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Denis Roegel
To cite this version:
Denis Roegel. A reconstruction of Schenmark’s table of factors (ca. 1780). [Research Report] 2011.
�hal-00654449�
Schenmark’s table of factors
(ca. 1780)
Denis Roegel
12 November 2011
This document is part of the LOCOMAT project:
http://locomat.loria.fr
Lund university [31, 2, 54]. Schenmark answered Euler’s call [14, pp. 177–179] for a table
of factors to a million. Euler’s call came shortly after Lambert’s call and at that time
there were still no published tables giving the factors up to a million.1
In his table, Lambert wrote that “it would in fact be desirable if we could obtain the
factors of a number from 1 to 1 000 000 and even beyond, merely by opening a table.” 2
Lambert possibly influenced Euler for his call.
Schenmark’s table must have been completed between 1775 and 1780, since Euler
published his article in 1775, and Heurlin mentions Schenmark’s work in 1780 [25, 1].
Schenmark was aided by five young mathematicians and students: Samuel Heurlin
(1744–1835)3
, Pehr Tegman (1757–1811)4
, Anders Lidtgren (1729–1815)5
, Johan
Cron-holm (1750–1809),6
and Enewald Widebeck (1752–1836)7
[31, p. 23]. The work was
completed in two years. Heurlin, one of the computers, wrote a description of Euler’s
method in 1780 [25].
There were three copies8
of Schenmark’s manuscript [49], [31, p. 23], [26, p. 76]. The
original copy is certainly the one in Stockholm. A copy of that manuscript was sent on
17 July 1782 by the Stockholm Academy to the French Academy and this copy is kept
at the library of the Institut in Paris. This copy was later used by Burckhardt as a basis
for his own table of factors [9], as indicated by a note on the first page of Schenmark’s
manuscript:
Une des épreuves de ma table des diviseurs a été comparée à ce travail de
Mr Schenmark, et j’ai marqué en marge de ce manuscrit, les fautes, que cette
comparaison m’a fait découvrir. Burckhardt.9
1Lambert’s table gave the smallest factor up to 102000. For an overview of the development of factor
tables, see Bullynck’s recent survey [5] and our other reconstructions.
2
Es wäre in der That erwünscht, wenn wir von 1 bis auf 1 000 000 und noch weiter die Theiler der
Zahlen durch blosses Aufschlagen einer Tafel haben könnten [27, p. 9].
3Heurlin was then lecturer in mathematics and physics in Lund, and later became professor and pastor
in Åseda in Småland [55, p. 368].
4Tegman was then a student and became professor of mathematics in 1787 in Lund [55, p. 182].
5Lidtgren was professor of mathematics and later became the first director of the astronomical
obser-vatory in Lund [55, p. 423].
6His dissertation Dissertatio gradualis de solidis Archimedæis was published in 1772. He later became
pastor [55, p. 368].
7Widebeck later became pastor in Jäder [55, pp. 426–427].
8We have consulted the copy at the library of the Institut in Paris (Ms. 922) and we obtained partial
copies of the manuscript located at the Center for history of science, The Royal Swedish Academy of
Sciences in Stockholm (MS Schenmark 14). It is a pleasure to thank the library of the Institut for making
it possible to examine their copy. At Stockholm, we received the help of Anne Miche de Malleray. The
copy at St. Petersburg has not been located, and we were in touch with Nikolai Nikolaev and Aleksej
Saveliev at the Scientific library of the State university of St. Petersburg, with Elena Gruzdeva at
the Russian Academy of Sciences in St. Petersburg who did not find a trace of the manuscript in the
inventory of the Euler and Lexell archives, and with I. M. Belaiev, also at the Russian Academy of
Sciences in St. Petersburg. We would also like to thank Ekaterina Lebedeva for her help in our Russian
correspondence.
9The errors found by Burckhardt are corrected with the pencil. For instance, on the last page of table,
1781 and Euler died there in 1783.
There had been plans to print Schenmark’s table by the Academy of sciences at
St. Petersburg, but Euler and Nicolas Fuss (Euler’s mathematical assistant) hesitated to
do so, as Bernoulli informed Fuss that Hindenburg had promised a table to 2 millions for
Easter 1782 [3, p. 31]. This table, however, never came to light.11
2
The structure of Schenmark’s manuscript
Schenmark’s table contains a preface (two pages, signed Nicolaus Schenmark), followed
by an introduction (24 pages), followed by three auxiliary tables. Next comes the main
table, covering 336 pages, and reproduced here.12
The main table is followed by two pages containing a list of the prime numbers from
1 000 003 to 1 007 977.
Finally, a (printed) copy of Heurlin’s article [25] on Euler’s method is inserted at the
end of the Institut copy.
2.1
Schenmark’s main table
Schenmark’s main table gives the smallest factor of all numbers N = 30q + r, with
q ≤ 33599 and r = 1, 7, 11, 13, 17, 19, 23, or 29. If a number is divisible by 2, 3, or 5, it
can of course not be found in the table, but the divisibility is almost immediate.
The greatest N is 1 007 999 and if N is not a prime number, its smallest factor has
at most three digits. The first number whose smallest factor has four digits is 10092 =
1 018 081.
Each page covers a range of 3000 integers and a range of 100 values of q. The first
page covers q = 0 to 99, the second page covers q = 100 to 199, and so on, and the last
page covers q = 33500 to 33599.
In his introduction, Schenmark gives the following example: find the smallest factor
of 214097 = 7136 × 30 + 17. One then checks the page headed 7100, line 36, column 17,
and finds 13, which is the answer sought.
As noted above by Burckhardt, Schenmark’s table contained errors which have been
corrected by Burckhardt.
for 1 007 579 = 33585 × 30 + 29, Schenmark had a prime, but this number is divisible by 653. A few lines
later, for 1 007 863 = 33595 × 30 + 13, Schenmark gave the divisor 379, which is incorrect.
10In Lambert’s correspondence, Johann Bernoulli writes that Lexell took it with him during his last trip
to St. Petersburg and presented it to the Academy of sciences [4, p. 140], [3]. Glaisher, who was unaware
of the existence of three manuscripts, erroneously interpreted this as meaning that the manuscript in
Paris was the one taken by Lexell to St. Petersburg, and that it must have arrived in Paris between 1785
and 1811 [23, p. 127].
11For the details of the correspondence between Lambert and Hindenburg about the tables of factors,
see our analysis of Felkel’s table [39], as well as Bullynck’s essay [5, pp. 191–192].
12The main table of the Institut manuscript contains two more pages which have been striked out.
The two pages both bear the heading 21200.
33300-33399 which is the last interval to reach one million. This corresponds to the
numbers 33300 × 30 + 1 = 999001 to 33399 × 30 + 29 = 1 001 999. Schenmark went
slightly beyond this limit.
Moreover, since his table went up to 1 002 000, Euler gave the list of all primes from
1 000 003 to 1 001 989 [14, p. 182–183]. This list was extended by Schenmark in accordance
with his extension of the main table to 1 008 000.
The main problem with Schenmark’s table is that the numbers N do not readily
appear in the table, and first need to be put in the form 30q + r. Almost all later tables
made it much easier to locate a given value in the table, although some tables then made
it more difficult to extract the sought factors.
2.2
The auxiliary tables
There are three auxiliary tables. The first table covers three pages. This table was
probably derived from the table given by Euler [14, p. 171–177]. For all primes from 7 to
1009, Euler gave the smallest q having a certain residue. For instance, for prime 271 and
residue 1, Euler gives the value 2448 and 2448 × 30 + 1 has 271 as its smallest factor. For
the same prime and residue 7, Euler gives 2502 and 2502 × 30 + 7 has 271 as its smallest
factor, and so on.
For prime p, Schenmark’s table does not give N , but N mod p. So, for 271, Schenmark
gives 2448 mod 271 = 9, 2502 mod 271 = 63, etc. Finally, Schenmark’s table gives the
first occurences of all factors 7 to 997. The first occurence of prime p is for the number
p2 = 30λ + r. This table gives the values of λ and r for each p. For instance, for p = 271,
we have λ = 2448 and r = 1. It can be shown that there are only two possible values for
r, namely 1 and 19. Schenmark gives these values in Roman numerals.
The second auxiliary table also covers three pages and is basically a multiplication
table for the odd integers from 1 to 99 not divisible by 5.
The third auxiliary table covers eight pages. It gives the positions of all smallest
factors 7 to 29 over a period which is multiple of 100. For instance, one of the subtable
for the factor 7 gives the locations of the factor 7 in the first column as 206, 213, 220,
227, . . . , 297, then in the second column as 203, 210, 217, . . . , 294, and so on. These
positions can be used as a sieve to fill a large part of the table.
3
The legacy of Schenmark’s table
Schenmark’s table was never printed, but other tables of more restricted scope appeared:
Felkel’s table to 408000 was published after Lambert’s call but before Schenmark’s
com-putation [15, 16, 17, 18]. Felkel’s table is interesting because its structure is the same
as that suggested by Euler, although this was only a coincidence. Vega’s 1782 and 1797
13In addition, we should remark that the periodicity of 30, resulting from the elimination of multiples
of 2, 3, and 5, was also used by Schaffgotsch in his article on the extension of Brancker’s table [53].
After that, Burckhardt, Crelle, Dase, Glaisher and others filled the next millions [8,
11, 12, 13, 19, 20, 21].
Schenmark’s table is not the only table of factors only in manuscript form, and we
hope that the still extant ones will someday be digitized or reconstructed for a better
comparison between all these endeavours.
Not all items of this list are mentioned in the text, and the sources which have not been
seen are marked so. We have added notes about the contents of the articles in certain
cases.
[1] Anonymous. Heurlin, Samuel. In Biographiskt Lexicon öfver namnkunnige Svenska
Män, volume 6, pages 133–138. Upsala: Wahlström & Låstbom, 1840.
[2] Anonymous. Schenmark, Nils. In Biographiskt Lexicon öfver namnkunnige Svenska
Män, volume 14, pages 45–47. Upsala: Wahlström & C., 1847.
[3] Extrait de la correspondance de M. Bernoulli. Nouveaux mémoires de l’Académie
royale des sciences et belles-lettres, année 1781, pages 31–35, 1783.
[4] Johann Bernoulli, editor. Johann Heinrich Lamberts deutscher gelehrter
Briefwechsel, volume 5. Berlin: Franz de la Garde, 1785–1787. [see pp. 140 and 166 for
Schenmark’s table]
[5] Maarten Bullynck. Factor tables 1657–1817, with notes on the birth of number
theory. Revue d’histoire des mathématiques, 16(2):133–216, 2010.
[6] Johann Karl Burckhardt. Table des diviseurs pour tous les nombres du deuxième
million, etc. Paris: Vve Courcier, 1814. [reconstructed in [33]]
[7] Johann Karl Burckhardt. Table des diviseurs pour tous les nombres du troisième
million, etc. Paris: Vve Courcier, 1816. [reconstructed in [34]]
[8] Johann Karl Burckhardt. Table des diviseurs pour tous les nombres des 1er, 2e et 3e
million, etc. Paris: Vve Courcier, 1817.
[9] Johann Karl Burckhardt. Table des diviseurs pour tous les nombres du premier
million, etc. Paris: Vve Courcier, 1817. [reconstructed in [32]]
[10] Ladislaus Chernac. Cribrum arithmeticum sive, tabula continens numeros primos,
a compositis segregatos, occurrentes in serie numerorum ab unitate progredientium,
usque ad decies centena millia, et ultra haec, ad viginti millia (1020000). Numeris
compositis, per 2, 3, 5 non dividuis, adscripti sunt divisores simplices, non minimi
tantum, sed omnino omnes. Deventer: J. H. de Lange, 1811. [reconstructed in [35]]
14Note on the titles of the works: Original titles come with many idiosyncrasies and features (line
splitting, size, fonts, etc.) which can often not be reproduced in a list of references. It has therefore
seemed pointless to capitalize works according to conventions which not only have no relation with the
original work, but also do not restore the title entirely. In the following list of references, most title
words (except in German) will therefore be left uncapitalized. The names of the authors have also been
homogenized and initials expanded, as much as possible.
The reader should keep in mind that this list is not meant as a facsimile of the original works. The
original style information could no doubt have been added as a note, but we have not done it here.
[12] Johann Martin Zacharias Dase. Factoren-tafeln für alle Zahlen der achten Million
etc. Hamburg: Perthes-Besser & Mauke, 1863. [reconstructed in [36]]
[13] Johann Martin Zacharias Dase and H. Rosenberg. Factoren-tafeln für alle Zahlen
der neunten Million etc. Hamburg: Perthes-Besser & Mauke, 1865. [reconstructed
in [37]]
[14] Leonhard Euler. De tabula numerorum primorum usque ad millionem et ultra
continuanda; in qua simul omnium numerorum non primorum minimi divisores
exprimantur. Novi Commentarii academiae scientiarum Petropolitanae,
19:132–183, 1775.
[15] Anton Felkel. Tafel aller einfachen Factoren der durch 2, 3, 5 nicht theilbaren
Zahlen von 1 bis 10 000 000. I. Theil. Enthaltend die Factoren von 1 bis 144000.
Wien: von Ehelenschen, 1776. [There is also a Latin edition [16] of this first part.]
[reconstructed in [39]]
[16] Anton Felkel. Tabula omnium factorum simplicum numerorum per 2, 3, 5 non
divisibilium, ab 1 usque 10 000 000. Pars I. Exhibens factores ab 1 usque 144000.
Wien: A. Gheleniana, 1777. [Latin version of [15].] [not seen] [reconstructed in [39]]
[17] Anton Felkel. Tabula factorum. Pars II. Exhibens factores numerorum ab 144001
usque 336000. Wien: A. Gheleniana, 1777? [reconstructed in [39]]
[18] Anton Felkel. Tabula factorum. Pars III. Exhibens factores numerorum ab 336001
usque 408000. Wien: A. Gheleniana, 1777? [reconstructed in [39]]
[19] James Glaisher. Factor table for the fourth million etc. London: Taylor and
Francis, 1879. [reconstructed in [41]]
[20] James Glaisher. Factor table for the fifth million etc. London: Taylor and Francis,
1880. [reconstructed in [40]]
[21] James Glaisher. Factor table for the sixth million etc. London: Taylor and Francis,
1883. [reconstructed in [42]]
[22] James Whitbread Lee Glaisher. Report of the committee on mathematical tables.
London: Taylor and Francis, 1873. [Also published as part of the “Report of the forty-third
meeting of the British Association for the advancement of science,” London: John Murray, 1874.
A review by R. Radau was published in the Bulletin des sciences mathématiques et
astronomiques, volume 11, 1876, pp. 7–27]
[23] James Whitbread Lee Glaisher. On factor tables, with an account of the mode of
formation of the factor table for the fourth million. Proceedings of the Cambridge
Philosophical Society, 3(4):99–138, 1878. [p. 127 on Schenmark’s table]
[25] Samuel Heurlin and Frederik Schlüter. De methodo Euleriana numeros primos et
compositorum factores in tabulas redigendi. Lund, 1780.
[26] Arne Holmberg. Pehr Wargentin och Kungl. Vetenskapsakademiens bibliotek.
Nordisk tidskrift för bok- och biblioteksväsen, 21:73–79, 1934. [Schenmark’s manuscript
is mentioned on page 76, where it is stated that there were three manuscripts, one being kept by
the Academy and the two others sent to Paris and Saint Petersburg]
[27] Johann Heinrich Lambert. Zusätze zu den Logarithmischen und Trigonometrischen
Tabellen zur Erleichterung und Abkürzung der bey Anwendung der Mathematik
vorfallenden Berechnungen. Berlin: Haude und Spener, 1770. [the table of factors was
reconstructed in [44]; [28] is a Latin translation of this book]
[28] Johann Heinrich Lambert and Anton Felkel. Supplementa tabularum
logarithmicarum et trigonometricarum. Lisbon, 1798. [Latin translation of [27]; the table
of factors was reconstructed in [43]]
[29] Derrick Norman Lehmer. Factor table for the first ten millions containing the
smallest factor of every number not divisible by 2, 3, 5, or 7 between the limits 0
and 10017000. Washington, D.C.: Carnegie Institution of Washington, 1909.
[reconstructed in [45]]
[30] Derrick Norman Lehmer. List of prime numbers from 1 to 10,006,721.
Washington, D.C.: Carnegie Institution of Washington, 1914. [reconstructed in [46]]
[31] Zacharias Nordmark. Åminnelse-tal, öfver Kongl. Vetenskaps Academiens
framledne Ledamot, Matheseos Professorn i Lund samt Ledamoten af där varande
Kongl. Physiographiska Sällskap, Herr Nils Schenmark, hållet för Kongl. vetenskaps
academien den 22 junii år 1791. Stockholm: Anders Zetterberg, 1791.
[32] Denis Roegel. A reconstruction of Burckhardt’s table of factors (first million, 1817).
Technical report, LORIA, Nancy, 2011. [This is a reconstruction of the table in [9].]
[33] Denis Roegel. A reconstruction of Burckhardt’s table of factors (second million,
1814). Technical report, LORIA, Nancy, 2011. [This is a reconstruction of the table in [6].]
[34] Denis Roegel. A reconstruction of Burckhardt’s table of factors (third million,
1816). Technical report, LORIA, Nancy, 2011. [This is a reconstruction of the table in [7].]
[35] Denis Roegel. A reconstruction of Chernac’s Cribrum arithmeticum (1811).
Technical report, LORIA, Nancy, 2011. [This is a reconstruction of [10].]
[36] Denis Roegel. A reconstruction of Dase’s table of factors (eighth million, 1863).
Technical report, LORIA, Nancy, 2011. [This is a reconstruction of the table in [12].]
[38] Denis Roegel. A reconstruction of Dase’s table of factors (seventh million, 1862).
Technical report, LORIA, Nancy, 2011. [This is a reconstruction of the table in [11].]
[39] Denis Roegel. A reconstruction of Felkel’s tables of primes and factors (1776).
Technical report, LORIA, 2011. [This is a reconstruction and an extension of Felkel’s
tables [15, 16, 17, 18].]
[40] Denis Roegel. A reconstruction of Glaisher’s table of factors (fifth million, 1880).
Technical report, LORIA, Nancy, 2011. [This is a reconstruction of the table in [20].]
[41] Denis Roegel. A reconstruction of Glaisher’s table of factors (fourth million, 1879).
Technical report, LORIA, Nancy, 2011. [This is a reconstruction of the table in [19].]
[42] Denis Roegel. A reconstruction of Glaisher’s table of factors (sixth million, 1883).
Technical report, LORIA, Nancy, 2011. [This is a reconstruction of the table in [21].]
[43] Denis Roegel. A reconstruction of Lambert and Felkel’s table of factors (1798).
Technical report, LORIA, Nancy, 2011. [This is a reconstruction of the table of factors
in [28].]
[44] Denis Roegel. A reconstruction of Lambert’s table of factors (1770). Technical
report, LORIA, Nancy, 2011. [This is a reconstruction of the table of factors in [27].]
[45] Denis Roegel. A reconstruction of Lehmer’s table of factors (1909). Technical
report, LORIA, Nancy, 2011. [This is a reconstruction of [29].]
[46] Denis Roegel. A reconstruction of Lehmer’s table of primes (1914). Technical
report, LORIA, Nancy, 2011. [This is a reconstruction of [30].]
[47] Denis Roegel. A reconstruction of Vega’s table of primes and factors (1782).
Technical report, LORIA, 2011. [This is a reconstruction of the table in [51].]
[48] Denis Roegel. A reconstruction of Vega’s table of primes and factors (1797).
Technical report, LORIA, Nancy, 2011. [This is a partial reconstruction of [52].]
[49] Nils Schenmark. Tabula, numerorum primorum et pro minimis divisoribus
compositorum, ad octo millia ultra millionem expedite inveniendis, ca. 1780. [Copies
of the manuscript at the library of the Institut in Paris, at the Royal Swedish Academy of
Sciences in Stockholm, and probably in St. Petersburg.]
[50] Paul Peter Heinrich Seelhoff. Geschichte der Factorentafeln. Archiv der
Mathematik und Physik, 70:413–426, 1884.
[51] Georg Vega. Vorlesungen über die Mathematik, volume 1. Wien: Johann Thomas
Edlen von Trattnern, 1782. [The tables of primes and factors are reconstructed in [47].]
[53] Franz Ernst von Schaffgotsch. Ein Gesetz, welches zu Fortsetzung der bekannten
Pellischen Tafeln dienet. Abhandlungen einer Privatgesellschaft in Böhmen,
5:354–382, 1782.
[54] Martin Weibull. Lunds universitets historia, 1668–1868, volume 1. Lund: C. W. K.
Gleerups, 1868. [pp. 181–182 on Schenmark]
[55] Martin Weibull and Elof Tegnér. Lunds universitets historia, 1668–1868, volume 2.
Lund: C. W. K. Gleerups, 1868. [p. 182 on Tegman, p. 368 on Heurlin and Cronholm,
p. 423 on Lidtgren, and pp. 426–427 on Widebeck]
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