A necessary and sufficient condition for transcendency
Math. Comp. 29 (1975), 145-153
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Abstract: As has been known for many years (see, e.g., K. Mahler, J. Reine Angew. Math., v. 166, 1932, pp. 118-150), a real or complex number is transcendental if and only if the following condition is satisfied.
To every positive number there exists a positive integer n and an infinite sequence of distinct polynomials at most of degree n with integral coefficients, such that
In the present note I prove a simpler test which makes the transcendency of
depend on the approximation behaviour of a single sequence
of distinct polynomials of arbitrary degrees with integral coefficients.
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American Mathematical Society