Bertelsen's Number

Bertelsen's number is an erroneous name erroneously given to the erroneous value of , where  is the prime counting function. This value is 56 lower than the correct value of . Ore (1988, p. 69) states that the erroneous value 478 originated in Bertelsen's application of Meissel's method in 1893 (MathPages; Prime Curios!). However, the incorrect value actually first appears in Meissel (1885) rather than Bertelsen in 1893, as correctly noted by Lagarias et al. 1985. (Note that MathPages incorrectly states that Lagarias et al. attribute the result to Bertelsen.)

Unfortunately, the incorrect value has continued to be propagated in modern works such as Hardy and Wright (1979, p. 9), Davis and Hersch (1981, p. 175; but actually given correctly in the table on p. 213), Sondheimer (1981), Kramer (1983), Ore (1988, p. 77), and Cormen et al. (1990).

REFERENCES:

Cormen, T. H.; Leiserson, C. E.; and Rivest, R. L. Introduction to Algorithms. Cambridge, MA: MIT Press, 1990.

Davis, P. J. and Hersch, R. The Mathematical Experience. Boston, MA: Birkhäuser, 1981.

Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.

Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, p. 171, 2003.

Kramer, E. E. Nature and Growth of Modern Mathematics, Vol. 1. Princeton, NJ: Princeton University Press, 1983.

Lagarias, J. C.; Miller, V. S. and Odlyzko, A. M. "Computing : The Meissel-Lehmer Method." Math. Comput. 44, 537-560, 1985.

MathPages. "Bertelsen's Number." https://www.mathpages.com/home/kmath049.htm.

Meissel, E. D. F. "Berechnung der Menge von Primzahlen, welche innerhalb der ersten Milliarde naturlicher Zahlen vorkommen." Math. Ann. 25, 251-257, 1885.

Ore, Ø. Number Theory and Its History. New York: Dover, 1988.

Prime Curios! "50847478." https://primes.utm.edu/curios/page.php/50847478.html. Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, p. 236, 1996.

Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, p. 11, 1994.

Sondheimer, E. Numbers and Infinity : A Historical Account of Mathematical Concepts. Cambridge, England: Cambridge University Press, 1981.