Euler's Sum of Powers Conjecture

Euler's Sum of Powers Conjecture proposed that for any positive integers, a sum of k-1 k-th power numbers could never equal a single k-th power number. For example, it suggested that the sum of four fourth powers could not equal a fifth power. This was an extension of Fermat’s Last Theorem. However, mathematicians eventually found counterexamples disproving the conjecture, showing that such equalities are possible in some cases. The conjecture was an important step in understanding the relationships between powers and sums, highlighting the complexity and surprises in number theory.