Fermat's Last Theorem: statement, history, and the modern proof
A readable account of Fermat's Last Theorem: its simple statement, the centuries-long search for a proof, key ideas and contributors, and the role of modern number theory in its resolution.
Overview
Fermat's Last Theorem is a classical assertion in number theory that is simple to state but famously difficult to prove. In plain terms it says that for any integer exponent n > 2, the Diophantine equation x^n + y^n = z^n has no solutions in positive integers x, y, z. For n = 1 the equation is trivially solvable and for n = 2 it admits infinitely many solutions (the Pythagorean triples such as 3^2 + 4^2 = 5^2). The theorem received its modern name because it was the last of Pierre de Fermat's claims about integers to be resolved.
Statement and elementary context
The statement can be expressed succinctly: there do not exist nonzero integers x, y, z and an integer n > 2 such that x^n + y^n = z^n. Although the claim concerns whole numbers and looks accessible to non-specialists, attempts to prove it led to developments across algebra, arithmetic geometry and the theory of elliptic curves. Its elementary cases are well understood: n = 2 is the classical theorem of Pythagoras and admits parametric descriptions; small exponents such as n = 3 and n = 4 were settled by individual proofs centuries ago. However, a general proof for all n resisted all efforts until the late twentieth century.
Historical development
Pierre de Fermat recorded the claim in the margin of a copy of Diophantus's Arithmetica in the 17th century, adding that he had discovered "a truly marvelous proof" which the margin was too small to contain. No convincing evidence has ever been found that Fermat had a valid proof for the general case. Over the next three and a half centuries many prominent mathematicians attacked the problem. Some important advances included special-case proofs (for specific exponents), methods developed by Sophie Germain, and Ernst Kummer's 19th-century innovations in algebraic number theory, which introduced ideal theory and the notion of regular primes to handle many cases.
Road to the final proof
The modern resolution did not follow from elementary manipulations of integers but from profound connections between seemingly different areas of mathematics. In the 1980s and early 1990s a sequence of ideas connected the nonexistence of solutions to x^n + y^n = z^n (for n > 2) to properties of elliptic curves and modular forms. Gerhard Frey suggested that a hypothetical solution would give rise to an elliptic curve with unusual properties. Building on this, Jean-Pierre Serre and Ken Ribet formulated and proved results that linked such elliptic curves to a general conjecture (now known as the modularity or Taniyama–Shimura conjecture) that predicted which elliptic curves arise from modular forms. In 1994 Andrew Wiles, working largely in secret for several years, proved enough of this conjecture for a large class of elliptic curves (the semistable ones) to imply Fermat's Last Theorem. A gap in his first announcement was later corrected with Richard Taylor, and the resulting work is widely accepted as a complete proof.
Key ideas and collaborators
- Frey curve: the process of associating an elliptic curve to a putative integer solution, which would have special arithmetic properties.
- Ribet's theorem: a bridge showing that Frey's construction would contradict modularity if a solution existed.
- Modularity theorem: the deep statement about elliptic curves and modular forms whose special cases were proven by Wiles and Taylor.
Important figures in the chain of ideas include Euler (who proved n = 3), Sophie Germain, Kummer, Frey, Serre, Ribet, Mazur, Wiles and Taylor. Each contributed tools or partial results that made the final proof possible.
Significance and legacy
Fermat's Last Theorem is celebrated not only because a long-standing puzzle was finally resolved, but because its resolution helped consolidate and motivate large areas of modern number theory. The proof showcased how abstract structures — elliptic curves, modular forms, Galois representations — can address concrete questions about integers. It also illustrated the collaborative nature of mathematical progress: centuries of partial results, novel definitions and new techniques combined to produce a single decisive advance. Beyond technical consequences, the story popularized mathematics and demonstrated how deep links between different fields can produce breakthroughs.
Distinctions and related remarks
Fermat's Last Theorem should not be confused with other results bearing Fermat's name, such as Fermat's Little Theorem, which concerns remainders in modular arithmetic. Although the final proof is rooted in advanced theory, the theorem's plain formulation remains accessible, which has helped it retain a special place in both mathematical culture and public imagination.
Designations
There are different names for this theorem. The most common one in German is Großer Fermatscher Satz and derived from it Großer Fermat in contrast to Kleiner Fermatschen Satz or Kleiner Fermat. Since no proof has been handed down by Fermat himself, strictly speaking it was initially only a conjecture. Therefore the term Fermat's conjecture is also used, but even before the proof was spoken of Fermat's theorem. To include Wiles, the finder of the proof, there is also talk of the Fermat-Wiles theorem. In English, the theorem is called Fermat's Last Theorem, which is sometimes (inaccurately) translated in German as Fermat's letzter Satz or Fermat's letztes Theorem.
Source
Probably between 1637 and 1643, an exact year cannot be given due to the following circumstances, Fermat wrote the following lines as a marginal note in his hand copy of the Arithmetika of Diophantos of Alexandria next to the 8th problem of the second (Greek) "book":
"Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duas ejusdem nominis fas est dividere: cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet."
"It is not possible, however, to decompose a cube into 2 cubes, or a biquadrate into 2 biquadrates, and in general to decompose a power, higher than the second, into 2 powers with the same exponent: I have discovered a truly wonderful proof of this, but this margin is too narrow here to contain it."
Since Fermat's hand copy of the Arithmetika was found by his son in his father's estate only after his death and he did not date his marginal notes, an exact date cannot be determined. But it is plausible to assume that Fermat 
had solved at least the case 
and perhaps also the case before he was tempted to make his equally famous as "careless" remark. Therefore the year of origin is more likely 1641 than 1637.
That Fermat had found a proof for the special case which he perhaps believed to be able to generalize, is obvious, because this special case is an easy conclusion from a theorem explicitly proved by him: Area trianguli rectanguli in numeris non potest quadratus. (The area of a Pythagorean triangle cannot be a square number.), which he wrote including proof in the margin next to the 26th problem of the 6th (Greek) "book" of Arithmetika. André Weil also proved convincingly that Fermat possessed all means to prove also the case with his method.
The theories used in Wiles' proof in 1995 were not even rudimentarily developed over 350 years earlier. This does not exclude with certainty that one day a simpler proof will be found, using more elementary means. But that Fermat could have found one is doubted by most number theorists today. The surest sign that Fermat soon realized that he had not found a proof after all is that he did not mention the theorem and a proof of it to any of his correspondents. Fermat's marginal remark, moreover, was intended only for himself. He could not have counted on its publication by his son Samuel.
Questions and answers
Q: What is Fermat's Last Theorem?
A: Fermat's Last Theorem (FLT) states that if n is a whole number larger than 2, then the equation x^n + y^n = z^n has no solutions when x, y and z are natural numbers. In other words, it is impossible to express in whole numbers two cubes which added equal a third cube or anything higher than squares.
Q: When was FLT written?
A: Pierre de Fermat wrote about FLT in 1637 inside his copy of a book called Arithmetica.
Q: What did Fermat say about the theorem?
A: He said "I have a proof of this theorem, but there is not enough space in this margin".
Q: How long did it take for FLT to be proven?
A: It took 357 years for FLT to be proven correctly; it was finally done in 1995.
Q: Do mathematicians think that Fermat had an actual proof of the theorem?
A: Most mathematicians do not think that Fermat actually had a margin proof of this theorem.
Q: What does the original problem state?
A: The original problem states that it is impossible to divide cubum autem (a cube) into two cubes or quadratoquadratum (a square-square) into two square-squares and generally nothing beyond squares can be divided into two of its same name, with demonstration being remarkable yet too large for the margin size.
Related articles
Author
AlegsaOnline.com Fermat's Last Theorem: statement, history, and the modern proof Leandro Alegsa
URL: https://en.alegsaonline.com/art/34029
Sources
- math.stanford.edu : "From Fermat to Wiles: Fermat's Last Theorem becomes a theorem"
- doi.org : 10.1007/PL00000079