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A reconstruction of Nevilles Farey series of order
1025(1950)
Denis Roegel
To cite this version:Denis Roegel. A reconstruction of Nevilles
Farey series of order 1025 (1950). [Research Report] 2011.
https://hal.inria.fr/hal-00654447https://hal.archives-ouvertes.fr
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A reconstruction of
Nevilles Farey series of order 1025
(1950)
Denis Roegel
22 October 2011
This document is part of the LOCOMAT
project:http://locomat.loria.fr
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1 Introduction
Eric Harold Neville (18891961) was born in London and entered
Trinity College in 1907.There he became acquainted with Bertrand
Russell and G. H. Hardy.
In 1914, visiting India, he was instrumental in persuading
Srinivasa Ramanujan to goback with him to England.
Neville worked on a variety of topics, and in particular on
Jacobian elliptic functions.He chaired the Mathematical Tables
Committee of the British Association for the Ad-vancement of
Science (BAAS) from 1931 to 1947. He contributed two sets of
tables,one on the Farey series of order 1025 (1950) [22]
reconstructed here, and another onrectangular-polar conversion
(1956) [23].
2 Farey series
2.1 Introduction
The Farey sequence Fn of order n is the increasing sequence of
fractionsp
qwith 0 p
q n. If three consecutive terms ai1bi1
, aibi
, ai+1bi+1
, of the sequence are given, the middle onecan be obtained by
the mediant property:
ai1 + ai+1bi1 + bi+1
=ai
bi
This property was described by John Farey (17661826) in 1816
[9].An algorithm for the construction of this sequence had already
been given by Haros
in 1802 [14], and the sequence is nowadays called the
Haros-Farey sequence [8].Farey sequences are related to the
construction of the Stern-Brocot tree of the ratio-
nals [6, 27]. A Farey sequence is sometimes called a Brocot
table [20].1 Such a table isuseful to find rational approximations
of a number.
2.2 The Farey series of order 1025
Nevilles main table [22] gives the fractions 0 < pq
1
1in increasing order and with
q 1025. Each fraction is given as a triplet (p, q, q p), which
allows to stop the tableat 1
2. The fractions following 1
2can be read backwards from the end of the table.
Each page contains 400 fractions. The last page contains 282
fractions. This tablecovers exactly 400 pages, and contains
therefore 159882 fractions. The total number offractions, when the
two parts of the table are separated, is 1598812+1 = 319765,
sincethe last fraction counts only for one. Each page also gives
the value at the end of theprevious page and the value at the
beginning of the following page.
1For an introduction to the construction of the Stern-Brocot
tree and Farey sequences, see [15, 16,12, 13].
i
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Apart from its use in finding rational approximations, this
table also gives solutionsof the Diophantine equation bx ay = 1 for
1 a b 1025, with one fraction being a
b
and the next one in the series xy. For instance, the solution of
657x 134y = 1 is found
in the second column of page 164, right after the entry
134657
, and is x = 103, y = 505.Using this table is however not that
convenient, because it appears difficult to locate
a given ratio in the main table. In his introduction, Neville
provides an additional table(not reproduced here), giving the
location of specific fractions, but one reviewer regrettedthat
decimal equivalents were not given at least for the last value on
each page.
2.3 The Farey series of order 50
The main table is followed by three smaller tables. The first
one gives the Farey seriesof order 50 over two pages. For each
fraction p
q, a triplet (p, q, q p) is given, as well as
five-digit decimal equivalents of pq
and qpq
.
2.4 The Farey series of order 64
The second appendix gives the Farey series of order 64 over one
page. For each fractionp
q, a triplet (p, q, q p) is given.
2.5 The Farey integer-series of order 100
The third appendix gives the denominators of the fractions of
the Farey series of order100.
2.6 Nevilles table of factors
This table gives the complete factorization of all integers from
1 to 1049.
3 Reconstruction
For the reconstruction, we have used Patracu and Patracus
algorithm. They gave avery simple way for the computation of the
next term of a sequence [24, 26]. If a
b< c
d< p
q
are three consecutive terms, we have a+pb+q
= cd
(mediant property) and there is a k such
that kc = a + p and kd = b + q. It is easy to show that k =
n+bd
where n is thegreatest possible denominator. Once k is computed,
p and q follow. This algorithm iseasily adapted for a decreasing
sequence.
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4 Acknowledgements
It is our pleasure to acknowledge the help of Scott Guthery who
first drew our attentionto Nevilles table.
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References
The following list covers the most important references2 related
to Nevilles table. Notall items of this list are mentioned in the
text, and the sources which have not been seenare marked so. We
have added notes about the contents of the articles in certain
cases.
[1] Raymond Clare Archibald. Scarce mathematical tables.
Mathematical Tables andother Aids to Computation, 1(2):6667, 1943.
[On Brocots original table.]
[2] Raymond Clare Archibald. Scarce mathematical tables.
Mathematical Tables andother Aids to Computation, 1(4):132, 1943.
[On Brocots original table.]
[3] Raymond Clare Archibald. E. H. Neville: The Farey Series of
Order 1025 (review).Mathematical Tables and other Aids to
Computation, 5(35):135139, July 1951.
[4] Paul T. Bateman. Review of E. H. Neville, The Farey series
of order 1025,displaying solutions of the diophantine equation bx
ay = 1. Bulletin of theAmerican Mathematical Society, 57(4):325326,
1951.
[5] Thomas Arthur Alan Broadbent. Obituary: Eric Harold Neville.
Journal of theLondon Mathematical Society, 37:479482, 1962.
[6] Achille Brocot. Calcul des rouages par approximation,
nouvelle mthode. Revuechronomtrique, 3:186194, 1860.
[7] Achille Brocot. Calcul des rouages par approximation,
nouvelle mthode. Paris,1862. [not seen]
[8] Cristian Cobeli and Alexandru Zaharescu. The Haros-Farey
sequence at twohundred years: A survey. Acta Universitatis
Apulensis, Mathematics-Informatics,5:138, 2003.
[9] John Farey. On a curious property of vulgar fractions.
Philosophical Magazine,47:385386, 1816.
[10] 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-thirdmeeting of the
British Association for the advancement of science, London: John
Murray, 1874.
2Note 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 thereforeseemed pointless to
capitalize works according to conventions which not only have no
relation with theoriginal work, but also do not restore the title
entirely. In the following list of references, most titlewords
(except in German) will therefore be left uncapitalized. The names
of the authors have also beenhomogenized 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. Theoriginal style information
could no doubt have been added as a note, but we have not done it
here.
iv
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A review by R. Radau was published in the Bulletin des sciences
mathmatiques et
astronomiques, volume 11, 1876, pp. 727]
[11] James Whitbread Lee Glaisher. Table, mathematical. In Hugh
Chisholm, editor,The Encyclopdia Britannica, 11th edition, volume
26, pages 325336. Cambridge,England: at the University Press,
1911.
[12] Scott B. Guthery. A motif of mathematics: history and
application of the mediantand the Farey sequence. Createspace,
2010.
[13] Godfrey Harold Hardy and Edward Maitland Wright. An
introduction to the theoryof numbers. Oxford: Clarendon Press,
1938.
[14] Charles Haros. Tables pour valuer une fraction ordinaire
avec autant de dcimalesquon voudra ; et pour trouver la fraction
ordinaire la plus simple, et qui approchesensiblement dune fraction
dcimale. Journal de lcole Polytechnique, 4:364368,1802.
[15] Brian Hayes. On the teeth of wheels. American Scientist,
88(4):296300,July-August 2000. [reprinted in [16]]
[16] Brian Hayes. Group theory in the bedroom, and other
mathematical diversions.New York: Hill and Wang, 2008. [This
collection of articles contains a reprint of [15]. [17] isa French
translation of this book.]
[17] Brian Hayes. Lhorloge de lternit. Paris: Vuibert, 2010.
[Translation of [16].]
[18] Walter James Langford, Thomas Arthur Alan Broadbent, and
Reuben LouisGoodstein. Obituary: Professor Eric Harold Neville,
M.A., B.Sc. The MathematicalGazette, 48(364):131145, May 1964.
[19] Roger Mansuy. Achille Brocot, mathmaticien ses heures,
2008. [available
onhttp://www.math.ens.fr/culturemath/histoire%20des%20maths]
[20] Henry Edward Merritt. Gear trains, including a Brocot table
of decimal equivalentsand a table of factors of all useful numbers
up to 200,000. London: Sir IsaacPitman & Sons, Ltd., 1947. [The
table of useful numbers was reconstructed in [25].]
[21] Eric Harold Neville. The structure of Farey series.
Proceedings of the LondonMathematical Society, 51 (series
2):132144, 1949.
[22] Eric Harold Neville. The Farey series of order 1025,
displaying solutions of theDiophantine equation bx ay = 1, volume 1
of Royal Society Mathematical Tables.Cambridge: University Press,
1950.
[23] Eric Harold Neville. Rectangular-polar conversion tables,
volume 2 of Royal SocietyMathematical Tables. Cambridge: University
Press, 1956. [not seen]
v
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[24] Corina E. Patracu and Mihai Patracu. Computing order
statistics in the Fareysequence. In Proceedings of the 6th
Algorithmic Number Theory Symposium(ANTS 2004), volume 3076 of
Lecture Notes in Computer Science, pages 358366.Springer, 2004.
[25] Denis Roegel. A reconstruction of Merritts table of useful
numbers (1947).Technical report, LORIA, 2011. [This is a
reconstruction of the table in [20].]
[26] Norman Routledge. Computing Farey series. The Mathematical
Gazette,92(523):5562, March 2008.
[27] Moritz Abraham Stern. ber eine zahlentheoretische Funktion.
Journal fr diereine und angewandte Mathematik, 55(3):193220,
1858.
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2011)
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Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 1025 (reconstruction, D. Roegel,
2011)
-
appendices
I
The Farey Series of Order with Decimal Equivalents
II
The Farey Series of Order
III
The Farey Integer-Series of Order
-
Nevilles Farey series of order 50 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 50 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey series of order 64 (reconstruction, D. Roegel,
2011)
-
Nevilles Farey integer-series of order 100 (reconstruction, D.
Roegel, 2011)
-
Nevilles auxiliary factor table (reconstruction, D. Roegel,
2011)
I. The Numbers 1549 factorized Completely
1 3 2 5 . 7 3 2
. 11 2. 13 . . 4 17 .2 19
2. . . 23 3. 2 . 3 2. 29
.. 31 5 . . . 2.2 37 . .
3. 41 .. 43 2. 2. . 47 4. 2
.2 . 2. 53 .3 . 3. . . 59
2.. 61 . 2. 6 . .. 67 2. .
.. 71 3.2 73 . .2 2. . .. 79
4. 4 . 83 2.. . . . 3. 89
.2. . 2. . . . 5. 97 .2 2.
2.2 101 .. 103 3. .. . 107 2.3 109
.. . 4. 113 .. . 2. 2. . .
3.. 2 . . 2. 3 .2. 127 7 .
.. 131 2.. . . 3. 3. 137 .. 139
2.. . . . 4.2 . . .2 2. 149
..2 151 3. 2. .. . 2.. 157 . .
5. . .4 163 2. .. . 167 3.. 2
.. 2. 2. 173 .. 2. 4. . . 179
2.2. 181 .. . 3. . .. . 2. 3.
.. 191 6. 193 . .. 2.2 197 .2. 199
3.2 . . . 2.. . . 2. 4. .
... 211 2. . . . 3.3 . . .
2.. . .. 223 5. 2.2 . 227 2.. 229
.. .. 3. 233 .2. . 2. . .. 239
4.. 241 .2 5 2. .2 .. . 3. .
.3 251 2.2. . . .. 8 257 .. .
2.. 2. . 263 3.. . .. . 2. 269
.3. 271 4. .. . 2. 2.. 277 . 2.
3.. 281 .. 283 2. .. .. . 5.2 2
.. . 2. 293 ..2 . 3. 3. . .
2..2 . . . 4. . .2. 307 2.. .
.. 311 3.. 313 . 2.. 2. 317 .. .
6. . .. . 2.4 2. . . 3. .
... 331 2. 2. . . 4.. 337 .2 .
2.. . .2. 3 3. .. . 347 2.. 349
.2. 3. 5. 353 .. . 2. .. . 359
3.2. 2 . .2 2.. . .. 367 4. 2.
.. . 2.. 373 .. .3 3. . .3. 379
2.. . . 383 7. .. . 2. 2. 389
... . 3.2 . . . 2.2. 397 . ..
4.2 401 .. . 2. 4. .. . 3.. 409
.. . 2. . .2. . 5. . .. 419
2... 421 . 2. 3. 2. .. . 2. ..
.. 431 4.3 433 .. .. 2. . .. 439
3.. 2.2 .. 443 2.. . . . 6. 449
.2.2 . 2. . . .. 3.. 457 . 3.
2.. 461 ... 463 4. .. . 467 2.2. .
.. . 3. . .. 2. 2.. 2. . 479
5.. . . .. 2.2 . .5 487 3. .
..2 491 2.. . .. 2.. 4. . .. 499
2.3 . . 503 3.2. . .. .2 2. 509
... . 9 3. . . 2.. . .. .
3.. 521 .2. 523 2. .2. . . 4.. 2
.. 2. 2.. . .. . 3. . . 2.
2.3. 541 . . 5. . ... 547 2. 2.
-
Nevilles auxiliary factor table (reconstruction, D. Roegel,
2011)
I. The Numbers 5001049 factorized Completely
2.3 . . 503 3.2. . .. .2 2. 509
... . 9 3. . . 2.. . .. .
3.. 521 .2. 523 2. .2. . . 4.. 2
.. 2. 2.. . .. . 3. . . 2.
2.3. 541 . . 5. . ... 547 2. 2.
.2. . 3.. . . .. 2. 557 .2. .
4.. .. . 563 2.. . . 4. 3. 569
... 571 2.. . .. 2. 6.2 577 .2 .
2.. . .. . 3. 2.. . 587 2..2 .
.. . 4. 593 .3. .. 2. . .. 599
3..2 601 .. 2. 2. .2 .. 607 5. ..
.. . 2.2. 613 . .. 3.. 617 .. 619
2.. 3. . . 4.. 4 . .. 2. .
.2.. 631 3. . . . 2.. 2. .. 2.
7. 641 .. 643 2.. .. .. 647 3.4 .
.2. .. 2. 653 .. . 4. 2. .. 659
2... 661 . .. 3. .. .2. . 2. .
.. . 5.. 673 . 3.2 2.2 677 .. .
3.. . .. 683 2.2. . .3 . 4. .
... 691 2. 2.. . . 3.. . . .
2.2. 701 .3. . 6. .. . . 2.. 709
.. 2. 3. . ... .. 2. . . 719
4.2. . .2 . 2. 2. ..2 727 3.. 6
.. . 2.. 733 . ..2 5. . .2. 739
2.. .. .. 743 3.. . . 2. 2.. .
..3 751 4. . .. . 2.3. 757 . ..
3.. 761 .. . 2. 2.. . . 8. 769
... . 2. 773 .2. 2. 3. .. . .
2... . .. 3. 4.2 . .. 787 2. .
.. . 3.2. . . .. 2. 797 ... .
5.2 2. . . 2.. .. .. . 3. 809
.4. 811 2.. . .. . 4.. . . 2..
2.. 821 .. 823 3. .2. .. 827 2.2. 829
.. . 6. 2. .. . 2.. 3. . 839
3... 2 . . 2. .2 .2. .2 4. .
.2. . 2.. 853 .. 2.. 3. 857 ... 859
2.. .. . 863 5.3 . . .2 2.. .
... . 3. 2. .. 3. 2.. 877 . .
4.. 881 .2.2 883 2.. .. . 887 3.. .
.. 4. 2. . .. . 7. .. . .
2.2.2 . .. .. 3. . .. 907 2. 2.
... 911 4.. . . .. 2. . .3. 919
3.. . . . 2... 2. . 2. 5. 929
... 2. 2. . . .. 3.2. 937 .. .
2.. 941 .. . 4. 3.. .. 947 2.. .
.2. . 3.. 953 .2. . 2. .. . .
6.. 2 .. 2. 2. . ... 967 3.2 ..
.. 971 2.5 . . .2. 4. 977 .. .
2..2 2. . 983 3.. . .. .. 2.. .
.2.. 991 5. . .. . 2.. 997 . 3.
3.3 .. .. . 2. .. . . 4.2. 1009
.. . 2.. 1013 ..2 .. 3. 2. . 1019
2... 1021 .. .. 10 2. .3. . 2. .3
.. 1031 3.. 1033 .. 2.. 2.. . .. 1039
4.. . . . 2.2. .. . . 3. 1049