Bertelsen's Number


The number of primes less than x is usually denoted by p(x). In Hardy & Wright's book "An Introduction to the Theory of Numbers", 5th edition, page 9, it says the vale of p(109) is 50,847,478. However, on page 179 of Paulo Ribenboim's book "The Book of Prime Number Records" it gives the value 50,847,534, which exceeds Hardy &Wright's count by 56. Since number theory is a somewhat empirical science, should we consider the possibility that the number of primes has increased since 1938 (when Hardy & Wright was first written)?


Looking for a tie-breaker, I remembered seeing a table of p(x) in "The Mathematical Experience" by Reuben and Hersh. Actually, the book mentions p(109) in two places. On page 175 it says p(109) is known to equal 50,847,478, in agreement with Hardy and Wright, whereas the table on page 213 lists p(109) as 50,847,534, in agreement with Ribenboim! Curious. At least two of these three books has an error in it. Perhaps Reuben and Hersh got their first value from Hardy and Wright, and then constructed their table from some other (more recent) source?


Out of curiosity I checked a few more books and found the same erroneous value of p (109) in each of the following:


An Introduction To the Theory of Numbers           Hardy/Wright       1938/1988

The Mathematical Experience                                   Davis/Hersh          1981/1981

Number Theory and Its History                                 Ore                          1948/1988

Elementär talteori                                                          Nagell                     1950/1950

Numbers and Infinity                                                   Sondheimer           1981/1981

Nature and Growth of Modern Mathematics           Kramer                   1970/1982

Introduction to Algorithms                                        Cormen, et al         1993/1993    


The error evidently goes back further then Hardy and Wright. In Oystein Ore's book "Number Theory and Its History" (1st Dover edition, 1988) the table on page 77 gives the old value p(109) = 50,847,478. Furthermore, it says the values of p(x) for x up to 107 were obtained by actual count from tables, whereas the values for x = 108 and 109 "are the values of p(x) calculated by Meissel and Bertelsen which we have mentioned previously."


On page 69 Ore discusses a formula for computing p(N) in terms of all the primes less than the square root of N. He goes on to say that Meissel (1870) improved the method and, through various shortcuts, succeeded in finding p(108) = 5,761,455.


These computations were continued by the Danish mathematician Bertelsen [Niels Peder Bertelsen (1869-1938)], who applied them for the determination of errors in tables. He announced the following result (1893):


                     p(109) = 50,847,478


which represents our most extended knowledge of the number of primes.


From this account one would infer that there was an error in Bertelsen's calculation back in 1893 (rather than just a typo), and that it has been propogated onwards, appearing in books printed nearly 100 years later. This strikes me as interesting, especially since Bertelsen applied his results "for the determination of errors in tables".


S. Sorrento then called my attention to the 1968 book "En bok om primtal" by Riesel, which gives the correct value, and remarks on the earlier incorrect value appearing in various texts. Then I heard about a paper entitled "Calculating p(x): The Meisell-Lehmer Method" by Lagarias, Miller, and Odlyzko, Math Comp, 1985, which reports that the correct value of p(109) is 50,847,534, but contrary to Ore's account it says the erroneous value 50,847,478 originated with Meisell in 1885, and was merely repeated by Bertelsen in 1893. Remarkably, the erroneous value (which I had dubbed "Bertelsen's number") was still being propagated in text books a century later, with at least seven citations in books published between 1938 and 1993.


Apparently there has been a long hazardous history of computing the number of primes. Often the greatest "known" value of p(x) has later turned out to be wrong. As another illustration of this, the value of p(1010) listed in the "modern" table in Reuben and Hersch is 455052512, which is the uppermost value they quote, exceeds by 1 the value in Ribenboim's Book of Prime Number Records (1988). For the record, based on Ribenboim's book, the number of primes of the first several orders of magnitude are listed (correctly I hope!) below:



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