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For the smallest positive integer m such that n divides m!, see Kempner function.

The Kempner number[1] is the sum of the series

It is named after Aubrey Kempner, who proved it transcendental in 1916.[2] It is an example of a number easy to prove transcendental which is not a Liouville number.[1]: §1 

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.[1][3][4][5][6][7][8][9]

Jeffrey Shallit has proven that it has a simple continued fraction expansion, obtainable by the following construction:[10]: Theorem 1 

  1. Start with the partial expansion [0, 1, 3].
  2. If the partial expansion is [a, b, …, y, z], replace it by [a, b, …, y, z + 1, z − 1, y, …, b].
  3. If this generated a zero, replace […, a, 0, b, …] by […, a + b, …].
  4. Repeat steps 2 and 3 indefinitely.

This generates the expansion (sequence A007400 in the OEIS)

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

  1. ^ a b c Adamczewski, Boris (2013). “The many faces of the Kempner number”. arXiv:1303.1685 [math.NT].; also published as Journal of Integer Sequences 16 (2013), article 13.2.15.
  2. ^ On Transcendental Numbers, A. J. Kempner, Transactions of the American Mathematical Society, 17, #4 (1916), pp. 476-482, MR 1501054, doi:10.1090/s0002-9947-1916-1501054-4.
  3. ^ Section 13.3, Automatic Sequences: Theory, Applications, Generalizations, Jean-Paul Allouche, Jeffrey Shallit, Cambridge University Press, 2003, ISBN 9780521823326, doi:10.1017/CBO9780511546563.
  4. ^ Note on a theorem of Kempner concerning transcendental numbers, H. Blumberg, Bulletin of the American Mathematical Society 32 (1926), pp. 351–356, doi:10.1090/s0002-9904-1926-04222-1.
  5. ^ Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen, K. Mahler, Mathematische Annalen 101 (1929), pp. 342–366, doi:10.1007/BF01454845. Corrigendum, 103 (1930), p. 532, doi:10.1007/BF01455708.
  6. ^ 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, doi:10.1017/S1446788700011472.
  7. ^ 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, doi:10.2307/2324154, JSTOR 2324154.
  8. ^ Theorem 1.1.2, Mahler Functions and Transcendence, Kumiko Nishioka, Berlin, Heidelberg: Springer-Verlag, 1996, ISBN 3-540-61472-9, doi:10.1007/BFb0093672. Volume 1631 of Lecture Notes in Mathematics.
  9. ^ “The Beginnings of Transcendental Numbers”, Michael Filaseta, lecture notes, Math 785, Transcendental Number Theory, Spring 2011, University of South Carolina. Accessed Jan. 22, 2026.
  10. ^ Simple continued fractions for some irrational numbers, Jeffrey Shallit, Journal of Number Theory, 11, #2 (May 1979), pp. 209-217, doi:10.1016/0022-314X(79)90040-4.