styles, and what problems might arise.” Most of the students knew Fermat’s Little Theorem, but when asked who could prove it, no participants raised their hands. Pablo then passed out handouts of three “alleged” proofs for the students to analyze. After a while, he asked, “Is there a proof you would favor or feel is more reasonable?” Noting the students’ silence, he added, “Or is this totally an alien language?” Encouraging students to speak, he noted that they were in a space where there was no need to feel foolish. “If anything, now is the time to speak up,” he said. Finally one student admitted, “I got lost around the third line of the first one.” With Pablo, the students reviewed and analyzed the three proofs together. Pablo mentioned that proofs are not always complete and urged students to think critically about missing steps and implied ones. At one point he pointed out the sentence, “We are done,” and noted that sometimes, proof writers like to take you close to the answer and tell you to take it from there. With Pablo’s guidance, students built connections between different parts of proofs and began to pinpoint their favorite proof.

After the students had gone through the proofs, Pablo asked the students to close their eyes as he rattled off a complicated equation. “Would you rather have heard that equation or seen it written on the board?” Unsurprisingly, everyone preferred to see it written. “We’re visual people, and it can be difficult expressing some things in words,” Pablo told students, explaining why proofs can be so difficult. Translating mathematical concepts into language means using one medium to describe another, and the fit isn’t always exact. “Part of the challenge is using words in the most concise way. It’s always difficult balancing being clear while not writing an encyclopedia. Some think it is elegant to not say much. Others think it is overly terse.”

Whatever style of proof they preferred, students in Pablo’s group enjoyed their conversation about writing and math, math and language!

One of the three proofs! Comprehensible or “totally an alien language”?

posted by Katherine Wang
Berkeley Connect Communications Assistant