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# High Schooler Develops Groundbreaking Proof for Puzzler That Long Stumped Math Experts

Nov 1, 2022

Thought Question: Reflect on a time when you were presented with a difficult question. How did you go about finding your answer?

What question can you not stop thinking about? For Daniel Larsen of Indiana, his was about math. He spent more than a year poring over an unproven theorem. In his senior year of high school, he developed his own mathematical proof for it. He emailed it to some of the top people working in number theory

Larsen’s question was about Carmichael numbers. These look like prime numbers. They aren't, though. A prime number is a number that can only be divided by 1 and itself. Carmichaels are made by multiplying at least three prime numbers together. The results often appear prime themselves. The smallest Carmichael is 561 (3 x 11 x 17).

Carmichael numbers were found by mathematicians over 100 years ago. Carmichaels got in the way as arithmetic addicts sought to identify prime numbers quickly. The quest has become more relevant in modern cryptography (the art of writing secret codes)! That’s because today’s most-used secret codes involve math with huge primes.

So, how are Carmichael numbers distributed along the number line? Larsen proved that they must always appear between X and 2X, as long as X is a number bigger than 561. Mathematicians hadn't figured that out yet.

Larson emailed his proof to experts. One of the people he emailed was Andrew Granville. In 1994, he helped prove there are infinitely many Carmichaels.

“It wasn’t the easiest read ever,” Granville told Quanta Magazine. “But … he wasn’t messing around.”

“(Larsen’s proof) changes a lot of things about how we might prove things about Carmichael numbers,” said one mathematician. He and others hope Larsen’s work will help reveal more about how these strange numbers behave.

Moral of the story? Don’t stop scratching at those questions that tickle your brain.

Photo by Clayton Robbins courtesy of Unsplash.

Question
After reading the story, which of the following questions is left unanswered? (Common Core RI.5.2; RI.6.2)
a. What is a prime number?
b. What is tricky about Carmichael numbers?
c. How will Daniel Larson continue his study of Carmichael numbers?
d. How many Carmichael numbers are there?
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