Channel: singingbanana

Category: Education

Tags: problemfermat's last theoremmathematicsmathsexponentsmathrationalirrationalnumber theory

Description: Fermat's Last Theorem states the equation x^n + y^n = z^n has no integer solutions for positive integer exponents greater than 2. However, Fermat's Last Theorem says nothing about exponents that are not positive integers. Note: x, y and z are meant to be positive integers, which I should have said in the video. Whoops. This video introduces some results for rational and irrational exponents. For rational exponents, k/m, k must be equal to 1 or 2. If we allow complex roots, then we have strange solutions with x=y=z and m divisible by 6. For irrational exponents no general results exist but we know there are infinitely many integer solutions, in this video I give a couple of examples. Many of the examples in this video, as well as the proof for rational exponents, were taken from this paper by Frank Morgan (2010) maa.org/sites/default/files/pdf/cmj_ftp/CMJ/May%202010/3%20Articles/1%20Morgan/Morgan9_5_09.pdf Here is another description of the same proof, with a bit more detail math.leidenuniv.nl/~hwl/PUBLICATIONS/1992d/art.pdf Another proof for rational exponents is here, as well as the result with complex roots, by Bennett, Glass, Székely (2004) digitalcommons.lmu.edu/cgi/viewcontent.cgi?referer=&httpsredir=1&article=1103&context=math_fac The result for rationals seems to have been first proven by R. Oblath. Quelques proprietes arithmetiques des radicaux (Hungarian). In Comptes Rendus du Premier Congres des Mathematiciens Hongrois, 27 Aout–2 Septembre 1950, pages 445–450. Akad´emiai Kiado, Budapest, 1952.