Period of a complex exponential signal

[lahanadar]

I have a simple complex exponential signal of the form x(t)=e^jωt. To find period of the signal I tested if x(t)=x(t+nT) for all n:

e^jωt=e^jω(t+nT) ⇒ e^jωnT=1=e^j2πk​

where n and k are integers. Then I find a general period expression as

T=2πk/ωn​

Period T means it is the least time a signal repeat itself. As an example, pick k=1, n=1, then T₁=2π/ω. Now pick k=3, n=2, then T₂=3π/ω. These two periods seem valid for the same signal x(t) in contrary it shouldn't, since a signal should have only one period. Am I missing something here?

 

[mfb]

If x(t)=x(t+nT) for all n, then x(t)=x(t+nT)=x(t+n(2T)) for all n by using the same step twice. If T is a period, then 2T is a period as well, and 3T and so on. Usually, just the smallest value is called period of a signal.