Euler's work on the solution to Diophantine equations marks the early history of algebraic number theory. A technique for finding integer solutions of equations of the form

was one particular problem considered by Euler.  After developing a method of solution, he specialized by setting a = 1, b = 2, and y =  1 to solve . Euler first factored this equation over the complex numbers:
                  ,
and then used the arithmetic properties of the set:

                                                     

to solve the equation.  (This was very bold of Euler since most mathematicians of that era were reluctant to use complex numbers in their research).  He made the following two assumptions:
(1) If  ,  ,    are in      with  ,   relatively prime (i.e. have no common factors) in   and  , then  ,   are cubes of numbers in  .

(2) If u and v are relatively prime integers, then u +  and u –  are relatively prime in    .

Under the above assumptions, Euler established that  and        for some integers a, b and therefore

so that  . Expanding the right-hand side of , and equating real and imaginary parts, yields  ; thus, it is easily deduced that b = 1, a = 1, z = 3, x = 5.

Using similar ideas, Euler also presented a proof of Fermat's Last Theorem for n = 3; that is, he claimed to have proved that there are no non-trivial integer solutions of  .  His proof involved numbers in

 but there were problems with his approach since there were several gaps and errors in this reasoning.  In fact, in this particular area of number theory, Euler did not justify many of his assumptions, and stated many claims without proof.  A deeper knowledge of factorization in algebraic number fields is required for a more complete treatment. However, this very subject is still of great research interest today.