Meaning: The quote you are referring to is known as Fermat's Last Theorem, and it has fascinated mathematicians for centuries. Pierre de Fermat, a French mathematician and lawyer, wrote this statement in the margin of his copy of the ancient Greek text Arithmetica by Diophantus around 1637. The theorem states that there are no three positive integers a, b, and c that satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2.

Fermat's Last Theorem is one of the most famous problems in the history of mathematics and remained unsolved for over 350 years. Many mathematicians attempted to prove the theorem, but it resisted all efforts until 1994 when Andrew Wiles, a British mathematician, finally presented a successful proof. Wiles' proof was a monumental achievement that required seven years of intensive work and drew upon advanced mathematical concepts and techniques developed in the 20th century.

The significance of Fermat's Last Theorem extends beyond its mathematical implications. The theorem's enduring mystery and the numerous failed attempts to prove it have captured the imagination of both mathematicians and the general public. The quest to solve the theorem has had a profound impact on the development of number theory and mathematical research, leading to the creation of new mathematical tools and approaches.

Fermat's Last Theorem belongs to a branch of mathematics known as number theory, which deals with the properties and relationships of numbers, particularly integers. The theorem itself is a statement about the behavior of whole numbers when raised to higher powers and the impossibility of finding certain types of solutions to specific equations.

The concept of "like powers" in the quote refers to the exponentiation of numbers, where a number is raised to a certain power. For example, in the equation a^n + b^n = c^n, the terms a^n, b^n, and c^n represent numbers raised to the nth power. Fermat's assertion that it is impossible for any number that is a power greater than the second to be written as a sum of two like powers is a bold claim about the nature of integer solutions to this specific type of equation.

Fermat's Last Theorem has connections to other areas of mathematics, such as algebraic geometry, modular forms, and elliptic curves, which were instrumental in Andrew Wiles' eventual proof. The theorem also inspired the development of new mathematical theories and tools, including the Taniyama-Shimura-Weil conjecture, which played a crucial role in Wiles' proof.

In conclusion, Fermat's Last Theorem is a historic mathematical problem that has captivated mathematicians for centuries. Its resolution by Andrew Wiles in 1994 marked a major milestone in the history of mathematics and had far-reaching implications for the field of number theory. The theorem's impact extends beyond its mathematical significance, as it has inspired new avenues of research and contributed to the development of advanced mathematical concepts and techniques. Fermat's Last Theorem stands as a testament to the enduring allure of mathematical puzzles and the power of human creativity and perseverance in solving them.