How Can Infinitely Many Primes Be Infinitely Far Apart — Quanta Magazine

My latest column for Quanta Magazine ties recent news about “digitally delicate” primes to some simple but fascinating results about prime numbers.

You may have noticed that mathematicians are obsessed with prime numbers. What draws them in? Maybe it’s the fact that prime numbers embody some of math’s most fundamental structures and mysteries. The primes map out the universe of multiplication by allowing us to classify and categorize every number with a unique factorization. But even though humans have been playing with primes since the dawn of multiplication, we still aren’t exactly sure where primes will pop up, how spread out they are, or how close they must be. As far as we know, prime numbers follow no simple pattern.

There’s a tension among the infinitude of prime numbers — that there will always be primes close together and primes far apart — that can also be seen among digitally delicate primes, primes that become composite if any digit is changed. It may come as a surprise that any digitally delicate primes exist at all, but that’s just the beginning of their story. Find out more at by reading the full article here, and be sure to check out the exercises!

The Mysterious Math of Perfection — Quanta Magazine

My latest column for Quanta Magazine explores the mathematics of perfect numbers. Humans have been studying perfect numbers for thousands of years, but we still don’t know if an odd perfect number exists!

Euclid laid out the basics of perfect numbers over 2,000 years ago, and he knew that the first four perfect numbers were 6, 28, 496 and 8,128. Since then, many more perfect numbers have been discovered. But, curiously, they’re all even. No one has been able to find an odd perfect number, and after thousands of years of unsuccessful searching, it might be tempting to conclude that odd perfect numbers don’t exist. But mathematicians haven’t been able to prove that either. How is it that we can know so much about even perfect numbers without being able to answer the simplest question about an odd one? 

With some basic number theory and an assist from a famous formula from Algebra class, we can get pretty far into the world of perfect numbers. So read the full article here, and be sure to stick around for the exercises at the end!

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