Nov 5, 2023 07:16 PM
The Hidden Connection That Changed Number Theory
https://www.quantamagazine.org/the-hidde...-20231101/
INTRO: There are three kinds of prime numbers. The first is a solitary outlier: 2, the only even prime. After that, half the primes leave a remainder of 1 when divided by 4. The other half leave a remainder of 3. (5 and 13 fall in the first camp, 7 and 11 in the second.) There is no obvious reason that remainder-1 primes and remainder-3 primes should behave in fundamentally different ways. But they do.
One key difference stems from a property called quadratic reciprocity, first proved by Carl Gauss, arguably the most influential mathematician of the 19th century. “It’s a fairly simple statement that has applications everywhere, in all sorts of math, not just number theory,” said James Rickards, a mathematician at the University of Colorado, Boulder. “But it’s also non-obvious enough to be really interesting.”
Number theory is a branch of mathematics that deals with whole numbers (as opposed to, say, shapes or continuous quantities). The prime numbers — those divisible only by 1 and themselves — are at its core, much as DNA is core to biology. Quadratic reciprocity has changed mathematicians’ conception of how much it’s possible to prove about them. If you think of prime numbers as a mountain range, reciprocity is like a narrow path that lets mathematicians climb to previously unreachable peaks and, from those peaks, see truths that had been hidden.
Although it’s an old theorem, it continues to have new applications. This summer, Rickards and his colleague Katherine Stange, together with two students, disproved a widely accepted conjecture about how small circles can be packed inside a bigger one. The result shocked mathematicians. Peter Sarnak, a number theorist at the Institute for Advanced Study and Princeton University, spoke with Stange at a conference soon after her team posted their paper. “She told me she has a counterexample,” Sarnak recalled. “I immediately asked her, ‘Are you using reciprocity somewhere?’ And that was indeed what she was using.’”
Patterns in Pairs of Primes
To understand reciprocity, you first need to understand modular arithmetic... (MORE - details)
A New Generation of Mathematicians Pushes Prime Number Barriers
https://www.quantamagazine.org/a-new-gen...-20231026/
INTRO: More than 2,000 years ago, the Greek mathematician Eratosthenes came up with a method for finding prime numbers that continues to reverberate through mathematics today. His idea was to identify all the primes up to a given point by gradually “sieving out” the numbers that aren’t prime. His sieve starts by crossing out all the multiples of 2 (except 2 itself), then the multiples of 3 (except 3 itself). The next number, 4, is already crossed out, so the next step is to cross out the multiples of 5, and so on. The only numbers that survive are primes — numbers whose only divisors are 1 and themselves.
Eratosthenes was focused on the full set of primes, but you can use variations on his sieve to hunt for primes with all kinds of special features. Want to find “twin primes,” which are only 2 apart, like 11 and 13 or 599 and 601? There’s a sieve for that. Want to find primes that are 1 bigger than a perfect square, like 17 or 257? There’s a sieve for that too.
Modern sieves have fueled many of the biggest advances in number theory on problems ranging from Fermat’s Last Theorem to the still unproved twin primes conjecture, which says that there are infinitely many pairs of twin primes. Sieve methods, the Hungarian mathematician Paul Erdős wrote in 1965, are “perhaps our most powerful elementary tool in number theory.”
Yet this power is constrained by mathematicians’ limited understanding of how primes are distributed along the number line. It’s simple to carry out a sieve up to some small number, like 100. But mathematicians want to understand the behavior of sieves when numbers get big. They can’t hope to list all the numbers that survive the sieve up to some extremely large stopping point. So instead, they try to estimate how many numbers are on that list... (MORE - details)
https://www.quantamagazine.org/the-hidde...-20231101/
INTRO: There are three kinds of prime numbers. The first is a solitary outlier: 2, the only even prime. After that, half the primes leave a remainder of 1 when divided by 4. The other half leave a remainder of 3. (5 and 13 fall in the first camp, 7 and 11 in the second.) There is no obvious reason that remainder-1 primes and remainder-3 primes should behave in fundamentally different ways. But they do.
One key difference stems from a property called quadratic reciprocity, first proved by Carl Gauss, arguably the most influential mathematician of the 19th century. “It’s a fairly simple statement that has applications everywhere, in all sorts of math, not just number theory,” said James Rickards, a mathematician at the University of Colorado, Boulder. “But it’s also non-obvious enough to be really interesting.”
Number theory is a branch of mathematics that deals with whole numbers (as opposed to, say, shapes or continuous quantities). The prime numbers — those divisible only by 1 and themselves — are at its core, much as DNA is core to biology. Quadratic reciprocity has changed mathematicians’ conception of how much it’s possible to prove about them. If you think of prime numbers as a mountain range, reciprocity is like a narrow path that lets mathematicians climb to previously unreachable peaks and, from those peaks, see truths that had been hidden.
Although it’s an old theorem, it continues to have new applications. This summer, Rickards and his colleague Katherine Stange, together with two students, disproved a widely accepted conjecture about how small circles can be packed inside a bigger one. The result shocked mathematicians. Peter Sarnak, a number theorist at the Institute for Advanced Study and Princeton University, spoke with Stange at a conference soon after her team posted their paper. “She told me she has a counterexample,” Sarnak recalled. “I immediately asked her, ‘Are you using reciprocity somewhere?’ And that was indeed what she was using.’”
Patterns in Pairs of Primes
To understand reciprocity, you first need to understand modular arithmetic... (MORE - details)
A New Generation of Mathematicians Pushes Prime Number Barriers
https://www.quantamagazine.org/a-new-gen...-20231026/
INTRO: More than 2,000 years ago, the Greek mathematician Eratosthenes came up with a method for finding prime numbers that continues to reverberate through mathematics today. His idea was to identify all the primes up to a given point by gradually “sieving out” the numbers that aren’t prime. His sieve starts by crossing out all the multiples of 2 (except 2 itself), then the multiples of 3 (except 3 itself). The next number, 4, is already crossed out, so the next step is to cross out the multiples of 5, and so on. The only numbers that survive are primes — numbers whose only divisors are 1 and themselves.
Eratosthenes was focused on the full set of primes, but you can use variations on his sieve to hunt for primes with all kinds of special features. Want to find “twin primes,” which are only 2 apart, like 11 and 13 or 599 and 601? There’s a sieve for that. Want to find primes that are 1 bigger than a perfect square, like 17 or 257? There’s a sieve for that too.
Modern sieves have fueled many of the biggest advances in number theory on problems ranging from Fermat’s Last Theorem to the still unproved twin primes conjecture, which says that there are infinitely many pairs of twin primes. Sieve methods, the Hungarian mathematician Paul Erdős wrote in 1965, are “perhaps our most powerful elementary tool in number theory.”
Yet this power is constrained by mathematicians’ limited understanding of how primes are distributed along the number line. It’s simple to carry out a sieve up to some small number, like 100. But mathematicians want to understand the behavior of sieves when numbers get big. They can’t hope to list all the numbers that survive the sieve up to some extremely large stopping point. So instead, they try to estimate how many numbers are on that list... (MORE - details)