**OR A LONG** time, prime numbers kept Daniel Larsen up at night. Prime numbers and their weird cousins, Carmichael numbers, which act like primes but aren’t. Daniel would lie in bed in his split-level house on a quiet street in Bloomington where his parents, math professors at IU, slept in a bedroom nearby, and think about a complex math problem concerning Carmichael numbers. It was a problem that his father, Dr. Michael Larsen, estimated Daniel had little chance of solving as it had stumped a trio of mathematicians, whose cumulative ages at the time of their breakthrough paper on the subject was 136.

Daniel was 17. A prime number. Two less than its twin prime number, 19, but more on that later.

“It seemed like a long shot to me,” recalls Michael Larsen. We’re talking math in the Larsens’ living room, a big, open space with a piano, lots of books, a bowl of seashells, and several menorahs on the fireplace mantle. “It wasn’t an absurdity, not hopeless, but I thought if he was going to work on a problem for a long time and get frustrated, it could be a negative experience. But he was into it. I didn’t tell him *not* to do it. He probably wouldn’t have listened to me if I had.”

This last bit makes Daniel laugh.

“You said I had a 10 percent chance!” he says. Daniel looks a bit like Harry Potter—thin, brown hair, glasses, black socks. He’s soft-spoken and shy, and laughs easily at himself. “That wasn’t really discouraging for me. I wanted to prove you wrong.”

Daniel is really good at things many people struggle with. Things like Rubik’s Cubes, chess, spelling bees, crossword puzzles, and math. When Daniel is interested in something, he takes a deep dive—and often succeeds at the highest level. Take crossword puzzles. Daniel was just 13 when he published a Tuesday crossword puzzle in *The New York Times*, the youngest constructor in the history of the newspaper. Since then, he has published 10 more. The* Times* pays $750 a puzzle.

“I think it’s very generous since I do it for fun,” he says.

And that’s the striking thing about all of Daniel’s creative projects. He pursues them—day and night—not because his parents nag or teachers egg him on, not for the celebrity or the cash, but because he finds them fun. And of all the challenges Daniel has pursued since what he calls “the olden days,” when he was a kid, the most fun—and wondrous—is math.

“I’ve been drawn to understanding how things work, what the fundamental structure and order of something is, and math is full of these beautiful pictures of how seemingly unrelated concepts are part of the same idea,” says the Bloomington High School South graduate who is headed to MIT in the fall. “That’s my favorite part of math: how you have these beautiful connections. There’s a great sense of harmony and unity.”

**MATH WAS IN** the air in the Larsen house when Daniel was growing up. Michael Larsen used to host Math Circle, a free Saturday-afternoon group for area kids, where he talked about oddball topics that would excite kids about math. Daniel would listen in, though he was only 4.

“The fact we were talking about math a lot when they were growing up had an influence, I am sure,” Michael says. “But children come with definite desires and interests and tastes. Your child wants to do what your child wants to do.”

What Daniel wanted to do was projects. At age 3, he would plant himself in front of classical music concerts on YouTube—and conduct. He took up violin at 5, piano at 6. When he could solve a Rubik’s Cube in 45 seconds, he moved on to larger cubes, four, five, and seven squares. He devised a Lego robot that would sort copper pennies from zinc ones. “It wasn’t very reliable,” he concedes, “but it was better than nothing.”

In middle school, Daniel twice flew to Washington, D.C., and competed in the Scripps National Spelling Bee. Despite wearing his lucky lion T-shirt, he never made it to the final round of 50. In seventh grade, he didn’t score well enough on the written test. The next year, he aced the written test, but misspelled a word that wasn’t on his study list of 5,000. Sheepishly, he tells me the word he missed, but asks me not to print it.

“It’s a bit embarrassing that it’s a word everybody can spell, but it wasn’t what I studied and I was under pressure,” Daniel says. “It was not my finest hour.”

I jot down the proper noun, seven-letters commonly slurred to sound more like six. Later, I realize that I’ve spelled it wrong.

THE GREAT crossword puzzle adventure began on his family’s yellow couch, where his parents and older sister, Anne, used to puzzle through puzzles together. “I wasn’t very good at solving them,” Daniel says. “So I thought if I can’t solve them, maybe I can make them, and then I can stump other people, instead of being stumped myself.”

Daniel began making 4-by-4 word squares, then his father devised a crossword puzzle computer program so Daniel could design online. For his 12th birthday, Daniel asked for a word list and wrote code to upgrade his software. His first puzzle submission to *The New York Times* had a Bertie and Jeeves theme, based on a favorite writer, P.G. Wodehouse. The* Times* wrote back: “This is solid work, but the theme doesn’t quite excite us enough to say yes.”

Seven more rejection letters followed. “It was very discouraging at first,” Daniel recalls. “I didn’t realize how competitive it was.”

His ninth attempt proved the charm. His theme was Elmer Fudd, and *Times* crossword puzzle editor Will Shortz was so delighted to learn Daniel was 13 years old that he wrote a story about him. Fifty-eight teenagers have published puzzles in the *Times*, including Daniel’s sister, Anne. Daniel was the youngest by half a year.

Daniel didn’t tell his friends, but his school caught wind of it and made an announcement. “It wasn’t a big deal,” Daniel says. “I don’t expect that record to last very long, but for now, I am happy with it.”

The math of crosswords is daunting. The* Times* accepts only seven puzzles from 200 weekly submissions. Daniel has submitted 60 puzzles for his 11 acceptances. Each puzzle takes him two to 50 hours to create. The English of crosswords is also daunting. It’s not enough to simply fill your grid. Answers should be witty, fun. If your crossword has a theme, it can’t be cliched or obscure. One of Daniel’s coolest puzzles had an alphabet theme, with phrases such as UVWAVE, and XYZAFFAIR, and his personal favorite, FILMNOIR, which has, amazingly, LMNO.

Daniel still struggles coming up with clues. “There are people who do almost all their clues, but I was never able to come up with very many good clues,” he says. “Most of the clues are theirs.”

His dad tries to help, to not much avail. “*The New York Times* tells us we’re not good at clues,” he says, with a chuckle. “It’s a knack none of us have really mastered.”

Daniel combs through his latest puzzle, checking off clues the editors kept.

Eight letters: Many national anthems: WARSONGS

Three letters: ____ Reactor (Iron Man’s power source.): ARC

Three letters: High degree: NTH

Five letters: Path an electron may take moving in a constant magnetic field: HELIX

Four letters: Big name in laptops: ACER

Three letters: Indifferent remark: MEH

Shortz, the *Times* puzzle master, says there’s nothing MEH about Daniel’s puzzles. “They are as good as a contributor of any age,” says the Crawfordsville native. “He especially likes theme-less puzzles, and he fills them with interesting, colorful, fresh, juicy vocabulary.”

Case in point, Daniel’s latest puzzle, which has a whopping eight 15-letter answers, such as A WALK IN THE WOODS, THE BACHELORETTE, CARE TO ELABORATE, RUSSIAN ROULETTE, SLEEVELESS DRESS, and mostly avoids crossword puzzle-ese, short, cliched words jammed with vowels.

“This is a gorgeously made puzzle,” Shortz says.

While most constructors have ditched graph paper and dictionaries for the computer and word databases, Shortz insists there’s nothing facile about designing a compelling puzzle and prevailing in the fierce competition for publication in the *Times*.

“Anyone can make a bad crossword,” he says. “It takes a special person to make a good one.”

** AND NOW WE** circle back to math with a quick lesson on prime numbers. For those who may have forgotten, a prime number is a positive number with only two factors, one and itself. 7 is a prime number because 7 and 1 are the only things you can multiply to get 7. Twelve is not because there are a few options (3 x 4, for example). Thousands of years ago, Euclid—the founder of geometry, born in 300 B.C.—proved there are infinitely many prime numbers. Interestingly, some prime numbers come in pairs just two apart, such 3 and 5, 5 and 7, 17 and 19. The spacing between these sets of “twins” becomes farther apart as the numbers grow. (The largest known twin primes have 388,342 digits.) But does the list of twin primes ever end? This unsolved question has a name that sounds like a Robert Ludlum thriller: the twin prime conjecture. While all this number theory might sound really abstract, prime numbers do have some practical, modern-day uses. The encryption algorithm RSA uses the magic of primes to keep banking, online shopping, and other communication secure.

Enter Daniel and a movie.

Five years ago, he watched the documentary *Counting from Infinity*, which tells the story of Yitang Zhang, an unknown mathematician who toiled for years on the twin prime conjecture, at one point working at Subway to support himself. (Zhang earned his Ph.D. at Purdue, but had a falling out with his adviser who did not write him a letter of recommendation for a job.) In 2013, Zhang proved there are infinitely many consecutive primes less than 70 million apart, a thrilling breakthrough in number theory that made him an instant math celebrity. He was later awarded the MacArthur “Genius” Grant.

“If you go up to unimaginably large numbers, primes that are 70 million apart are like right next door in the grand scheme of things,” Daniel says. The grand scheme of things is where Daniel spends a lot of his time.

Daniel tried to read Zhang’s paper. Impenetrable. But then another mathematician wrote a clearer proof, whittling 50 pages down to 15. Daniel followed his curiosity to an important 1994 paper by W.R. Alford, Andrew Granville, and Carl Pomerance, which proved that, just like primes, there are an infinite number of Carmichael numbers. The authors also raised the question of whether Bertrand’s Postulate—for every number there is a prime between it and its double—could be proven for Carmichael numbers.

Daniel thought: “*That can’t be that hard*, so I just started thinking about it.” He laughs. “It turned out to not be so easy.”

Daniel tackled the question in his free time.

For six months, Michael Larsen watched his son persevere. “You need to get to the point where it’s under your skin and driving you nuts, if you want to work at maximum effectiveness,” he says. “It’s probably true for most people in other things as well. Daniel became obsessed with this problem.”

Last summer, Daniel visited his grandparents in Massachusetts. Except for breaks to watch Olympic swimming, Daniel was on the computer, his spirits rising and falling with each breakthrough and setback.

“He wouldn’t give up,” recalls his grandfather, David Larsen. “That’s part of his nature.”

Daniel’s sister, Anne, encouraged him, though she had her limits. “He would talk about it to anyone who would listen, which was not really me,” she says. Anne also heads to MIT this fall, to pursue a Ph.D. in math. “I don’t want to talk about this at breakfast.”

After roughly 300 hours of work, Daniel finally proved there is a Carmichael number between any number and its double. (His actual finding has a few caveats and complications too arcane for a reporter who never made it past calculus to explain.) He sent the paper to British mathematician Andrew Granville, co-author of that 1994 paper on Carmichael numbers published in *Annals of Mathematics*, a premier journal in the field.

“It’s amazing, to be honest,” Granville says. “It’s hard, technical stuff, really quite deep. The level of detail and depth is unbelievable for a high school student.”

The writing did need work. Granville made suggestions. “Daniel came back a few weeks later and he had far surpassed anything I’d expected, or even thought of myself,” Granville says. “He took everything I said and ran a long distance with it. He’s very, very good at taking advice and thinking about that advice. He seems to have everything going for him.”

Granville says the paper would make a good Ph.D. thesis. Indeed, Daniel is currently taking two graduate-level math courses at IU, one with Dr. Noah Snyder, who says in his 10 years on campus, he can only remember one other high school student taking classes at this level: Anne Larsen, Daniel’s sister.

“It’s difficult to make long-term predictions, but I think it’s clear that if Daniel decides he wants to stay in mathematics or closely related sciences, he’s likely to be quite successful,” Snyder says. “He is very impressive.”

** IT WAS** Daniel’s grandfather, David, who suggested Daniel submit his proof to the Regeneron Science Talent Search, formerly the Westinghouse, the nation’s oldest and most prestigious high school science contest. The Larsen family has a long history with the contest. David applied but didn’t win. Daniel’s grandmother, Suzanne Larsen, received an honorable mention. Daniel’s father applied but was disqualified when a faculty recommender missed the deadline.

Nearly 1,900 students apply to the Regeneron each year, an elaborate process, similar to applying to college. In January, Daniel learned he was one of 40 finalists and flew to Washington, D.C., for a week of interviews and events. The top 10 winners were announced in a formal ceremony—ballgowns and tuxes—which was livestreamed.

Anne followed the frantic family texts about sartorial stress. “It was kind of funny,” she says. “My little brother is up there wearing a tuxedo. He’s never worn a tuxedo in his life. His roommate figured out his cufflinks.”

After an hour of speeches and windup, the “class” of 40 finalists stood shoulder to shoulder on stage as the emcee opened the 10th-place envelope and worked his way up.

“I am standing on stage thinking, *This is nice, but I am not going to get any**thing*,” Daniel says. “Then they called me for fourth place.”

It was the highest award for a math project.

His prize: a check for $100,000.

His grandparents were thrilled. “It was so improbable that a high school kid could solve a problem that very good mathematicians had not been able to crack in 27 years,” David says. “When I heard that he had done it, I was very, very happy. The prize is nice and the excitement of number four is nice, but the achievement of his effort was the thing that I was most excited by.”

Daniel’s paper was recently accepted by the journal *International Mathematics Research Notices*. It is his second publishing credit.

**IT’S COMFORTING** in such moments to remember that even the most remarkable people have shortcomings. I ask Daniel what he’s not good at.

“It feels like I struggle with everything,” he says. “Do you mean school subjects?”

I shrug. “I mean like making your bed.”

“Oh, that, for sure. Being organized, in general. Using my time in efficient ways.” He clasps his hands and stretches. “I find I take the path of least resistance often. If there’s a situation that I am unhappy about, I might not be as active in dealing with it as I should be.”

I press for an example. Sometimes when he’s cold, he can’t be bothered to fetch his coat. The other day, he couldn’t read the blackboard, but didn’t move closer. “There was a problem and there was a solution. I guess laziness is what I’m trying to say.”

**I HAD ALMOST** finished a draft of this profile when I thought, *Gee, maybe I should actually see Daniel’s paper.* He emailed it to me. The shock was not that it was 26 pages. (As an English professor, I see work 10 times that long.) The shock is that it was 26 pages of math—math like I have never seen before, a menagerie of strange symbols like an oversized, spikey E and swooping F like a crane’s neck. An alphabet soup of algebraic letters—some bolded, others in italics—intermix with *max* and *log* and *cos*, all stacked in elaborate fractions that squeeze under square-root signs like people hiding from the rain. Then in wanders an upside-down horseshoe and the IU logo and a yoga pose, while dozens of greater-than and less-than signs swim about like lost fish.

The most complex equations stretch five lines long.

Like a paragraph. A paragraph of math.

These massive fractions build an argument. You can tell this by the English transition words, *therefore *and *hence* and *it follows,* and the paper grows more emphatic with the arrival on page 18 of a stampede of oversized II, and the square-root signs swell and more cranes fly in, followed by a wacky symbol with three horizontal lines like a triple bunk beds or a cheese sandwich, and then the exponents rise up, stretching like a delicate tree limb of baby leaves, until on the final page, the equations shorten—*hence* and *therefore*—and Daniel modestly concludes: “In fact, Theorem 2 is effective.”

Stop.

It’s a symphony. A leaf under a microscope. A dictionary in Italian. If this is the blur Daniel was struggling to bring into focus, of course he couldn’t be bothered to fetch his coat.

**AFTER A COUPLE** hours at the Larsens’ house, I’m seeing math everywhere. How many books? The probability that one spring leaf will fall off a tree. Michael laughs and says he probably sees more math in the world than the average person, but math “isn’t the key to every lock.”

Daniel interjects, explaining this is one reason he’s taking a break from crosswords. “I’d be watching a movie and somebody would say something, and I’d think, *That should be on my word list.* I don’t want to be carrying that around all the time.”

I save a couple big-sky questions for last. What’s going to be the biggest problem of your generation? Daniel’s answer relates to why he dislikes cellphones.

“Everything feels very disconnected these days,” he says. “Because we’re interacting with people online, there is less sense of a community and common purpose, and you see people being more selfish because they are not as connected with their environment, and I think that’s a major problem. There might be more existential problems, but this is a meta problem, in that it stops us from addressing other problems.”

He sees this as another wondrous aspect of math. Math can build consensus, a rare feat these days.

“If two people have a mathematical disagreement, almost always they can konk it out and one of them will say, ‘Okay, you’re right,’ or maybe they were both right but misunderstanding each other.” Daniel tips his head, wistful. “In the rest of the world, it’s very hard for people to agree on very basic things. I do think the principle of being able to agree on stuff is important.”

Daniel is Jewish, and I ask him how someone devoted to proofs sustains religious faith. He jumps to physics. While human life is relatively mundane, he’s astonished by the delicate balance of physical laws that make life and matter possible. “Intelligent design,” I say. He agrees. “So God is a mathematician?”

“That’s how I like to think about it,” he says.

“So you’re doing God’s work?” I ask.

He begs off with a laugh. “I don’t like taking it that far.”

**MY LAST** question is one I’m rather proud of: “If DANIEL LARSEN is the crossword puzzle answer, what would be the clue?”

The lion face on Daniel’s shirt stretches, almost seeming to blink. Daniel likes lions because they are majestic, and there is something majestic about Daniel, a young man who is exceptional but humble, who uses his free time to puzzle through problems most of us can barely fathom.

After a moment, Daniel gives up. “I should be able to think of something clever, but I can’t,” he says.

Even a puzzle master is sometimes stumped.

Later, I ask Daniel’s father the same question. It takes Michael Larsen only a minute to think up his crossword clue for DANIEL LARSEN, one that *The New York Times* would not accept, but that surely comes from the heart: “Person I will miss very much when he goes away next year.”

Will Shortz also plays the game. “What jumps to mind when I hear the name Daniel Larsen? Brilliant, young, math-loving Hoosier. If I was writing a crossword clue for him, I’d say, ‘Youngest-ever *New York Times* crossword contributor.’ That’s an interesting fact that makes him unique in the world.”