From the point of view of the logical development of number theory as a mathematical theory, the principle of induction is used to prove essentially every nontrivial elementary statement in number theory. Basically, you can hardly get started without induction.

For example, if one defines the natural numbers in Peano's manner, as the unique inductive structure $\langle \mathbb{N},S,0\rangle$, where $S$ is the injective successor function and so on, then one typically defines addition as repeated application of the successor: $x+0=x$, $x+S(y)=S(x+y)$, and multiplication as repeated addition: $x\cdot 0=0$, $x\cdot S(y)=x\cdot y+x$. In this development, one uses induction to prove that:

- addition is commutative
- multiplication is commutive
- distributivity of multiplication over addition
- every fraction can be placed in lowest terms
- Euclidean algorithm
- $\sqrt{2}$ is irrational
- every number is a unique product of primes

And on and on with all the basic facts.

If instead of the second-order Peano characterization of the natural numbers, one works rather in the first-order theory PA, then the induction axiom is the central axiom, used in essentially every nontrival argument, since the theory stated without any induction axiom is extremely weak and is not able to prove much

Meanwhile, the logicians study the theories that arise when one restricts the induction principle. For example, PA proves the consistency of $\Sigma_n$-induction, for any particular finite $n$, and one can use this to show that PA is not finitely axiomatizable, if consistent, for then it would prove its own consistency, contrary to the incompleteness theorem.