What is fermat's theorem: Definition and 16 Discussions

The works of the 17th-century mathematician Pierre de Fermat engendered many theorems. Fermat's theorem may refer to one of the following theorems:

Fermat's Last Theorem, about integer solutions to an + bn = cn
Fermat's little theorem, a property of prime numbers
Fermat's theorem on sums of two squares, about primes expressible as a sum of squares
Fermat's theorem (stationary points), about local maxima and minima of differentiable functions
Fermat's principle, about the path taken by a ray of light
Fermat polygonal number theorem, about expressing integers as a sum of polygonal numbers
Fermat's right triangle theorem, about squares not being expressible as the difference of two fourth powers

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  1. C

    Why Can We Take Limits of Both Sides? [Answered]

    For this, Does someone please know why we are allowed to take limits of both side [boxed in orange]? Also for the thing boxed in pink, could we not divide by -h if ##h > 0##? Many thanks!
  2. M

    Find the units digit of ## 3^{100} ## by the use of Fermat's theorem

    Consider modulo ## 10 ##. Then ## 10=5\cdot 2 ##. Applying the Fermat's theorem produces: ## 3^{4}\equiv 1\pmod {5} ##. This means ## (3^{4})^{25}=3^{100}\equiv 1\pmod {5} ##. Observe that ## 3\equiv 1\pmod {2}\implies 3^{100}\equiv 1\pmod {2} ##. Now we have ## 5\mid (3^{100}-1) ## and ##...
  3. F

    MHB Fermat's Theorem: Did Fermat Have a Proof?

    In your opinion did Fermat have a proof for his theorem?
  4. E

    MHB Inverse Theorem and Fermat's Theorem (Iso point)

    Hello, I've been using Caratheodory's Lemma to prove the Inverse Function Theorem and Fermat's Theorem. I have managed to prove both of them, I would just like someone to look over my proof and tell me if I'm missing anything (i.e. should I clarify any parts of my proof). So here goes: Inverse...
  5. I

    MHB Fermat's theorem (stationary points) of higher dimensions

    Look at this page and the Proof part, Fermat's theorem (stationary points) - Wikipedia, the free encyclopedia How to change the proof 2 into a proof of higher dimensions or can you give a proof of Fermat's theorem of higher dimensions?
  6. B

    Secant line in Fermat's Theorem

    I'm trying to understand something in Fermat's Theorem. I can't really phrase it in words, but I will write what my textbook says. Apparently if \lim_{x→c}\frac{f(x)-f(c)}{x-c} > 0 then there exists an open interval (a,b) containing c such that \frac{f(x)-f(c)}{x-c} > 0 for all...
  7. B

    Simple proof of Fermat's theorem?

    Can someone point out the error in the following "proof": Prove a^n + b^n =/ c^n for n>2, a,b,c>1 (=/ means not equal to) Let b=xa where x>1 and is from the set of real numbers generated by fractions, such that b is an integer so: a^n + (xa)^n =/ c^n Expanding a^n + x^n.a^n =/...
  8. E

    Fermat's theorem applied to multivariate functions

    Fermat's theorem provides that, if a function f(x) has a local max or min at a, and if f'(a) exists, then f'(a)=0. I was wondering whether a similar theory exists for a function f(x,y) or f(x,y,z) etc.
  9. K

    Proving Fermat's Theorem (1): a^(n-1) = a (mod n)

    (1). Prove the following statements. (a). When n = 2p, where p is an odd prime, then a^(n-1) = a (mod n) for any integer a. (b). For n = 195 = 3 * 5 *13, we have a^(n-2) = a (mod n) for any integer a If I am correct Fermat's Theorem comes into play. a)n=2p a^(2p-1) a^(2p)*(1/a)...
  10. dkotschessaa

    Is the work on Fermat's Theorem really done?

    I'm still relatively new to mathematics, in the sense of studying it with any degree of seriousness, so I have a question related to the general field of mathematics and a little bit on it's history. I haven't read Simon Singh's book yet,but a I understand the story on Fermat's Last Theorem...
  11. W

    Could someone explain to me Fermat's theorem?

    I DO NOT like this least time hocus pocus. I prefer the idea of causality. I just CANNOT stomach this idea. Here are my arguments (italicized text) almost verbatim from my notes against what I read (in bold). Someone please explain to me the whys and hows. Arguments against Fermat: "Given...
  12. E

    Fermat's Theorem: Proven by Euler and By Me?

    I'm talking about neither his "last theorem" nor his "little theorem", but another one. He suggested that x^2+2=y^3 can only have one solution (if we're dealing in natural numbers), which was (5,3). Euler reproved the theorem since, like so many others of his, the proof was lost. I can't...
  13. A

    Extension of Fermat's theorem?

    Hi! I assume you all know Fermat's last theorem. Well, has anyone considered the following extension to it? Assuming we're just using integers: We know that x1^n + x2^n = y^n has no solution for n > 2. However, what about this? For which values of k does x1^n+x2^n+...+xk^n = z^n...
  14. G

    Fermat's theorem question.

    Edit in new post. Or Bumping with the edited change.
  15. U

    Fermat's Theorem: A Math Problem and the Smart Boy Who Proved It Wrong

    from: http://www.math.utah.edu/~cherk/puzzles.html I am stumped, I noticed the pattern in the digits of the numbers, but I do not see how I can link that to the possibility of forming such a statement with those numbers when n is greater than 2.
  16. T

    Fermat's theorem disproved (not really)

    I read a book on Fermat's last theorem (a^n + b^n = c^n has no integer solutions for n > 2) last summer and I found this while trying to find the actual proof: http://home.mindspring.com/~jbshand/ferm.html [Broken]. It is a funny read if you have the time.