- Fermat is considered to be one of the 'fathers' of analytic geometry. (Along with Rene' Descartes.)
- Fermat along with Blaise Pascal is also considered to be one of the founders of probability theory.
- Fermat also made contributions in the field of optics and provided a law on light travel
- Fermat's most important work was done in the development of modern number theory which was one of his favorite areas in math. He is best remembered for his number theory, and for Fermat's Last Theorem.
- Fermat’s Last Theorem: (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most famous theorems in the history of mathematics.
- Fermat's spiral: (also known as a parabolic spiral) follows the equation r=+0^1/2 in polar coordinates(the more general Fermat's spiral follows r 2 = a 2θ.) It is a type of Archimedean spiral. The mesh of spirals occurs in Fibonacci numbers because divergence (angle of succession in a single spiral arrangement) approaches the golden ratio. The shape of the spirals depends on the growth of the elements generated sequentially. In mature-disc phyllotaxis, when all the elements are the same size, the shape of the spirals is that of Fermat spirals. That is because Fermat's spiral traverses equal annuli in equal turns. The full model proposed by H Vogel in 1979 is r=c square root of n and 0=n X 137.50 where θ is the angle, r is the radius or distance from the center, and n is the index number of the floret and c is a constant scaling factor. The angle 137.508° is the golden angle which is approximated by ratios of Fibonacci numbers. Fermat's spiral has also been found to be an efficient layout for the mirrors of concentrated solar power plants.