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Saturday, October 13, 2018

Pierre de Fermat and number theory


Photo source: Wikimedia Commons


Pierre de Fermat (1607-1665) was a French mathematician best known for Fermat's Last Theorem and his contributions to number theory and calculus. Physicist Isaac Newton said,


"I had hints of this method [of fluxions] from Fermat's way of drawing tangents, and by applying it to abstract equations, directly and invertedly, I made it general." (Quoted in Isaac Newton by Louis Trenchard More)


Regarding Fermat's Last Theorem, Wikipedia says,


"In number theory, Fermat's Last Theorem... states that no three positive integers A, B, and C satisfy the equation A^N + B^N = C^N for any integer value of N greater than 2. The cases N=1 and N=2 have been known to have infinitely many solutions since antiquity." (Wikipedia: Fermat's Last Theorem, 8.11.21 UTC 03:24)


Mathematician Andrew Wiles wrote the proof to Fermat's Last Theorem in 1993, which was regarded as a major event in the history of mathematics. Wiles said,


"The best problem I ever found, I found in my local public library. I was just browsing through the section of math books and I found this one book, which was all about one particular problem - Fermat's Last Theorem." (Nova Interview, 2000)


The rest of this post is some quotes from Fermat.


Number theory


"There is scarcely any one who states purely arithmetical questions, scarcely any one who understands them. Is this not because arithmetic has been treated up to this time geometrically rather than arithmetically? ... If they succeed in discovering the proof or solution, they will acknowledge that questions of this kind are not inferior to the more celebrated ones from geometry either for depth or difficulty or method of proof..." (Letter to Frenicle de Bessy, 1657)


"It is impossible for any number which is a power greater than the second to be written as a sum of two like powers." (


"To divide a cube into two other cubes, a fourth power, or in general any power whatever into two powers of the same denomination above the second is impossible..." (


"...given any number which is not a square, there also exists an infinite number of squares such that when multiplied into the given number and unity is added to the product, the result is a square." (Letter to Frenicle de Bessy, 1657)