Quadratic Reciprocity

“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.”

“One key difference stems from a property called quadratic reciprocity, first proved by Carl Gauss”

“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”

“Soon after Gauss published the first proof of quadratic reciprocity in 1801, mathematicians tried to extend the idea beyond squares. “Why not third powers or fourth powers? They imagined maybe there’s a cubic reciprocity law or quartic reciprocity law,” said Keith Conrad, a number theorist at the University of Connecticut.”

“In 1832, Gauss proved a quartic reciprocity law for the complex integers that bear his name.”

“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.”

“Then, in the 1920s, Emil Artin, one of the founders of modern algebra, discovered what Conrad calls the “ultimate reciprocity law.” All the other reciprocity laws could be seen as special cases of Artin’s reciprocity law.”

“Quadratic reciprocity is used in areas of research as diverse as graph theory, algebraic topology and cryptography.”