This is pretty much asked for every chapter, so I'll take the torch on this one. What exactly is the point of quadratic reciprocity? I can't exactly imagine a lot of people running into problems in which they need to figure out what numbers, if any, when squared are congruent to 5 mod 17, for example. Just a thought.

You know, Brad, thats a really good question! I actually completely agree with you…the only thing I've come up with is that some people that are crazy about math (such as ourselves) enjoy computing these problems….idk!

Actually, it is because Gauss had already accomplished so much that he decided to prove this theorem to make future mathematicians feel inferior to his extraordinary intelligence.

Applications of quadratic residues

**Acoustics**

Sound diffusers have been based on number-theoretic concepts such as primitive roots and quadratic residues.

**Graph theory**

Paley graphs are dense undirected graphs, one for each prime p ≡ 1 (mod 4), that form an infinite family of conference graphs, which yield an infinite family of symmetric conference matrices.

Paley digraphs are directed analogs of Paley graphs, one for each p ≡ 3 (mod 4), that yield antisymmetric conference matrices.

The construction of these graphs uses quadratic residues.

**Cryptography**

The fact that finding a square root of a number modulo a large composite n is equivalent to factoring (which is widely believed be a hard problem) has been used for constructing cryptographic schemes such as the Rabin cryptosystem and the oblivious transfer. The quadratic residuosity problem is the basis for the Goldwasser-Micali cryptosystem.

The discrete logarithm is a similar problem that is also used in cryptography.

**Primality testing**

Euler's criterion is a formula for the Legendre symbol (a|p) where p is prime. If p is composite the formula may or may not compute (a|p) correctly. The Solovay-Strassen primality test for whether a given number n is prime or composite picks a random a and computes (a|n) using a modification of Euclid's algorithm,[34] and also using Euler's criterion.[35] If the results disagree, n is composite; if they agree, n may be composite or prime. For a composite n at least 1/2 the values of a in the range 2, 3, …, n − 1 will return "n is composite"; for prime n none will. If, after using many different values of a, n has not been proved composite it is called a "probable prime".

The Miller-Rabin primality test is based on the same principles. There is a deterministic version of it, but the proof that it works depends on the generalized Riemann hypothesis; the output from this test is "n is definitely composite" or "either n is prime or the GRH is false". If the second output ever occurs for a composite n, then the GRH would be false, which would have implications through many branches of mathematics.

**Integer factorization**

In § VI of the Disquisitiones Arithmeticae[36] Gauss discusses two factoring algorithms that use quadratic residues and the law of quadratic reciprocity.

Several modern factorization algorithms (including Dixon's algorithm, the continued fraction method, the quadratic sieve, and the number field sieve) generate small quadratic residues (modulo the number being factorized) in an attempt to find a congruence of squares which will yield a factorization. The number field sieve is the fastest general-purpose factorization algorithm known.

In addition, Quadratic residues can and has been used in applications to acoustics, like diffusion. I found this on wikipedia too.

Quadratic-Residue Diffusors

MLS based diffusors are superior to geometrical diffusors in many respects; they have limited bandwidth. The new goal was to find a new surface geometry that would combine the excellent diffusion characteristics of MLS designs with wider bandwidth. A new design was discovered, called a quadratic-residue diffusor. Today the quadratic residue diffuser or Schroeder diffuser is still widely used. **Quadratic-Residue Diffusors can be designed to diffuse sound in either one or two directions**. They too suffer from "flat plate" frequencies, but at a higher frequencies than MLS diffusers.[citation needed] Fractal constructions can be used to extend bandwidth.

Primitive-Root Diffusors

**Are based on a number theoretic sequence**. Although they produce a notch in the scattering response, in reality the notch is over too narrow a bandwidth to be useful. In terms of performance, they are very similar to Quadratic-Residue Diffusors.

So, they do come in handy for some people!