Although he claimed to have a general proof of his conjecture, Fermat left no details of his proof, and no proof by him has ever been found. Attempts to prove it prompted substantial development in number theory, and over time Fermat's Last Theorem gained prominence as an unsolved problem in mathematics.

al. actually proved was far deeper and more mathematically interesting than its famous corollary, Fermat's last theorem, which demonstrates that in many cases the value of a mathematical problem is best measured by the depth and breadth of the tools that are developed to solve it.

Abstract. Fermat's Last Theorem (FLT), (1637), states that if n is an integer greater than 2, then it is impossible to find three natural numbers x, y and z where such equality is met being (x,y)>0 in xn+yn=zn.

Professor Who Solved Fermat's Last Theorem Wins Math's Abel Prize. Mathematics professor Andrew Wiles has won a prize for solving Fermat's Last Theorem. He's seen here with the problem written on a chalkboard in his Princeton, N.J., office, back in 1998.

The 7 Unsolved Mathematical Problems

- Poincaré Conjecture.
- Birch and Swinnerton-Dyer Conjecture.
- Hodge Conjecture.
- Navier–Stokes Existence and Smoothness.
- P Versus NP Problem.
- Riemann Hypothesis.
- Yang-Mills Existence and Mass Gap.

With its phenomenal size of 200 terabytes—the equivalent of all of the digital texts held by the Library of Congress—it is the longest mathematical proof ever produced. Three American-British computer scientists have just announced its creation using a supercalculator.

Fermat's last theorem, also called Fermat's great theorem, the statement that there are no natural numbers (1, 2, 3,) x, y, and z such that x^{n} + y^{n} = z^{n}, in which n is a natural number greater than 2.

A theorem is a statement that can be demonstrated to be true by accepted mathematical operations and arguments. In general, a theorem is an embodiment of some general principle that makes it part of a larger theory. The process of showing a theorem to be correct is called a proof.

Definition: Number theory is a branch of pure mathematics devoted to the study of the natural numbers and the integers. It is the study of the set of positive whole numbers which are usually called the set of natural numbers.

Fermat's little theorem states that if p is a prime number, then for any integer a, the number a ^{p} – a is an integer multiple of p. a^{p} ≡ a (mod p). Special Case: If a is not divisible by p, Fermat's little theorem is equivalent to the statement that a ^{p}^{-}^{1}-1 is an integer multiple of p.

Pierre de Fermat, (born Aug, Beaumont-de-Lomagne, France—died Janu, Castres), French mathematician who is often called the founder of the modern theory of numbers. Together with René Descartes, Fermat was one of the two leading mathematicians of the first half of the 17th century.

So to prove Fermat's last theorem, Wiles had to prove the Taniyama-Shimura conjecture. Proving the Taniyama-Shimura conjecture was an enormous task, one that many mathematicians considered impossible. Wiles decided that the only way he could prove it would be to work in secret at his Princeton home.

The mathematical methods of probability arose in the investigations first of Gerolamo Cardano in the 1560s (not published until 100 years later), and then in the correspondence Pierre de Fermat and Blaise Pascal (1654) on such questions as the fair division of the stake in an interrupted game of chance.

He is best known for proving Fermat's Last Theorem, for which he was awarded the 2016 Abel Prize and the 2017 Copley Medal by the Royal Society. He was appointed Knight Commander of the Order of the British Empire in 2000, and in 2018 was appointed as the first Regius Professor of Mathematics at Oxford.

So, here are top 10 most tough topics of mathematics that usually students struggle with:

- Algebra:
- Calculus:
- Geometry and topology:
- Combinatory:
- Logic:
- Number Theory:
- Dynamical Systems & Differential Equations:
- Mathematical Physics:

The Riemann hypothesis – an unsolved problem in pure mathematics, the solution of which would have major implications in number theory and encryption – is one of the seven $1 million Millennium Prize Problems. First proposed by Bernhard Riemann in 1859, the hypothesis relates to the distribution of prime numbers.

Math seems difficult because it takes time and energy. Many people don't experience sufficient time to "get" math lessons, and they fall behind as the teacher moves on. Many move on to study more complex concepts with a shaky foundation. We often end up with a weak structure that is doomed to collapse at some point.

Integers. The set of integers is represented by the letter Z. An integer is any number in the infinite set, Integers are sometimes split into 3 subsets, Z^{+}, Z^{-} and 0. Z^{+} is the set of all positive integers (1, 2, 3,), while Z^{-} is the set of all negative integers (..., -3, -2, -1).

Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom, theorem, and proof.

The typical order of math classes in high school is:

- Algebra 1.
- Geometry.
- Algebra 2/Trigonometry.
- Pre-Calculus.
- Calculus.

It also uses standard constructions of modern algebraic geometry, such as the category of schemes and Iwasawa theory, and other 20th-century techniques which were not available to Fermat. Together, the two papers which contain the proof are 129 pages long, and consumed over seven years of Wiles's research time.

Pythagorean theorem, the well-known geometric theorem that the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse (the side opposite the right angle)—or, in familiar algebraic notation, a^{2} + b^{2} = c^{2}.

Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states: Every even integer greater than 2 is the sum of two primes.

An axiom is a statement that is considered to be true, based on logic; however, it cannot be proven or demonstrated because it is simply considered as self-evident. A theorem, by definition, is a statement proven based on axioms, other theorems, and some set of logical connectives.

Here's the difference. A theorem is a mathematical statement that has been proven on the basis of previously established statements (often other theorems). A law in science is a demonstrated statement based on repeated observations.

A number is perfect if all of its factors, including 1 but excluding itself, perfectly add up to the number you began with. 6, for example, is perfect, because its factors — 3, 2, and 1 — all sum up to 6. 28 is perfect too: 14, 7, 4, 2, and 1 add up to 28.

The set of integers is often denoted by a boldface letter 'Z' ("Z") or blackboard bold. (Unicode U+2124 ℤ) standing for the German word Zahlen ([ˈtsaːlən], "numbers"). ℤ is a subset of the set of all rational numbers ℚ, which in turn is a subset of the real numbers ℝ. Like the natural numbers, ℤ is countably infinite.

Number theory used to be considered the purest of pure math. The best known application of number theory is public key cryptography, such as the RSA algorithm. Public key cryptography in turn enables many technologies we take for granted, such as the ability to make secure online transactions.

Fermat's little theorem is a fundamental theorem in elementary number theory, which helps compute powers of integers modulo prime numbers. It is a special case of Euler's theorem, and is important in applications of elementary number theory, including primality testing and public-key cryptography.

The generalization of Fermat's theorem is known as Euler's theorem. In general, Euler's theorem states that, “if p and q are relatively prime, then ”, where φ is Euler's totient function for integers. That is, is the number of non-negative numbers that are less than q and relatively prime to q.

Fermat's principle, also known as the principle of least time, is the link between ray optics and wave optics. In its original "strong" form, Fermat's principle states that the path taken by a ray between two given points is the path that can be traversed in the least time.