7 added 94 characters in body

Here are a few examples. In some, number theory provided an essential motivation. In the others, it plays a more direct role.

1) Are there nonisometric Riemannian manifolds that are isospectral (eigenvalues of the Laplacian match, including multiplicities)? An example was given by Milnor in the 1960s, which depended on prior work of Witt involving theta-functions (modular forms) of lattices. In the 1980s, Sunada created examples systematically by exploiting the analogy with the number theorist's construction of pairs of nonisomorphic number fields that have the same zeta-function. These number field pairs are found with Galois theory (find a finite group $G$ admitting a pair of nonconjugate subgroups having appropriate properties and then find a Galois extension of the rationals with Galois group isomorphic to $G$). There is a well-known analogy between Galois theory and covering spaces, and Sunada used this to translate the group-theoretic conditions for Galois groups into the setting of Riemannian manifolds. For more on this story, see the Wikipedia page here, where you'll see that the nonisometric isospectral pairs found between the work of Milnor and Sunada were closely related to other parts of number theory (quaternion algebras over the rationals).

2) Lens spaces are distinguished from each other using quadratic residues.

3) Knot theory uses continued fractions. See one of the answers to the MO question here. (Some of the other answers to that question could also be regarded as more applications of number theory, to the extent that you consider finite continued fractions to be number theory.)

4) The construction of Ramanujan graphs uses number theory. Also look here.

5) Frobenius proved that the only ${\mathbf R}$-central division algebras that are finite-dimensional are ${\mathbf R}$ and the quaternions. If you want to see infinitely many other examples of noncommutative division rings that are finite-dimensional over their centers, especially if you want examples that are more than 4-dimensional, you probably should learn number theory since the simplest examples come from cyclic Galois extensions of the rationals. Verifying the examples really work requires knowing a rational number is not a norm from a particular number field, and that amounts to showing a certain Diophantine equation has no rational solutions.

6) The classical induction theorems of Artin and Brauer about representations of finite groups were motivated by the desire to prove Artin's conjecture on Artin $L$-functions. Although number theory appears in the proof in the context of algebraic integers, the main point I want to make is that a conjecture from number theory provided an essential motivation to imagine the theorems might be true in the first place.

7) Several concepts of general importance in mathematics were originally developed within number theory. The most prominent example is ideals, which were first defined by Dedekind in his work on algebraic number theory. The first examples of finite abelian groups were unit groups mod $m$ and class groups of quadratic forms. The first finitely generated abelian groups to be studied as such were unit groups in number fields (Dirichlet's unit theorem). The first application of the pigeonhole principle was in Dirichlet's proof of the solvability of Pell's equation. The motivation for Steinitz's 1910 paper setting out a general theory of fields was Hensel's creation of $p$-adic numbers.

6 added 700 characters in body

Here are a few examples. In some, number theory provided an essential motivation. In the others, it plays a more direct role.

1) Are there nonisometric Riemannian manifolds that are isospectral (eigenvalues of the Laplacian match, including multiplicities)? An example was given by Milnor in the 1960s. In the 1980s, Sunada created examples systematically by exploiting the analogy with the number theorist's construction of pairs of nonisomorphic number fields that have the same zeta-function. These number field pairs are found with Galois theory (find a finite group $G$ admitting a pair of nonconjugate subgroups having appropriate properties and then find a Galois extension of the rationals with Galois group isomorphic to $G$). There is a well-known analogy between Galois theory and covering spaces, and Sunada used this to translate the group-theoretic conditions for Galois groups into the setting of Riemannian manifolds. For more on this story, see the Wikipedia page here, where you'll see that the nonisometric isospectral pairs found between the work of Milnor and Sunada were closely related to other parts of number theory (quaternion algebras over the rationals).

2) Lens spaces are distinguished from each other using quadratic residues.

3) Knot theory uses continued fractions. See one of the answers to the MO question here. (Some of the other answers to that question could also be regarded as more applications of number theory, to the extent that you consider finite continued fractions to be number theory.)

4) The construction of Ramanujan graphs uses number theory. Also look here.

5) Frobenius proved that the only ${\mathbf R}$-central division algebras that are finite-dimensional are ${\mathbf R}$ and the quaternions. If you want to see infinitely many other examples of noncommutative division rings that are finite-dimensional over their centers, especially if you want examples that are more than 4-dimensional, you probably should learn number theory since the simplest examples come from cyclic Galois extensions of the rationals. Verifying the examples really work requires knowing a rational number is not a norm from a particular number field, and that amounts to showing a certain Diophantine equation has no rational solutions.

6) The classical induction theorems of Artin and Brauer about representations of finite groups were motivated by the desire to prove Artin's conjecture on Artin $L$-functions. Although number theory appears in the proof in the context of algebraic integers, the main point I want to make is that a conjecture from number theory provided an essential motivation to imagine the theorems might be true in the first place.

7) Several concepts of general importance in mathematics were originally developed within number theory. The most prominent example is ideals, which were first defined by Dedekind in his work on algebraic number theory. The first examples of finite abelian groups were unit groups mod $m$ and class groups of quadratic forms. The first finitely generated abelian groups to be studied as such were unit groups in number fields (Dirichlet's unit theorem). The first application of the pigeonhole principle was in Dirichlet's proof of the solvability of Pell's equation. The motivation for Steinitz's 1910 paper setting out a general theory of fields was Hensel's creation of $p$-adic numbers.

5 added 550 characters in body

Here are a few examples. In some, number theory provided an essential motivation. In the others, it plays a more direct role.

1) Are there nonisometric Riemannian manifolds that are isospectral (eigenvalues of the Laplacian match, including multiplicities)? An example was given by Milnor in the 1960s. In the 1980s, Sunada created examples systematically by exploiting the analogy with the number theorist's construction of pairs of nonisomorphic number fields that have the same zeta-function. These number field pairs are found with Galois theory (find a finite group $G$ admitting a pair of nonconjugate subgroups having appropriate properties and then find a Galois extension of the rationals with Galois group isomorphic to $G$). There is a well-known analogy between Galois theory and covering spaces, and Sunada used this to translate the group-theoretic conditions for Galois groups into the setting of Riemannian manifolds. For more on this story, see the Wikipedia page here, where you'll see that the nonisometric isospectral pairs found between the work of Milnor and Sunada were closely related to other parts of number theory (quaternion algebras over the rationals).

2) Lens spaces are distinguished from each other using quadratic residues.

3) Knot theory uses continued fractions. See one of the answers to the MO question here. (Some of the other answers to that question could also be regarded as more applications of number theory, to the extent that you consider finite continued fractions to be number theory.)

4) The construction of Ramanujan graphs uses number theory. Also look here.

5) Frobenius proved that the only ${\mathbf R}$-central division algebras that are finite-dimensional are ${\mathbf R}$ and the quaternions. If you want to see infinitely many other examples of noncommutative division rings that are finite-dimensional over their centers, especially if you want examples that are more than 4-dimensional, you probably should learn number theory since the simplest examples come from cyclic Galois extensions of the rationals. Verifying the examples really work requires knowing a rational number is not a norm from a particular number field, and that amounts to showing a certain Diophantine equation has no rational solutions.

6) The classical induction theorems of Artin and Brauer about representations of finite groups were motivated by the desire to prove Artin's conjecture on Artin $L$-functions. Although number theory appears in the proof in the context of algebraic integers, the main point I want to make is that a conjecture from number theory provided an essential motivation to imagine the theorem theorems might be true in the first place.(I honestly don't know how anyone would ever motivate these induction theorems without knowing about the Artin conjecture.)

4 added 100 characters in body
3 added 819 characters in body; added 2 characters in body; deleted 7 characters in body