The Kempner number; also published as Journal of Integer Sequences 16 (2013), article 13.2.15. is the sum of the series
It is named after Aubrey Kempner, who proved it transcendental in 1916.On Transcendental Numbers, A. J. Kempner, Transactions of the American Mathematical Society, 17, #4 (1916), pp. 476-482, , . It is an example of a number easy to prove transcendental which is not a Liouville number. Properties By definition, the binary expansion of the Kempner number has zeroes everywhere except at places which are powers of two: κ = 0.110100010000000100000000000000010000000000000000000000000000000100... (base two.) Since the first proof of transcendence by Kempner, many other proofs have been given; see the references.Section 13.3, Automatic Sequences: Theory, Applications, Generalizations, Jean-Paul Allouche, Jeffrey Shallit, Cambridge University Press, 2003, ISBN 9780521823326, .Note on a theorem of Kempner concerning transcendental numbers, H. Blumberg, Bulletin of the American Mathematical Society 32 (1926), pp. 351–356, .Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen, K. Mahler, Mathematische Annalen 101 (1929), pp. 342–366, . Corrigendum, 103 (1930), p. 532, .Algebraic independence properties of the Fredholm series, J. H. Loxton and A. J. van der Poorten, Journal of the Australian Mathematical Society, Series A, 26, #1 (1978), pp. 31–45, .An "Oceans of zeros" proof that a certain non-Liouville number is transcendental, M. J. Knight, The American Mathematical Monthly, 98, #10 (December 1991), pp. 947–949, , .Theorem 1.1.2, Mahler Functions and Transcendence, Kumiko Nishioka, Berlin, Heidelberg: Springer-Verlag, 1996, ISBN 3-540-61472-9, . Volume 1631 of Lecture Notes in Mathematics. "The Beginnings of Transcendental Numbers", Michael Filaseta, lecture notes, Math 785, Transcendental Number Theory, Spring 2011, University of South Carolina. Accessed Jan. 22, 2026.
Jeffrey Shallit has proven that it has a simple continued fraction expansion, obtainable by the following construction:Simple continued fractions for some irrational numbers, Jeffrey Shallit, Journal of Number Theory, 11, #2 (May 1979), pp. 209-217, . Start with the partial expansion . If the partial expansion is , replace it by . If this generated a zero, replace by . Repeat steps 2 and 3 indefinitely. This generates the expansion
After the first partial quotients, the remainders are all 2, 4 or 6. Since this continued fraction has bounded partial quotients, the Kempner number has irrationality measure 2.
References