Fermat numbers are coprime


Let Fm and Fn two Fermat numbers, and assume m<n. Let d a positivePlanetmathPlanetmath common divisorMathworldPlanetmathPlanetmath of Fn and Fm, that is


If dFm then dF1F2Fn-1 since some factor must be Fm itself. But Fn-F1F2Fn-1=2 and so d2. Since d is odd, we must have d=1.

Therefore, the greatest common divisorMathworldPlanetmath of any two Fermat numbers must be 1.


Title Fermat numbers are coprime
Canonical name FermatNumbersAreCoprime
Date of creation 2013-03-22 14:51:24
Last modified on 2013-03-22 14:51:24
Owner yark (2760)
Last modified by yark (2760)
Numerical id 5
Author yark (2760)
Entry type Theorem
Classification msc 11A51