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.

## Patterns in Pairs of Primes

To understand reciprocity, you first need to Phone Number List understand modular arithmetic. Modular operations rely on calculating remainders when you’re 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. Just as with the integers, these number systems can have perfect squares —numbers that are the product of another number times itself.

### The Power of Generalization

Soon after Gauss published the first proof BYB Directory 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. But they got stuck, Conrad said, “because there’s no easy pattern.” This changed once Gauss brought reciprocity into the realm of complex numbers.