Nobody would ever accuse me of being a math whiz, though I do feel I have skills some of my younger acquaintances lack: I eschew a calculator on occasion to make sure the various cortices of my brain responsible for addition, subtraction, multiplication, and division get a little workout.
You just never know when your battery may run down while you’re stranded on a desert island and have an immediate need to balance your checkbook.
Oh, and there’s another mathematical task that I’ve mastered.
An older friend told me not long ago that physicians who are concerned that a patient may be in the early stages of dementia will test mental agility by asking said patient to subtract backwards from 100—by 8s.
I’ve gotten quite facile at that effort—and have moved on to 7s with similar success.
Why am I telling you all this? Because I’m inviting you to join me in exploring a story that involves number theory—a deep dive that I have no business whatsoever attempting.
But my friend Allan, who excels in math, knows I like quirky stories and thoughtfully sent me this one, which appeared in Popular Mechanics.
So once again, Annie plunges ahead—undeterred by her total lack of experience. I promise, though, that I won’t go beyond what I understand, which means there won’t be a whole lot of number theory here.
I hope not too many of you will be disappointed by this limitation…
Christopher Havens (Chris) is the name of the guy in the title. He’d made quite a mess of his life: a high school dropout, he had become a homeless drug addict.
And then in 2010, he committed murder, killing a 25-year-old man with whom he’d been feuding; there was apparently a methamphetamine connection.
He was convicted and sentenced to serve 25 years in the Washington Department of Corrections.
He didn’t immediately transform himself into a model prisoner—far from it. Soon after being incarcerated in 2011, his behavior earned him a place in solitary confinement.
That’s where and when he “got math”—recognized his love for mathematics.
He’d spend as many as 10 hours a day studying, and then entered the Intensive Transition Program, ITP, which he described (with a vulgarity) as
“a one-year program which helps people get their minds right. It’s designed to effectively aid you into ‘taking your head from your backside.’
“This was my schedule. Eat, math, remove my head from my backside, brush, rinse, repeat. It was an important time in my life.”
After that, he sought information to help him with his studies by writing the following letter:
“To Whom It May Concern:
I’m interested in finding more information on a subscription to Annals of Mathematics for personal use.
I’m currently serving 25 years in the Washington Department of Corrections and I’ve decided to use this time for self-betterment. I’m studying calculus and number theory.
As numbers have become my passion, can you please send me any information on your mathematic journal. Thank you…Chris Havens.
P.S.: I am self-teaching myself (sic) and often get hung on problems for long periods of time.
Is there anyone I could correspond with, provided I send self-addressed stamped envelopes?
There are no teachers here who can help me so I often spend hundreds on books that may or may not contain the help I need. Thank you.”
(It isn’t clear where he gets the “hundreds” he spends on books, but that’s not central to our story.)
The letter circuitously reached a man named Matthew Cargo in January, 2013. Cargo, then production editor for Mathematical Sciences Publications, forwarded it to two mathematicians: the parents of his partner, Marta Cerrruti.
Cerruti wrote about the results of that connection in phys.Org.
“Initially, my father, Umberto Cerruti, a number theorist who was a professor of mathematics at the University of Torino, Italy, agreed to help Havens simply because we asked him.
“My father thought that Havens was likely one of the many cranks that fall in love with numbers and come up with a flawed theory. To test him, he gave Havens a problem to solve.
“In return, my father received a 120-centimetre-long piece of paper in the mail, and on it was a long and complicated formula. My father entered the formula into his computer and to his surprise, the results were correct!
“After this, my father invited Havens to work on a problem involving continued fractions he was working on.”
OK, here’s where I have to stretch a bit. Continued fractions, which Euclid discovered in 300 BC, enable the expression of all numbers through sequences of integer numbers, which are positive and negative numbers expressed without fractions.
Cerruti gives pi as an example: the ratio of a circle’s diameter and circumference is written as 3.14159…
“The sequence of numbers after the initial digit continues forever and is totally chaotic. But written as a continued fraction its expression is simple and beautiful.”
Although this particular aesthetic is beyond my ken, Cerruti’s point is that continued fractions show the power of number theory.
Much of this is theoretical, pure math, but number theory plays an important role in cryptography today.
Even I understand that its many uses are applicable in both financial institutions and the military, among other areas. It’s the reason you can buy something online or pay your bills in what are for the most part secure transactions.
Havens was so motivated that he started a Prison Mathematics Project in which he explained math to other inmates.
The group met biweekly and used books sent by Cerruti’s father. That change from the man who’d been in solitary confinement struck me as extremely impressive.
And the story gets dramatically better:
From his prison cell, Havens made a significant contribution to the field of continued fractions—so significant that he is named the first author in an article published in the January, 2020, issue of the journal Research in Number Theory.
(If you want to read the article, you’ll have to plunk down $99 for it.)
He reached this lofty spot using the highly sophisticated tools of pen and paper, “exchanging ideas with his co-authors in Italy though hard-copy letters mailed across the ocean,” Cerruti writes.
Cerruti describes Havens’ pathbreaking work: He “showed for the first time some regularities in the approximation of a vast class of numbers.”
In essence, he provided number theorists with an important new tool whose applications may be vast. According to Cerruti,
“finding new ways for writing numbers is one of the most important problems for a number theorist, although the results may not have an immediate application.”
She points out that some supercomputers are employed solely for computing pi digits into the trillions.
Havens’ achievement has also had a huge impact on his own outlook and future plans.
He’s studying for an associate degree through the mail and is determined to complete bachelors and graduate degrees in math and to pursue a career once he’s released from prison.
And he hopes to turn the Prison Mathematics Project he founded into a nonprofit organization to help inmates with a talent for math.
He told Cerruti:
“I definitely have plotted out a long term life plan to accommodate paying a debt that has no price. I know this path is permanent…and there never is a day that it’s finally paid off.
“But this longevity in debt is not bad. It’s inspiration. Maybe this will sound stupid, but I serve my time in the company of the soul of my victim. I dedicate a lot of my biggest accomplishments to him.”