wp-plugin-hostgator domain was triggered too early. This is usually an indicator for some code in the plugin or theme running too early. Translations should be loaded at the init action or later. Please see Debugging in WordPress for more information. (This message was added in version 6.7.0.) in /home4/scienrds/scienceandnerds/wp-includes/functions.php on line 6114ol-scrapes domain was triggered too early. This is usually an indicator for some code in the plugin or theme running too early. Translations should be loaded at the init action or later. Please see Debugging in WordPress for more information. (This message was added in version 6.7.0.) in /home4/scienrds/scienceandnerds/wp-includes/functions.php on line 6114Source:https:\/\/www.quantamagazine.org\/the-hidden-connection-that-changed-number-theory-20231101\/#comments<\/a><\/br> 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.<\/p>\n One key difference stems from a property called quadratic reciprocity, first proved by Carl Gauss, arguably the most influential mathematician of the 19th century. \u201cIt\u2019s a fairly simple statement that has applications everywhere, in all sorts of math, not just number theory,\u201d said James Rickards<\/a>, a mathematician at the University of Colorado, Boulder. \u201cBut it\u2019s also non-obvious enough to be really interesting.\u201d<\/p>\n Number theory is a branch of mathematics that deals with whole numbers (as opposed to, say, shapes or continuous quantities). The prime numbers \u2014 those divisible only by 1 and themselves\u00a0\u2014\u00a0are at its core, much as DNA is core to biology. Quadratic reciprocity has changed mathematicians\u2019 conception of how much it\u2019s 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.<\/p>\n Although it\u2019s an old theorem, it continues to have new applications. This summer, Rickards and his colleague Katherine Stange<\/a>, together with two students, disproved a widely accepted conjecture<\/a> about how small circles can be packed inside a bigger one. The result shocked mathematicians. Peter Sarnak<\/a>, a number theorist at the Institute for Advanced Study and Princeton University, spoke with Stange at a conference soon after her team posted<\/a> their paper. \u201cShe told me she has a counterexample,\u201d Sarnak recalled. \u201cI immediately asked her, \u2018Are you using reciprocity somewhere?\u2019 And that was indeed what she was using.\u2019\u201d<\/p>\n To understand reciprocity, you first need to understand modular arithmetic. Modular operations rely on calculating remainders when you\u2019re dividing by a number called the modulus. For example, 9 modulo 7 is 2, because if you divide 9 by 7, you are left with a remainder of 2. In the modulo 7 number system, there are 7 numbers: {0, 1, 2, 3, 4, 5, 6}. You can add, subtract, multiply and divide these numbers.<\/p>\n Just as with the integers, these number systems can have perfect squares \u2014numbers that are the product of another number times itself. For example, 0, 1, 2 and 4 are the perfect squares modulo 7 (0 \u00d7 0 = 0, 1 \u00d7 1 = 1, 2 \u00d7 2 = 4, and 3 \u00d7 3 = 2 mod 7). Every ordinary square will be equal to either 0, 1, 2 or 4 modulo 7. (For example, 6 \u00d7 6 = 36 = 1 mod 7.)\u00a0 Because modular number systems are finite, perfect squares are more common.<\/p>\n Quadratic reciprocity stems from a relatively straightforward question. Given two primes p<\/em> and q<\/em>, if you know that p<\/em> is a perfect square modulo q<\/em>, can you say whether or not q<\/em> is a perfect square modulo p<\/em>?<\/p>\n It turns out that as long as either p<\/em> or q<\/em> leaves a remainder of 1 when divided by 4, if p<\/em> is a perfect square modulo q<\/em>, then q<\/em> is also a perfect square modulo p<\/em>. The two primes are said to reciprocate.<\/p>\n On the other hand, if both of them leave a remainder of 3 (like, say, 7 and 11) then they don\u2019t reciprocate: If p <\/em>is a square modulo q<\/em>, that means that q<\/em> will not be a square modulo p<\/em>. In this example, 11 is a square modulo 7, since 11 = 4 mod 7 and we already know that 4 is one of the perfect squares modulo 7. It follows that 7 is not a square modulo 11. If you take the list of ordinary squares (4, 9, 16, 25, 36, 49, 64, \u2026) and look at their remainders modulo 11, then 7 will never appear.<\/p>\n This, to use a technical term, is really weird!<\/p>\n Like many mathematical ideas, reciprocity has been influential because it can be generalized.<\/p>\n Soon after Gauss published the first proof of quadratic reciprocity in 1801, mathematicians tried to extend the idea beyond squares. \u201cWhy not third powers or fourth powers? They imagined maybe there\u2019s a cubic reciprocity law or quartic reciprocity law,\u201d said Keith Conrad<\/a>, a number theorist at the University of Connecticut.<\/p>\n But they got stuck, Conrad said, \u201cbecause there\u2019s no easy pattern.\u201d This changed once Gauss brought reciprocity into the realm of complex numbers, which add the square root of minus 1, represented by i<\/em>, to ordinary numbers. He introduced the idea that number theorists could analyze not only ordinary integers but other integer-like mathematical systems, like so-called Gaussian integers, which are complex numbers whose real and imaginary parts are both integers.<\/p>\n With Gaussian integers, the whole notion of what counts as prime changed. For example, 5 is no longer prime, because 5 = (2 + i<\/em>) \u00d7 (2 \u2212 i<\/em>). \u201cYou have to start over like you\u2019re in elementary school again,\u201d Conrad said. In 1832, Gauss proved a quartic reciprocity law for the complex integers that bear his name.<\/p>\n Suddenly, mathematicians learned to apply tools like modular arithmetic and factorization to these new number systems. Quadratic reciprocity was the inspiration, according to Conrad.<\/p>\n Patterns that had been elusive without complex numbers now started to emerge. By the mid-1840s Gotthold Eisenstein and Carl Jacobi had proved the first cubic reciprocity laws.<\/p>\n Then, in the 1920s, Emil Artin, one of the founders of modern algebra, discovered what Conrad calls the \u201cultimate reciprocity law.\u201d All the other reciprocity laws could be seen as special cases of Artin\u2019s reciprocity law.<\/p>\n A century later, mathematicians are still devising new proofs of Gauss\u2019s first quadratic reciprocity law and generalizing it to novel mathematical contexts. Having many distinct proofs can be useful. \u201cIf you want to extend the result to a new setting, maybe one of the arguments will easily carry over, while the other ones won\u2019t,\u201d Conrad said.<\/p>\n Quadratic reciprocity is used in areas of research as diverse as graph theory, algebraic topology and cryptography. In the latter, an influential public key encryption algorithm developed in 1982 by Shafi Goldwasser<\/a> and Silvio Micali<\/a> hinges on multiplying two large primes p<\/em> and q<\/em> together and outputting the result, N<\/em>, along with a number, x<\/em>, which is not a square modulo N<\/em>. The algorithm uses N<\/em> and x<\/em> to encrypt digital messages into strings of larger numbers. The only way to decrypt this string is to decide whether or not each number in the encrypted string is a square modulo N<\/em> \u2014 virtually impossible without knowing the values of the primes p<\/em> and q<\/em>.<\/p>\n And of course, quadratic reciprocity crops up repeatedly within number theory. For instance, it can be used to prove that any prime number equal to 1 modulo 4 can be written as the sum of two squares (for example, 13 equals 1 modulo 4, and 13 = 4 + 9 = 22<\/sup> + 32<\/sup>). By contrast, primes equal to 3 modulo 4 can never be written as the sum of two squares.<\/p>\n Sarnak noted that reciprocity might be used to solve open questions, like figuring out which numbers can be written as the sum of three cubes. It\u2019s known that numbers that are equal to 4 or 5 modulo 9 are not equal to the sum of three cubes, but others remain a mystery. (In 2019, Andrew Booker generated headlines<\/a> when he discovered that (8,866,128,975,287,528)\u00b3 + (\u22128,778,405,442,862,239)\u00b3 + (\u22122,736,111,468,807,040)\u00b3 = 33.)<\/p>\n For all its many applications, and many different proofs, there is something about reciprocity that remains a mystery, Stange said.<\/p>\n \u201cWhat often happens with a mathematical proof is you can follow every step; you can believe that it\u2019s true,\u201d she said. \u201cAnd you can still come out the other end feeling like, \u2018But why?\u2019\u201d<\/p>\n Understanding, at a visceral level, what makes 7 and 11 different from 5 and 13 might be forever beyond reach. \u201cWe can only juggle so many levels of abstraction,\u201d she said. \u201cIt shows up all over the place in number theory \u2026 and yet it\u2019s just a step beyond what feels like you could really just know.\u201d<\/p>\n Quanta\u00a0is conducting a series of surveys to better serve our audience. Take our\u00a0<\/i>mathematics reader survey<\/i><\/a>\u00a0and you will be entered to win free\u00a0<\/i>Quanta\u00a0merch.<\/i><\/p>\n<\/div>\n <\/br><\/br><\/br><\/p>\n
\nThe Hidden Connection That Changed Number Theory<\/br>
\n2023-11-02 21:58:40<\/br><\/p>\nPatterns in Pairs of Primes<\/strong><\/h2>\n
The Power of Generalization<\/strong><\/h2>\n
Why Reciprocity Is So Useful<\/strong><\/h2>\n