Elementary number theory results that are not generalized by ring or group theory

I've taken an undergraduate course in ring and group theory, but haven't studied number theory formally. I've noticed that many important results in number theory have been generalized in group/ring theory (e.g. Lagrange's Theorem generalizing Fermat's Little Theorem).

My question is, are there any results in elementary number theory that have not been generalized using elementary group and ring theory?

Solution 1:

Problems for integers whose generalizations in algebraic number theory have not been developed (or at least considered) are rare. The only examples I know of are:

1. Ancient problems without a strong relation to modern algebraic techniques. This includes the existence of odd perfect numbers, and compositeness of Fermat numbers $2^{2^k}+1$. Keep in mind, though, that most of the very old problems have been understood or finished off by modern techniques, such as the congruent number problem and Fermat's Last Theorem being reduced to facts about elliptic curves, or the probabilistic approach to the distribution of special types of primes (twin, Mersenne, k-term progressions, etc), and these better-understood problems are more amenable to generalization beyond the integers.

2. Questions of combinatorial nature, such as covering congruences or arithmetic progressions (van der Waerden theorems). Here the problems can be generalized to other rings, but most interest has focused on the integer case.

3. Uses of number theory in logic and computer science, such as Goedel coding, complexity of Presburger arithmetic, or other problems where iterated exponentiation appears. Algebraic number theory is primarily about problems described by polynomial equations, and the reduction of exponential Diophantine equations to polynomial ones (as in the solution of Hilbert's 10th problem) is not direct enough to allow the use of the standard algebraic tools.

For everything else, where standard algebraic techniques or analytic ones (zeta functions, Diophantine approximation, sieves, transcendence theory, complex analysis, etc) are applicable to a problem over the integers, there has usually been an effort to search for analogues of those methods in algebraic number theory, p-adic number theory, geometry over finite fields, and complex geometry. This applies to all the major techniques and theories developed since 1800. So to find examples one needs to move away from "central core" number theory and look at questions that are not, so far, primarily studied by the methods of algebraic number theory and algebraic geometry.

Solution 2:

I contend that every result that could reasonably be called "elementary" number theory also holds for $\mathbb{F}_q[x]$, the ring of polynomials in one variable over a finite field. Quadratic reciprocity fits the bill here, for example. This is the subject of at least one book.