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Applied Mathematics

Computation of tangent, Euler, and Bernoulli numbers

Profile image of Thomas J BuckholtzThomas J Buckholtz

1967, Math. Comp

Abstract

Some elementary methods are described which may be used to calculate tangent numbers, Euler numbers, and Bernoulli numbers much more easily and rapidly on electronic computers than the traditional recurrence relations which have been used for over a century. These methods have been used to prepare an accompanying table which extends the existing tables of these numbers. Some theorems about the periodicity of the tangent numbers, which were suggested by the tables, are also proved.

Figures (13)
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 4. Periodicity of the Sequences. Examination of the tables produced by the com- puter program shows that the unit’s digits of the nonzero tangent numbers repeat endlessly in the pattern 2, 6, 2, 6, 2, 6, starting with 73; furthermore the two least significant digits ultimately form a repeating period of length 10: 16, 72, 36, 92, 56, 12, 76, 32, 96, 52, 16, 72, .... The three least significant digits have a period of length 50, and for four digits the period-length is 250. These empirical observa- tions suggest that theoretical investigation of period-length might prove fruitful. THEOREM 1. Let p be an odd prime, and let d be the pertod-length of the sequence
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 4. Periodicity of the Sequences. Examination of the tables produced by the com- puter program shows that the unit’s digits of the nonzero tangent numbers repeat endlessly in the pattern 2, 6, 2, 6, 2, 6, starting with 73; furthermore the two least significant digits ultimately form a repeating period of length 10: 16, 72, 36, 92, 56, 12, 76, 32, 96, 52, 16, 72, .... The three least significant digits have a period of length 50, and for four digits the period-length is 250. These empirical observa- tions suggest that theoretical investigation of period-length might prove fruitful. THEOREM 1. Let p be an odd prime, and let d be the pertod-length of the sequence
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

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References (8)

  1. Thomas Clausen, "Theorem," Asir. Nachrichten, v. 17, 1840, cols. 351-352.
  2. S. A. Joffe, "Calculation of the first thirty-two Eulerian numbers from central differences of zero," Quart. J. Math., v. 47, 1916, pp. 103-126.
  3. S. A. Joffe, "Calculation of eighteen more, fifty in all, Eulerian numbers from central differences of zero," Quart. J. Math., v. 48, 1917-1920, pp. 193-271.
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  5. Niels Nielsen, Traité Élémentaire des Nombres de Bernoulli, Paris, 1923.
  6. J. Peters & J. Stein, Zehnstellige Logarithmentafel, Berlin, 1922.
  7. S. Z. Serebrennikoff, "Tables des premiers quatre vingt dix nombres de Bernoulli," Mém. Acad. St. Petersbourg 8, v. 16, 1905, no. 10, pp. 1-8.
  8. K. G. C. von Staudt, "Beweis eines Lehrsatzes die Bernoullischen Zahlen betreffend," J.für Math., v. 21, 1840, pp. 372-374.

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