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Some Transcendence Results from a Harmless Irrationality Theorem |
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PP: 91-96 |
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doi:10.18576/jant/050201
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Author(s) |
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F. M. S. Lima,
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Abstract |
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| 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 non-zero 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 well-known 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 non-null 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. |
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