




Some Transcendence Results from a Harmless Irrationality Theorem 

PP: 9196 

doi:10.18576/jant/050201


Author(s) 

F. M. S. Lima,


Abstract 

The arithmetic nature of values of some functions such as sinz, cosz, sinhz, coshz, ez, and lnz, is a relevant topic in number theory. For instance, all those functions return transcendental values for nonzero algebraic values of z (z6= 1 in the case of lnz). On the other hand, not even an irrationality proof is known for some numbers like ln π , π +e and π e, though it is wellknown that at least one of the last two numbers is irrational. In this note, we ﬁrst generalize the last result, showing that at least one of the sum and product of any two transcendental numbers is transcendental. We then use this to show that, given any nonnull complex number z6= 1/e, at least two of the numbers lnz, z+e and ze are transcendental. It is also shown that coshz, sinhz and tanhz return transcendental values for all z = r lnt, r∈Q, r6= 0, t being any transcendental number. The analogue for common trigonometric functions is also proved. 




