Meaning: The quote you provided is often referred to as Fermat's Last Theorem, and it has a rich and intriguing history in the field of mathematics. Pierre de Fermat, a French lawyer and amateur mathematician, is best known for his work in number theory and is considered one of the most influential mathematicians of the 17th century. The quote reflects Fermat's claim that it is impossible to divide a cube into two cubes, a fourth power into fourth powers, or generally any power beyond the square into like powers. He also mentioned that he had found a remarkable demonstration but did not provide any details.

Fermat's Last Theorem is one of the most famous problems in the history of mathematics and remained unsolved for over 350 years. The theorem states that there are no three positive integers a, b, and c that satisfy the equation an + bn = cn for any integer value of n greater than 2. In other words, there are no solutions to the equation x^n + y^n = z^n when n is an integer greater than 2.

Fermat's claim was famously jotted down in the margin of his copy of the ancient Greek text "Arithmetica" by Diophantus. The note, which became known as Fermat's Last Theorem, sparked intense interest and debate among mathematicians for centuries. Fermat's statement was not accompanied by a proof, and he provided no details of his remarkable demonstration, leaving mathematicians puzzled and intrigued for generations to come.

The theorem gained widespread attention and became a focal point for many mathematicians who attempted to prove or disprove Fermat's claim. Countless mathematicians over the centuries tried to solve the problem, but it remained elusive, earning the title of "Fermat's Last Theorem" due to its resistance to solution.

The first significant progress toward proving Fermat's Last Theorem came in the 19th century when mathematicians such as Ernst Kummer and Peter Gustav Lejeune Dirichlet made important contributions to the field of number theory. Kummer's work on ideal numbers and the theory of regular primes paved the way for future advancements in algebraic number theory, which would become crucial in the eventual proof of Fermat's Last Theorem.

In 1994, Andrew Wiles, a British mathematician, presented a proof of Fermat's Last Theorem after working on it in secret for seven years. Wiles' proof was a monumental achievement and involved advanced concepts from algebraic geometry and modular forms. His proof involved connecting elliptic curves with modular forms, a groundbreaking approach that ultimately led to the resolution of Fermat's Last Theorem.

Wiles' proof was met with widespread acclaim and marked a historic moment in the field of mathematics, as it finally put an end to the centuries-old conjecture. The proof brought closure to a problem that had puzzled and inspired mathematicians for generations, and it solidified Andrew Wiles' place as one of the most prominent figures in modern mathematics.

In conclusion, Fermat's Last Theorem, as expressed in the quote you provided, reflects a profound and enigmatic claim made by Pierre de Fermat in the 17th century. The theorem remained unsolved for over three centuries and captivated the minds of mathematicians around the world. Andrew Wiles' eventual proof of the theorem in 1994 marked a monumental achievement in the history of mathematics and brought closure to a problem that had long eluded resolution. Fermat's Last Theorem stands as a testament to the enduring pursuit of knowledge and the remarkable discoveries that can arise from the depths of mathematical inquiry.