## Transcendental continued fractions.(English)Zbl 0531.10035

Let $$A=a_ 1+\frac{1}{a_ 2+}\frac{1}{a_ 3+}...$$, $$B=b_ 1+\frac{1}{b_ 2+}\frac{1}{b_ 3+}...$$. The author proves Theorem 1: Suppose that there exists a real number $$r>1$$ such that $$r^{-1}a_ n\geq b_ n\geq a^{n-1}_{n-1}$$ holds for $$n=1,2,3,...$$. Then A and B are algebraically independent. Next he proves a theorem from which he deduces the following Corollary: Let $$a_ n\geq a^ 2_{n-1}$$, $$b_ n\geq b^ 2_{n-1}$$ hold for all $$n\geq 3$$ and $$\log a_ n\quad \log b_ n=o(\log \min(a_{n+1},b_{n+1}))$$ and $$\log b_{n+1}=o(b_ n\quad \log b_ n)$$ as $$n\to \infty$$. Then $$A^ B$$ is transcendental. A striking example is this: Let C and $$a_ 1$$ be positive integers. Put $$a_{n+1}=C^{a_ n}$$ for $$n=1,2,...$$. Then $$A^ A$$ is transcendental.
Reviewer: K.Ramachandra

### MSC:

 11J81 Transcendence (general theory) 11J70 Continued fractions and generalizations 11J85 Algebraic independence; Gel’fond’s method

### Keywords:

transcendence; continued fractions; algebraic independence
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### References:

 [1] Baker, A., A sharpening of the bounds for linear forms in logarithms, Acta Arith., 21, 117-129 (1972) · Zbl 0244.10031 [2] Baker, A., The theory of linear forms in logarithms, (Baker, A.; Masser, D. W., Transcendence Theory: Advances and Applications (1977), Academic Press: Academic Press London), 1-27 · Zbl 0361.10028 [3] Bijlsma, A., On the simultaneous approximation of $$a, b$$ and $$a^b$$, Compositio Math., 35, 99-111 (1977) · Zbl 0355.10025 [4] Durand, A., Indépendance algébrique de nombres complexes et critére de transcendance, Compositio Math., 35, 259-267 (1977) · Zbl 0372.10022 [5] Leveque, W. J., (Topics in Number Theory, Vol. 2 (1956), Addison-Wesley: Addison-Wesley Reading, Mass) · Zbl 0070.03803 [6] Nettler, G., On transcendental numbers whose sum, difference, quotient and product are transcendental numbers, Math. Student, 41, 339-348 (1973) · Zbl 0336.10023 [7] Nettler, G., Transcendental continued fractions, J. Number Theory, 13, 456-462 (1981) · Zbl 0464.10023 [8] Perron, O., Die Lehre von den Kettenbrüchen (1929), Chelsea: Chelsea New York · JFM 55.0262.09 [9] Van der Poorten, A. J., On Baker’s inequality for linear forms in logarithms, (Math. Proc. Cambridge Philos. Soc., 80 (1976)), 223-248 · Zbl 0341.10030
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