Observations

## Record 232-digit number from cryptography challenge factored

A team of researchers has successfully factored a 232-digit number into its two composite prime-number factors, but too late to claim a \$50,000 prize once attached to the achievement. The number, RSA-768, was part of a cryptography challenge that technically ended in 2007 that had been sponsored by RSA Laboratories, a prominent computer-security firm. RSA-768, [...]

Roots of Unity

## Another Reason to Love the Number Seven

The world’s favorite number is seven, at least if the result of a poll conducted by Alex Bellos is to be believed. Some people like it because it is prime, some because they have a lot of sevens in their birthdates. But I went to a talk by 2014 Fields Medalist Manjul Bhargava that gave [...]

Roots of Unity

## Extrapolation Gone Wrong: the Case of the Fermat Primes

Samuel Arbesman recently wrote about incorrect mathematical conjectures. I wanted to add one of my favorites, which came up in my math history class a couple weeks ago. Unlike the disproven conjectures Arbesman wrote about, which fail only for very large numbers, this one fails at 5. Pierre de Fermat was an amateur number theorist [...]

Roots of Unity

## The Most Mathematically Perfect Day of the Year

Whether you write it 6/28 or 28/6, today is a perfect day. A perfect number is a number that is the sum of its factors besides itself, and 6 (1+2+3) and 28 (1+2+4+7+14) are the first two perfect numbers. Hence, June 28 is a perfect day. Perfect numbers are few and far between, so don’t [...]

Roots of Unity

## Graham’s Number Is Too Big for Me to Tell You How Big It Is

I was going to write an April Fool’s Day post with the title “Mathematicians Declare Graham’s Number Equal to Infinity.” Graham’s number is really big, but of course, it’s precisely 0% as big as infinity. On the other hand, everything we touch is finite, so in some sense, Graham’s number is probably “close enough” to [...]

Roots of Unity

## Goldbach Variations

On Monday, Harald Helfgott of the École Normale Supériure in Paris posted a proof of one of the oldest open problems in number theory to the preprint repository arxiv. The ternary Goldbach conjecture, like so many questions in number theory, is easy to state but hard to prove. Every odd number greater than 5 can [...]

Roots of Unity

## 91 Is April Fooling You

Rather appropriately, April 1st is the 91st day of the year, at least in non-leap years such as 2013. 91 might look innocent, but it’s a sneaky little number because 91=7×13. That might not seem sneaky to you, but I’m here to tell you why it is. Every whole number can be broken down into [...]

Roots of Unity

## Where in the World Are the Odd Perfect Numbers?

The launch of Roots of Unity was just in time for this year’s Joint Mathematics Meetings, a mathapalooza put on by the two largest professional mathematical societies (the American Mathematical Society and the Mathematical Association of America). Nearly 6,000 of my closest mathematical friends are here with me in sunny San Diego taking in the [...]