Daniel Larsen ticked a little differently as a 13-year-old: The teenager from Bloomington, Indiana, liked to puzzle over puzzles like Scrabble – a game that has been known for its elderly fans at least since Loriot's “Schwanzhund” – and even created his own crossword.

He sent them to major newspapers, including the New York Times (NYT).

Eight times he tried. And eight times he was kindly told that his riddles didn't really suit the readership: too difficult, the words he was looking for too unfamiliar.

Not discouraged by this, however, he wrote a computer program that helped him improve his puzzle.

The ninth submission then brought the long-awaited success: Larsen became the youngest puzzler in NYT history in 2017.

Stubborn perseverance paired with an almost tireless tolerance for frustration: Daniel Larsen already amazed people as a child.

Last summer, now 17 and in his senior year of high school, these traits enabled him to devise a complicated proof from number theory: he was able to show that there is always a Carmichael number in the interval between x and 2x are.

Here, x represents any natural number that is sufficiently large.

Carmichael numbers are objects that are also called pseudoprimes because they satisfy certain relations that apply to prime numbers without actually being prime numbers.

With this proof, Larsen clearly shifted the previously assumed lower limit for the number of these numbers in a certain interval and also added an upper limit.

Above all, however, their distribution has now become clearer: Carmichael numbers, which were believed to have huge gaps between them along the number line, reliably appear again and again.

Larsen published his proof in the journal International Mathematics Research Notices (IMRN).

Impressive performance

The fact that a student manages to go through the review process in a journal that is respected by mathematicians such as IMRN and to publish there is "that's great," thinks Valentin Blomer.

The mathematician conducts research at the University of Bonn in the field of analytical number theory.

Blomer is also impressed by Larsen's argument because it is "technically difficult".

"We have little on hand to describe Carmichael numbers." That makes the strange numbers elusive.

So what are Carmichael numbers?

In order to understand this, one must first consider how prime numbers (2, 3, 7, 11, ...) can be characterized - and to do this, jump to 17th-century France.

At that time, the mathematician Pierre de Fermat was working on those numbers that are only divisible by themselves and one, and wondered how one could test whether a number was prime or composite, i.e. whether it could be broken down into further factors.

Then, in 1640, Fermat had an idea: if p is a prime number, then ap–a is equal to an integer multiple of p, where a is any natural number.

For the prime number 7, for example, this so-called little Fermat theorem is fulfilled: 27–2=126 and thus a multiple of 7, 37–3 is also a multiple of 7, and so on,

A primality test with gaps

A perfect prime number test was born - that was thought until the beginning of the 20th century when the American mathematician Robert Carmichael discovered a number that outsmarted Fermat's criterion: 561. The number 561 does fulfill the requirement that a561–a is a multiple of 561.

But 561 is still composite, it can be factored by 3∙11∙17.

For Fermat's little theorem, this means that any number that satisfies the test is not necessarily prime, but may belong to its imitators: to the Carmichael numbers.

It later turned out that at least three prime numbers are always needed as factors in order to generate such a pseudoprime number.