A reconstruction of Bojko's table of quarter-squares (1909)

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A reconstruction of Bojko’s table of quarter-squares

(1909)

Denis Roegel

To cite this version:

Denis Roegel. A reconstruction of Bojko’s table of quarter-squares (1909). [Research Report] 2013.

�hal-00880837�

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A reconstruction of

Bojko’s table of quarter-squares

(1909)

Denis Roegel

2013

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1 The method of quarter-squares

The method of quarter-squares is a multiplication method which makes use of the identity

ab

=

1

4 (a + b)

2

− (a − b)

2

.

If we possess a table of squares, and wish to compute the product ab, it is then

sufficient to compute a + b, a − b, to look up the table for these two values, to subtract

the values read in the table, and to divide the result by 4. This seems complex, but for

large numbers it is more efficient than to compute directly the product.

1

There have been a number of tables of squares around, in particular those of Ludolf

published in 1690 [35], and those published by Séguin in 1801 and giving the squares

up to 10000. But these authors did mostly not have multiplications in mind, although

Ludolf showed how squares could be used to compute a multiplication. At the beginning

of the 19th century, the method of quarter-squares was mentioned in passing by Laplace

in 1809 [14, p. 261] and Gergonne in 1816 [17, pp. 159–160], but they did not produce

tables.

2 The first tables of quarter-squares (1817)

In 1817, Voisin [55] and Bürger [9] published independently the first tables of

quarter-squares. Voisin and Bürger both understood that a table of squares could be useful for

multiplications, but they went one step beyond, and removed the need to divide the

difference of the squares by 4. Their tables gave the quarter-squares for all integers up

to 19999 (Bürger) and 20000 (Voisin).

More exactly, when a number is odd, its quarter-square is of the form n + 0.25 and

Voisin and Bürger only gave the value n. So if a and b are both odd or both even, the

values of the quarter-squares are given exactly. If one of the integers is odd and the other

is even, both quarter-squares are in default by 0.25, but the difference of these two values

is correct. So, although the tables of Voisin and Bürger are not totally accurate for a

table of squares, they serve the purpose of a table of multiplication. If one wished to use

them as a table of squares, one would need to remember to add 1 to the result multiplied

by 4, whenever the value entered is odd.

A table of quarter-squares is also very advantageous with respect to space. Such a

table is a linear table, with only one entry. With slightly more work than a mere table

of multiplication, it is possible with a table of quarter-squares to do computations that

would require many thousands of pages with a conventional table of multiplication, such

as Crelle’s table [12].

In 1820, in the second edition of his Philosophy of arithmetic [34, pp. 246–257], John

Leslie described tables of quarter-squares, and in particular Voisin’s table, of which he

gave an excerpt up to 2000. Leslie expressed the hope that someone would compute a

table up to 200000.

1

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Table

Range

Pages

Density

Voisin (1817)

20000

123

162.6 Bürger (1817)

20000

80

250.0 Centnerschwer (1825)

20000

40

500.0 Merpaut (1832)

40000

400

100.0 Kulik (1851)

30000

40

750.0 Laundy (1856)

100000

200

500.0 Blater (1887)

200000

200 1000.0

Bojko (1909)

20000

20 1000.0

Plassmann (1933)

20009

200

100.0 Table 1: A comparison of the main tables of quarter-squares. The density is the ratio

of the range by the number of pages. The tables which pack the greatest ranges in the

smallest number of pages are those of Blater and Bojko.

3 Later tables of quarter-squares

A number of other tables of quarter-squares were published during the 19th century

(table 1). Centnerschwer [10] published a table of quarter-squares up to 20000 in 1825,

Kulik in 1833 and 1851 (up to 30000) [27, 28], Merpaut in 1832 (up to 40000) [38],

Laundy in 1856 [32], and Blater in 1887 [1]. Laundy’s table gave the quarter-squares

up to 100000, whereas Blater extended it to 200000. In 1836, Galbraith published a

small table of quarter-squares in a more general collection of tables [16]. This table was

probably borrowed from Voisin’s table.

In the preface of his table of quarter-squares [32, p. iv], Laundy also mentions that

Peter Gray had in his possession a manuscript of a table of quarter-squares extending to

200000 by Shortrede, but that table was never published.

There have also been several applications of quarter-squares for mechanical aids. In

1829, Schiereck invented a calculating machine based on quarter-squares [57] and in 1841

he devised an instrument for the measurement of the area of triangles, useful in surveying,

and also based on quarter-squares [52, 24].

In a book published in 1857, Edward Sang also suggested the use of quarter-squares

for multiplications [11, pp. 50–51].

The use of quarter-squares declined in the first half of the 20th century [37], but Bojko

in 1909 [5] and Plassmann in 1933 [40] each published tables of quarter-squares up to

20000. In the 1950s, however, new applications of the quarter-squares method for

multi-plication on analog devices surfaced, and this prompted their use for the implementation

of multiplication in digital processors [25, 26].

Finally, we mention that tables of triangular numbers can also be used in a way similar

to quarter-squares in order to facilitate the multiplications [13].

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4 Bojko

Josef Bojko (1881– 19??) was an engineer from Novoselytsia, formerly in Russia and now

in Ukraine. He defended his PhD

2

in Zurich in 1908 and published his dissertation on a

new table of quarter-squares in 1909 [5, 4, 58].

He also published books on practical or mental calculation in 1907 and 1920 [8, 6],

several articles on mathematics [3, 2], as well as on the wiring of electrical motors [7].

5 Structure of Bojko’s table and reconstruction

Bojko’s table of quarter-squares is very compact and gives the quarter-squares of all

integers from 0 to 20000 on only 20 pages. When looking for the quarter-square of a

number x, this number is split in two parts x = N |n, where n are the last two digits. For

instance, x = 5134 is split into N = 51 and n = 34. Each double page of Bojko’s table

covers a range of values of N and n. The left-hand pages cover even values 0 ≤ N ≤ 98, as

well as odd values 101 ≤ N ≤ 199. The right-hand pages cover odd values of 1 ≤ N ≤ 99,

as well as even values 100 ≤ N ≤ 198. There are 10 values of n on each page, but these

values depend on whether N > 99 or N < 100. When N > 99, Bojko calls the prefix N

1

and the suffix n

1

, so that on the first double page 0 ≤ n ≤ 9 and 100 ≥ n

1

≥ 91, on the

second double page 10 ≤ n ≤ 19 and 90 ≥ n

1

≥ 81, on the third double page 20 ≤ n ≤ 29

and 80 ≥ n

1

≥ 71, . . . , and on the last double page 90 ≤ n ≤ 99 and 10 ≥ n

1

≥ 1.

The value of the quarter-square of N |n (or N

1

|n

1

) is split in three parts A|B|C (or

A

1

|B

1

|C). For instance, the quarter-square of 5134 is 65|894|89, where A is found at the

intersection of column A and line 51, B is found at the intersection of line 51 and column

34 (read at the top), and C is found at the bottom of column 34 (read at the top). The

intersection of line 51 and column 34 actually gives “4 894”, which should be read “894,”

the bold digit being ignored.

The quarter-square of 14864 is 552|346|24, where A

1

is found at the intersection of

column A

1

and line 148, B

1

is found at the intersection of line 148 and column 64 (read

at the bottom), and C is found at the bottom of column 64 (read at the bottom). The

intersection of line 148 and column 64 actually gives “3 946”, which should be read “346,”

this time the bold digit being used, because N > 99. Each cell contains two values which

share the same last two digits. That this is possible is shown below.

A number of entries of Bojko’s table contain asterisks. For instance, at the

intersec-tion of line 61 and column 32 (read at the top), the value given is “8* 003.” Such an

asterisk then also appears on all the values towards the right of the same line. The values

immediately to the left are “9 942” and “8 972.” The value of B has actually overflowed,

so that the value of A must be incremented. The quarter-square of 6132 is therefore

94|003|56 and not 93|003|56.

Asterisks may also precede bold digits, in which case they signify an overflow for

values of B

1

. For instance, at the intersection of line 144 and column 67 (read at the

bottom), we have the value “*2 535”, so that the quarter-square of 14467 is 523|235|22

and not 522|235|22.

2

He was registered under number 17028 in 1906 and left the University of Zurich on 23 November 1908, see http://www.matrikel.uzh.ch/active/static/2740.htm.

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Left-hand pages give the quarter-squares of 200p+n, where 0 ≤ p ≤ 49. Let us denote

q(x) the quarter-square of x. Then, q(200p+n) =

40000p2+n₄2+400pn

= 10000p

2

_+100pn+

n2

4

.

In other words, the last two digits of q(x) are the last two digits of

n2

4

, which explains

that C is a value common to a whole column. Similarly, right-hand pages give the

squares of 200p + 100 + n for 0 ≤ p ≤ 49 and the last two digits of the

quarter-square are those of

n2+200n

4

. In addition, left-hand pages give the quarter-squares of

x

= 200p + 100 + (100 − n) for 50 ≤ p ≤ 99, and q(x) =

40000(p+1)2+n2−400(p+1)n

4

whose last

two digits are the same as those of the quarter-square of 200p + n. Similarly, right-hand

pages also give the quarter-squares of x = 200p + (100 − n) for 50 ≤ p ≤ 99, and the last

two digits of the quarter-squares are those of

n2−200n

4

, whose last two digits are identical

to those of

n2+200n 4

.

All that remains to be shown is that the last two digits of B are identical to the last

two digits of B

1

for overlapping cells. In other words, on left-hand pages, x

1

= 200p + n

is shared with x

2

= 100(199 − 2p) + (100 − n) = 20000 − 200p − n = 20000 − x

1

. On

right-hand pages, x

1

= 100(2p + 1) + n is shared with x

2

= 100(198 − 2p) + (100 − n) = 20000 −

200p − 100 − n = 20000 − x

1

. Consequently, q(x

2

) =

20000

2_+x2

1−40000x1

4

= 10000z + q(x

1

)

where z is some integer. Therefore, the quarter-squares of x

1

and of x

2

have the same

last four digits.

Bojko’s table may contain a few errors. For instance, on the first page of the table,

the second column to the right contains 939 when it should actually be 940. Therefore,

when computing the quarter-square of 19391, Bojko returns 939|027|20, instead of the

correct 940|027|20.

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References

The following list covers the most important references

3

related to Bojko’s table. 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] Joseph Blater. Tafel der Viertel-Quadrate aller ganzen Zahlen von 1 bis 200000

welche die Ausführung von Multiplikationen, Quadrirungen und das Ausziehen der

Quadratwurzel bedeutend erleichtert und durch vorzügliche Correctheit fehlerlose

Resultate verbürgt. Wien: Alfred Hölder, 1887.

[reconstructed in [43]]

[2] Josef Bojko. Beitrag zum Ausziehen höherer Wurzeln. Zeitschrift für Mathematik

und Physik, 57:373–375, 1909.

[3] Josef Bojko. Eine neue Näherungskonstruktion für π. Zeitschrift für Mathematik

und Physik, 57:196, 1909.

[4] Josef Bojko. Eine neue Tafel der Viertelquadrate. Zeitschrift für Mathematik und

Physik, 57:194–196, 1909.

[5] Josef Bojko. Neue Tafel der Viertelquadrate aller natürlichen Zahlen 1 bis 20000

zur Bildung aller möglichen Produkte im Bereiche 1 × 1 bis 10000 × 10000. Zürich:

E. Speidel, 1909.

[6] Josef Bojko. Lehrbuch der Rechenvorteile. Schnellrechnen und Rechenkunst.

Leipzig und Berlin: B. G. Teubner, 1920.

[2nd edition in 1926] [not seen]

[7] Josef Bojko. Schaltungsschemata für zwei- und dreiphasige Stabrotore : Entwurf

und Rekonstruktion. München: Oldenbourg, 1924.

[not seen]

[8] Josef Bojko and Eugen Wendling. Neues System zum technischen Kopfrechnen. Die

Quadratbildung der Zahlen 1 bis 1250. Zürich: E. Speidel, 1907.

[not seen]

[9] Johann Anton Philipp Bürger. Tafeln zur Erleichterung in Rechnungen für den

allgemeinen Gebrauch eingerichtet. Karlsruhe: D. R. Marx, 1817.

[10] J. J. Centnerschwer. Neu erfundene Multiplikations- und Quadrat-Tafeln,

vermittelst welcher man die Produkte aller vierziffrigen und die Wurzeln aller

fünfziffrigen Zahlen sehr leicht finden kann, wie auch zur Erleichterung anderer

mathematischen Rechnungen. Berlin: Maurer, 1825.

3_{Note 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.

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[11] Alex D. D. Craik. The logarithmic tables of Edward Sang and his daughters.

Historia Mathematica, 30(1):47–84, February 2003.

[12] August Leopold Crelle. Rechentafeln, welche alles Multipliciren und Dividiren mit

Zahlen unter Tausend ganz ersparen, bei grösseren Zahlen aber die Rechnung

erleichtern und sicherer machen. Berlin: Maurerschen Buchhandlung, 1820.

[2 volumes, reconstructed in [41]]

[13] Elie de Joncourt. De natura et præclaro usu simplicissimæ speciei numerorum

trigonalium. The Hague: Husson, 1762.

[14] Pierre-Simon de Laplace. Mémoire sur divers points d’analyse. Journal de l’École

Polytechnique, 8:229–265, 1809.

[15] George Carey Foster. The Method of Quarter-Squares. Nature, 40(1042):593, 1889.

[answer to Glaisher’s article [20]]

[16] William Galbraith. New and concise general tables for computing the obliquity of

the ecliptic, converting mean solar into sidereal time determining the equation to

equal attitudes etc. Edinburgh: Blackwood, Stirling and Kenney, 1836.

[not seen]

[17] Joseph Diaz Gergonne. Sur divers moyens d’abréger la multiplication. Annales de

mathématiques pures et appliquées, 7(6):157–166, 1816.

[18] 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]

[19] James Whitbread Lee Glaisher. On multiplication by a table of single entry.

Philosophical magazine, 6(38):331–347, 1878.

[20] James Whitbread Lee Glaisher. The method of quarter squares. Nature,

40(1041):573–576, 1889.

[see Foster’s article [15]]

[21] James Whitbread Lee Glaisher. The method of quarter squares. Nature,

41(1045):9, 1889.

[22] James Whitbread Lee Glaisher. The method of quarter squares. Journal of the

Institute of Actuaries, 28:227–235, 1890.

[reprinted from Nature]

[23] James Whitbread Lee Glaisher. Table, mathematical. In Hugh Chisholm, editor,

The Encyclopædia Britannica, 11th edition, volume 26, pages 325–336. Cambridge,

England: at the University Press, 1911.

[24] Barbara Haeberlin and Stefan Drechsler. Die Viertelquadratemethode und das

Pediometer des Joseph Friedrich Schiereck. In Werner H. Schmidt and Werner

Girbardt, editors, 4. Symposium zur Entwicklung der Rechentechnik, Universität

Greifswald. Greifswald: University, 2009.

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[25] Totadri Jayashree and Dhruba Basu. On binary multiplication using the quarter

square algorithm. IEEE Transactions on Computers, C-25:957–960, September

1976.

[26] Everett L. Johnson. A digital quarter square multiplier. IEEE Transactions on

Computers, C-29:258–261, March 1980.

[27] Jakob Philipp Kulik. Toasirtafeln, zur leichtern Berechnung des Längen-

Flächen-und Kubik-Inhaltes Flächen-und der verschiedenen Münz- Maß- Flächen-und Gewichts-Beträge.

Prague: J. L. Eggenberger, 1833.

[28] Jakob Philipp Kulik. Neue Multiplikationstafeln : ein unentbehrliches Hülfsmittel

für Jedermann, um schnell, sicher und ohne Ermüdung zu rechnen. Leipzig:

Friedrich Fleischer, 1851.

[29] Harold D. Larsen. Pseudo logarithms. The Mathematics Teacher, 52(1):2–6,

January 1959.

[30] Review of Samuel Linn Laundy’s “A table of quarter-squares of all integer numbers

up to 100,000, by which the product of two factors may be found by the aid of

addition and subtraction alone”. The Assurance magazine, and journal of the

Institute of Actuaries, 6(4):234–237, July 1856.

[31] Samuel Linn Laundy. On a method of finding the product of two factors by means

of the addition and subtraction of natural numbers. The Assurance Magazine, and

Journal of the Institute of Actuaries, 6:121–129, April 1856.

[32] Samuel Linn Laundy. Table of quarter-squares of all integer numbers, up to

100,000, by which the product of two factors may be found by the aid of addition

and subtraction alone. London: Charles and Edwin Layton, 1856.

[33] Samuel Linn Laundy. On a method of using the “Table of quarter squares”. The

Assurance Magazine, and Journal of the Institute of Actuaries, 9(2):112–115, July

1860.

[34] John Leslie. The philosophy of arithmetic. Edinburgh: William and Charles Tait,

1820.

[pp. 245–257 on tables of quarter squares]

[35] Hiob Ludolf. Tetragonometria tabularia. Leipzig: Groschian, 1690.

[other editions were published in 1709 and 1712] [reconstructed in [48]]

[36] David D. McFarland. Quarter-squares revisited: earlier tables, division of labor in

table construction, and later implementations in analog computers. Technical

report, California Center for Population Research, UC Los Angeles, 2007.

[37] David D. McFarland. Tables of quarter-squares, sociologic[al] applications, and

contributions of George W. Jones. Technical report, California Center for

Population Research, UC Los Angeles, 2007.

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[38] J. M. Merpaut-Duzélidest. Tables arithmonomiques, fondées sur le rapport du

rectangle au carré ; ou, le calcul réduit à son dernier degré de simplification.

Vannes, 1832.

[not seen] [reconstructed in [49]]

[39] Quarter squares. The Pentagon: A Mathematics Magazine for Students,

10(2):102–103, 1951.

[40] Josef Plassmann. Tafel der Viertel-Quadrate aller Zahlen von 1 bis 20009 zur

Erleichterung des Multiplizierens vierstelliger Zahlen. Leipzig: Max Jänecke, 1933.

[41] Denis Roegel. A reconstruction of Crelle’s Rechentafeln (1820). Technical report,

LORIA, 2011.

[This is a reconstruction of [12].]

[42] Denis Roegel. A reconstruction of Kulik’s table of multiplication (1851). Technical

report, LORIA, Nancy, 2011.

[43] Denis Roegel. A reconstruction of Blater’s table of quarter-squares (1887).

Technical report, LORIA, Nancy, 2013.

[44] Denis Roegel. A reconstruction of Bürger’s table of quarter-squares (1817).

Technical report, LORIA, Nancy, 2013.

[45] Denis Roegel. A reconstruction of Centnerschwer’s table of quarter-squares (1825).

Technical report, LORIA, Nancy, 2013.

[46] Denis Roegel. A reconstruction of Joncourt’s table of triangular numbers (1762).

Technical report, LORIA, Nancy, 2013.

[47] Denis Roegel. A reconstruction of Laundy’s table of quarter-squares (1856).

Technical report, LORIA, Nancy, 2013.

[48] Denis Roegel. A reconstruction of Ludolf’s Tetragonometria tabularia (1690).

Technical report, LORIA, Nancy, 2013.

[49] Denis Roegel. A reconstruction of Merpaut’s table of quarter-squares (1832).

Technical report, LORIA, Nancy, 2013.

[50] Denis Roegel. A reconstruction of Plassmann’s table of quarter-squares (1933).

Technical report, LORIA, Nancy, 2013.

[51] Denis Roegel. A reconstruction of Voisin’s table of quarter-squares (1817).

Technical report, LORIA, Nancy, 2013.

[52] Joseph Friedrich Schiereck. Beschreibung des Pediometers, eines Instruments, um

den Flächeninhalt in Karten ohne Rechnung zu erhalten. Polytechnisches Journal,

82:251–265, 1841.

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[53] James Joseph Sylvester. Note on a formula by aid of which and of table of single

entry the continued product of any set of numbers (or at least a given constant

multiple thereof) may be effected by additions and subtractions only without the

use of logarithms. Philosophical magazine, 7:430–436, 1854.

[54] James Joseph Sylvester. On multiplication by aid of a table of single entry. The

Assurance Magazine, and Journal of the Institute of Actuaries, 4:236–238, 1854.

[55] Antoine Voisin. Tables de multiplication, ou, logarithmes des nombres entiers

depuis 1 jusqu’à 20,000, etc. Paris: Didot, 1817.

[56] Stephan Weiss. Die Multipliziertafeln: ihre Ausgestaltung und Verwendung, 2003.

[available at http://www.mechrech.info/publikat/MTafel1.pdf]

[57] Stephan Weiss. Die Rekonstruktion der Rechenmaschine von Schiereck 1829, 2006.

[58] P. Werkmeister. Review of “J. Bojko: Neue Tafel der Viertelquadrate etc.”.

(13)

Bojko’s table of quarter-squares (reconstruction, D. Roegel, 2013)

N n 00 01 02 03 04 05 06 07 08 09 A B 0 0 *0₀₀₀ 9₀₀₀ 8₀₀₀ 7₀₀₀ 6₀₀₀ 5₀₀₀ 4₀₀₀ 3₀₀₀ 2₀₀₀ 1₀₀₀ ₉₉₉ ₁₉₉ 2 0 *1₁₀₀ *0₁₀₁ 9₁₀₂ 8₁₀₃ 7₁₀₄ 6₁₀₅ 5₁₀₆ 4₁₀₇ 3₁₀₈ 2₁₀₉ ₉₇₉ ₁₉₇ 4 0 *4₄₀₀ *3₄₀₂ *2₄₀₄ *1₄₀₆ *0₄₀₈ 9₄₁₀ 8₄₁₂ 7₄₁₄ 6₄₁₆ 5₄₁₈ ₉₅₉ ₁₉₅ 6 0 9₉₀₀ 8₉₀₃ 7₉₀₆ 6₉₀₉ 5₉₁₂ 4₉₁₅ 3₉₁₈ 2₉₂₁ 1₉₂₄ 0₉₂₇ ₉₄₀ ₁₉₃ 8 1 *6₆₀₀ *5₆₀₄ *4₆₀₈ *3₆₁₂ *2₆₁₆ *1₆₂₀ *0₆₂₄ 9₆₂₈ 8₆₃₂ 7₆₃₆ ₉₂₀ ₁₉₁ 10 2 *5₅₀₀ *4₅₀₅ *3₅₁₀ *2₅₁₅ *1₅₂₀ *0₅₂₅ 9₅₃₀ 8₅₃₅ 7₅₄₀ 6₅₄₅ ₉₀₁ ₁₈₉ 12 3 *6₆₀₀ *5₆₀₆ *4₆₁₂ *3₆₁₈ *2₆₂₄ *1₆₃₀ *0₆₃₆ 9₆₄₂ 8₆₄₈ 7₆₅₄ ₈₈₂ ₁₈₇ 14 4 9₉₀₀ 8₉₀₇ 7₉₁₄ 6₉₂₁ 5₉₂₈ 4₉₃₅ 3₉₄₂ 2₉₄₉ 1₉₅₆ 0₉₆₃ ₈₆₄ ₁₈₅ 16 6 *4₄₀₀ *3₄₀₈ *2₄₁₆ *1₄₂₄ *0₄₃₂ 9₄₄₀ 8₄₄₈ 7₄₅₆ 6₄₆₄ 5₄₇₂ ₈₄₅ ₁₈₃ 18 8 *1₁₀₀ *0₁₀₉ 9₁₁₈ 8₁₂₇ 7₁₃₆ 6₁₄₅ 5₁₅₄ 4₁₆₃ 3₁₇₂ 2₁₈₁ ₈₂₇ ₁₈₁ 20 10 *0₀₀₀ 9₀₁₀ 8₀₂₀ 7₀₃₀ 6₀₄₀ 5₀₅₀ 4₀₆₀ 3₀₇₀ 2₀₈₀ 1₀₉₀ ₈₀₉ ₁₇₉ 22 12 *1₁₀₀ *0₁₁₁ 9₁₂₂ 8₁₃₃ 7₁₄₄ 6₁₅₅ 5₁₆₆ 4₁₇₇ 3₁₈₈ 2₁₉₉ ₇₉₁ ₁₇₇ 24 14 *4₄₀₀ *3₄₁₂ *2₄₂₄ *1₄₃₆ *0₄₄₈ 9₄₆₀ 8₄₇₂ 7₄₈₄ 6₄₉₆ 6₅₀₈ ₇₇₃ ₁₇₅ 26 16 9₉₀₀ 8₉₁₃ 7₉₂₆ 6₉₃₉ 5₉₅₂ 4₉₆₅ 3₉₇₈ 2₉₉₁ 2*₀₀₄ 1*₀₁₇ ₇₅₆ ₁₇₃ 28 19 *6₆₀₀ *5₆₁₄ *4₆₂₈ *3₆₄₂ *2₆₅₆ *1₆₇₀ *0₆₈₄ 9₆₉₈ 9₇₁₂ 8₇₂₆ ₇₃₈ ₁₇₁ 30 22 *5₅₀₀ *4₅₁₅ *3₅₃₀ *2₅₄₅ *1₅₆₀ *0₅₇₅ 9₅₉₀ 9₆₀₅ 8₆₂₀ 7₆₃₅ ₇₂₁ ₁₆₉ 32 25 *6₆₀₀ *5₆₁₆ *4₆₃₂ *3₆₄₈ *2₆₆₄ *1₆₈₀ *0₆₉₆ *0₇₁₂ 9₇₂₈ 8₇₄₄ ₇₀₄ ₁₆₇ 34 28 9₉₀₀ 8₉₁₇ 7₉₃₄ 6₉₅₁ 5₉₆₈ 4₉₈₅ 4*₀₀₂ 3*₀₁₉ 2*₀₃₆ 1*₀₅₃ ₆₈₈ ₁₆₅ 36 32 *4₄₀₀ *3₄₁₈ *2₄₃₆ *1₄₅₄ *0₄₇₂ 9₄₉₀ 9₅₀₈ 8₅₂₆ 7₅₄₄ 6₅₆₂ ₆₇₁ ₁₆₃ 38 36 *1₁₀₀ *0₁₁₉ 9₁₃₈ 8₁₅₇ 7₁₇₆ 6₁₉₅ 6₂₁₄ 5₂₃₃ 4₂₅₂ 3₂₇₁ ₆₅₅ ₁₆₁ 40 40 *0₀₀₀ 9₀₂₀ 8₀₄₀ 7₀₆₀ 6₀₈₀ 6₁₀₀ 5₁₂₀ 4₁₄₀ 3₁₆₀ 2₁₈₀ ₆₃₉ ₁₅₉ 42 44 *1₁₀₀ *0₁₂₁ 9₁₄₂ 8₁₆₃ 7₁₈₄ 7₂₀₅ 6₂₂₆ 5₂₄₇ 4₂₆₈ 3₂₈₉ ₆₂₃ ₁₅₇ 44 48 *4₄₀₀ *3₄₂₂ *2₄₄₄ *1₄₆₆ *0₄₈₈ *0₅₁₀ 9₅₃₂ 8₅₅₄ 7₅₇₆ 6₅₉₈ ₆₀₇ ₁₅₅ 46 52 9₉₀₀ 8₉₂₃ 7₉₄₆ 6₉₆₉ 5₉₉₂ 5*₀₁₅ 4*₀₃₈ 3*₀₆₁ 2*₀₈₄ 2*₁₀₇ ₅₉₂ ₁₅₃ 48 57 *6₆₀₀ *5₆₂₄ *4₆₄₈ *3₆₇₂ *2₆₉₆ *2₇₂₀ *1₇₄₄ *0₇₆₈ 9₇₉₂ 9₈₁₆ ₅₇₆ ₁₅₁ 50 62 *5₅₀₀ *4₅₂₅ *3₅₅₀ *2₅₇₅ *2₆₀₀ *1₆₂₅ *0₆₅₀ 9₆₇₅ 9₇₀₀ 8₇₂₅ ₅₆₁ ₁₄₉ 52 67 *6₆₀₀ *5₆₂₆ *4₆₅₂ *3₆₇₈ *3₇₀₄ *2₇₃₀ *1₇₅₆ *0₇₈₂ *0₈₀₈ 9₈₃₄ ₅₄₆ ₁₄₇ 54 72 9₉₀₀ 8₉₂₇ 7₉₅₄ 6₉₈₁ 6*₀₀₈ 5*₀₃₅ 4*₀₆₂ 3*₀₈₉ 3*₁₁₆ 2*₁₄₃ ₅₃₂ ₁₄₅ 56 78 *4₄₀₀ *3₄₂₈ *2₄₅₆ *1₄₈₄ *1₅₁₂ *0₅₄₀ 9₅₆₈ 8₅₉₆ 8₆₂₄ 7₆₅₂ ₅₁₇ ₁₄₃ 58 84 *1₁₀₀ *0₁₂₉ 9₁₅₈ 8₁₈₇ 8₂₁₆ 7₂₄₅ 6₂₇₄ 6₃₀₃ 5₃₃₂ 4₃₆₁ ₅₀₃ ₁₄₁ 60 90 *0₀₀₀ 9₀₃₀ 8₀₆₀ 7₀₉₀ 7₁₂₀ 6₁₅₀ 5₁₈₀ 5₂₁₀ 4₂₄₀ 3₂₇₀ ₄₈₉ ₁₃₉ 62 96 *1₁₀₀ *0₁₃₁ 9₁₆₂ 8₁₉₃ 8₂₂₄ 7₂₅₅ 6₂₈₆ 6₃₁₇ 5₃₄₈ 4₃₇₉ ₄₇₅ ₁₃₇ 64 102 *4₄₀₀ *3₄₃₂ *2₄₆₄ *1₄₉₆ *1₅₂₈ *0₅₆₀ 9₅₉₂ 9₆₂₄ 8₆₅₆ 7₆₈₈ ₄₆₁ ₁₃₅ 66 108 9₉₀₀ 8₉₃₃ 7₉₆₆ 6₉₉₉ 6*₀₃₂ 5*₀₆₅ 4*₀₉₈ 4*₁₃₁ 3*₁₆₄ 2*₁₉₇ ₄₄₈ ₁₃₃ 68 115 6₆₀₀ 5₆₃₄ 4₆₆₈ 4₇₀₂ 3₇₃₆ 2₇₇₀ 2₈₀₄ 1₈₃₈ 0₈₇₂ 0₉₀₆ ₄₃₅ ₁₃₁ 70 122 *5₅₀₀ *4₅₃₅ *3₅₇₀ *3₆₀₅ *2₆₄₀ *1₆₇₅ *1₇₁₀ *0₇₄₅ 9₇₈₀ 9₈₁₅ ₄₂₁ ₁₂₉ 72 129 6₆₀₀ 5₆₃₆ 4₆₇₂ 4₇₀₈ 3₇₄₄ 2₇₈₀ 2₈₁₆ 1₈₅₂ 0₈₈₈ 0₉₂₄ ₄₀₉ ₁₂₇ 74 136 9₉₀₀ 8₉₃₇ 7₉₇₄ 7*₀₁₁ 6*₀₄₈ 5*₀₈₅ 5*₁₂₂ 4*₁₅₉ 3*₁₉₆ 3*₂₃₃ ₃₉₆ ₁₂₅ 76 144 *4₄₀₀ *3₄₃₈ *2₄₇₆ *2₅₁₄ *1₅₅₂ *0₅₉₀ *0₆₂₈ 9₆₆₆ 9₇₀₄ 8₇₄₂ ₃₈₃ ₁₂₃ 78 152 *1₁₀₀ *0₁₃₉ 9₁₇₈ 9₂₁₇ 8₂₅₆ 7₂₉₅ 7₃₃₄ 6₃₇₃ 6₄₁₂ 5₄₅₁ ₃₇₁ ₁₂₁ 80 160 *0₀₀₀ 9₀₄₀ 8₀₈₀ 8₁₂₀ 7₁₆₀ 7₂₀₀ 6₂₄₀ 5₂₈₀ 5₃₂₀ 4₃₆₀ ₃₅₉ ₁₁₉ 82 168 *1₁₀₀ *0₁₄₁ 9₁₈₂ 9₂₂₃ 8₂₆₄ 8₃₀₅ 7₃₄₆ 6₃₈₇ 6₄₂₈ 5₄₆₉ ₃₄₇ ₁₁₇ 84 176 *4₄₀₀ *3₄₄₂ *2₄₈₄ *2₅₂₆ *1₅₆₈ *1₆₁₀ *0₆₅₂ 9₆₉₄ 9₇₃₆ 8₇₇₈ ₃₃₅ ₁₁₅ 86 184 9₉₀₀ 8₉₄₃ 7₉₈₆ 7*₀₂₉ 6*₀₇₂ 6*₁₁₅ 5*₁₅₈ 5*₂₀₁ 4*₂₄₄ 3*₂₈₇ ₃₂₄ ₁₁₃ 88 193 6₆₀₀ 5₆₄₄ 4₆₈₈ 4₇₃₂ 3₇₇₆ 3₈₂₀ 2₈₆₄ 2₉₀₈ 1₉₅₂ 0₉₉₆ ₃₁₃ ₁₁₁ 90 202 5₅₀₀ 4₅₄₅ 3₅₉₀ 3₆₃₅ 2₆₈₀ 2₇₂₅ 1₇₇₀ 1₈₁₅ 0₈₆₀ 0₉₀₅ ₃₀₂ ₁₀₉ 92 211 6₆₀₀ 5₆₄₆ 4₆₉₂ 4₇₃₈ 3₇₈₄ 3₈₃₀ 2₈₇₆ 2₉₂₂ 1₉₆₈ 1*₀₁₄ ₂₉₁ ₁₀₇ 94 220 9₉₀₀ 8₉₄₇ 7₉₉₄ 7*₀₄₁ 6*₀₈₈ 6*₁₃₅ 5*₁₈₂ 5*₂₂₉ 4*₂₇₆ 4*₃₂₃ ₂₈₀ ₁₀₅ 96 230 *4₄₀₀ *3₄₄₈ *2₄₉₆ *2₅₄₄ *1₅₉₂ *1₆₄₀ *0₆₈₈ *0₇₃₆ 9₇₈₄ 9₈₃₂ ₂₆₉ ₁₀₃ 98 240 *1₁₀₀ *0₁₄₉ 9₁₉₈ 9₂₄₇ 8₂₉₆ 8₃₄₅ 7₃₉₄ 7₄₄₃ 6₄₉₂ 6₅₄₁ ₂₅₉ ₁₀₁ B1 A1 100 99 98 97 96 95 94 93 92 91 n1 N1 C 00 00 01 02 04 06 09 12 16 20

(14)

Bojko’s table of quarter-squares (reconstruction, D. Roegel, 2013)

N n 00 01 02 03 04 05 06 07 08 09 A B 1 0 *0₀₂₅ 9₀₂₅ 8₀₂₆ 7₀₂₆ 6₀₂₇ 5₀₂₇ 4₀₂₈ 3₀₂₈ 2₀₂₉ 1₀₂₉ ₉₈₉ ₁₉₈ 3 0 *2₂₂₅ *1₂₂₆ *0₂₂₈ 9₂₂₉ 8₂₃₁ 7₂₃₂ 6₂₃₄ 5₂₃₅ 4₂₃₇ 3₂₃₈ ₉₆₉ ₁₉₆ 5 0 *6₆₂₅ *5₆₂₇ *4₆₃₀ *3₆₃₂ *2₆₃₅ *1₆₃₇ *0₆₄₀ 9₆₄₂ 8₆₄₅ 7₆₄₇ ₉₄₉ ₁₉₄ 7 1 *2₂₂₅ *1₂₂₈ *0₂₃₂ 9₂₃₅ 8₂₃₉ 7₂₄₂ 6₂₄₆ 5₂₄₉ 4₂₅₃ 3₂₅₆ ₉₃₀ ₁₉₂ 9 2 *0₀₂₅ 9₀₂₉ 8₀₃₄ 7₀₃₈ 6₀₄₃ 5₀₄₇ 4₀₅₂ 3₀₅₆ 2₀₆₁ 1₀₆₅ ₉₁₁ ₁₉₀ 11 3 *0₀₂₅ 9₀₃₀ 8₀₃₆ 7₀₄₁ 6₀₄₇ 5₀₅₂ 4₀₅₈ 3₀₆₃ 2₀₆₉ 1₀₇₄ ₈₉₂ ₁₈₈ 13 4 *2₂₂₅ *1₂₃₁ *0₂₃₈ 9₂₄₄ 8₂₅₁ 7₂₅₇ 6₂₆₄ 5₂₇₀ 4₂₇₇ 3₂₈₃ ₈₇₃ ₁₈₆ 15 5 *6₆₂₅ *5₆₃₂ *4₆₄₀ *3₆₄₇ *2₆₅₅ *1₆₆₂ *0₆₇₀ 9₆₇₇ 8₆₈₅ 7₆₉₂ ₈₅₄ ₁₈₄ 17 7 *2₂₂₅ *1₂₃₃ *0₂₄₂ 9₂₅₀ 8₂₅₉ 7₂₆₇ 6₂₇₆ 5₂₈₄ 4₂₉₃ 4₃₀₁ ₈₃₆ ₁₈₂ 19 9 *0₀₂₅ 9₀₃₄ 8₀₄₄ 7₀₅₃ 6₀₆₃ 5₀₇₂ 4₀₈₂ 3₀₉₁ 3₁₀₁ 2₁₁₀ ₈₁₈ ₁₈₀ 21 11 *0₀₂₅ 9₀₃₅ 8₀₄₆ 7₀₅₆ 6₀₆₇ 5₀₇₇ 4₀₈₈ 3₀₉₈ 3₁₀₉ 2₁₁₉ ₈₀₀ ₁₇₈ 23 13 *2₂₂₅ *1₂₃₆ *0₂₄₈ 9₂₅₉ 8₂₇₁ 7₂₈₂ 6₂₉₄ 6₃₀₅ 5₃₁₇ 4₃₂₈ ₇₈₂ ₁₇₆ 25 15 *6₆₂₅ *5₆₃₇ *4₆₅₀ *3₆₆₂ *2₆₇₅ *1₆₈₇ *1₇₀₀ *0₇₁₂ 9₇₂₅ 8₇₃₇ ₇₆₄ ₁₇₄ 27 18 *2₂₂₅ *1₂₃₈ *0₂₅₂ 9₂₆₅ 8₂₇₉ 7₂₉₂ 7₃₀₆ 6₃₁₉ 5₃₃₃ 4₃₄₆ ₇₄₇ ₁₇₂ 29 21 *0₀₂₅ 9₀₃₉ 8₀₅₄ 7₀₆₈ 6₀₈₃ 5₀₉₇ 5₁₁₂ 4₁₂₆ 3₁₄₁ 2₁₅₅ ₇₃₀ ₁₇₀ 31 24 *0₀₂₅ 9₀₄₀ 8₀₅₆ 7₀₇₁ 6₀₈₇ 6₁₀₂ 5₁₁₈ 4₁₃₃ 3₁₄₉ 2₁₆₄ ₇₁₃ ₁₆₈ 33 27 *2₂₂₅ *1₂₄₁ *0₂₅₈ 9₂₇₄ 8₂₉₁ 8₃₀₇ 7₃₂₄ 6₃₄₀ 5₃₅₇ 4₃₇₃ ₆₉₆ ₁₆₆ 35 30 *6₆₂₅ *5₆₄₂ *4₆₆₀ *3₆₇₇ *2₆₉₅ *2₇₁₂ *1₇₃₀ *0₇₄₇ 9₇₆₅ 8₇₈₂ ₆₇₉ ₁₆₄ 37 34 *2₂₂₅ *1₂₄₃ *0₂₆₂ 9₂₈₀ 8₂₉₉ 8₃₁₇ 7₃₃₆ 6₃₅₄ 5₃₇₃ 4₃₉₁ ₆₆₃ ₁₆₂ 39 38 *0₀₂₅ 9₀₄₄ 8₀₆₄ 7₀₈₃ 7₁₀₃ 6₁₂₂ 5₁₄₂ 4₁₆₁ 3₁₈₁ 3₂₀₀ ₆₄₇ ₁₆₀ 41 42 *0₀₂₅ 9₀₄₅ 8₀₆₆ 7₀₈₆ 7₁₀₇ 6₁₂₇ 5₁₄₈ 4₁₆₈ 3₁₈₉ 3₂₀₉ ₆₃₁ ₁₅₈ 43 46 *2₂₂₅ *1₂₄₆ *0₂₆₈ 9₂₈₉ 9₃₁₁ 8₃₃₂ 7₃₅₄ 6₃₇₅ 5₃₉₇ 5₄₁₈ ₆₁₅ ₁₅₆ 45 50 *6₆₂₅ *5₆₄₇ *4₆₇₀ *3₆₉₂ *3₇₁₅ *2₇₃₇ *1₇₆₀ *0₇₈₂ *0₈₀₅ 9₈₂₇ ₅₉₉ ₁₅₄ 47 55 *2₂₂₅ *1₂₄₈ *0₂₇₂ 9₂₉₅ 9₃₁₉ 8₃₄₂ 7₃₆₆ 6₃₈₉ 6₄₁₃ 5₄₃₆ ₅₈₄ ₁₅₂ 49 60 *0₀₂₅ 9₀₄₉ 8₀₇₄ 7₀₉₈ 7₁₂₃ 6₁₄₇ 5₁₇₂ 4₁₉₆ 4₂₂₁ 3₂₄₅ ₅₆₉ ₁₅₀ 51 65 *0₀₂₅ 9₀₅₀ 8₀₇₆ 8₁₀₁ 7₁₂₇ 6₁₅₂ 5₁₇₈ 5₂₀₃ 4₂₂₉ 3₂₅₄ ₅₅₄ ₁₄₈ 53 70 *2₂₂₅ *1₂₅₁ *0₂₇₈ *0₃₀₄ 9₃₃₁ 8₃₅₇ 7₃₈₄ 7₄₁₀ 6₄₃₇ 5₄₆₃ ₅₃₉ ₁₄₆ 55 75 *6₆₂₅ *5₆₅₂ *4₆₈₀ *4₇₀₇ *3₇₃₅ *2₇₆₂ *1₇₉₀ *1₈₁₇ *0₈₄₅ 9₈₇₂ ₅₂₄ ₁₄₄ 57 81 *2₂₂₅ *1₂₅₃ *0₂₈₂ *0₃₁₀ 9₃₃₉ 8₃₆₇ 7₃₉₆ 7₄₂₄ 6₄₅₃ 5₄₈₁ ₅₁₀ ₁₄₂ 59 87 *0₀₂₅ 9₀₅₄ 8₀₈₄ 8₁₁₃ 7₁₄₃ 6₁₇₂ 6₂₀₂ 5₂₃₁ 4₂₆₁ 3₂₉₀ ₄₉₆ ₁₄₀ 61 93 *0₀₂₅ 9₀₅₅ 8₀₈₆ 8₁₁₆ 7₁₄₇ 6₁₇₇ 6₂₀₈ 5₂₃₈ 4₂₆₉ 3₂₉₉ ₄₈₂ ₁₃₈ 63 99 *2₂₂₅ *1₂₅₆ *0₂₈₈ *0₃₁₉ 9₃₅₁ 8₃₈₂ 8₄₁₄ 7₄₄₅ 6₄₇₇ 6₅₀₈ ₄₆₈ ₁₃₆ 65 105 6₆₂₅ 5₆₅₇ 4₆₉₀ 4₇₂₂ 3₇₅₅ 2₇₈₇ 2₈₂₀ 1₈₅₂ 0₈₈₅ 0₉₁₇ ₄₅₅ ₁₃₄ 67 112 *2₂₂₅ *1₂₅₈ *0₂₉₂ *0₃₂₅ 9₃₅₉ 8₃₉₂ 8₄₂₆ 7₄₅₉ 6₄₉₃ 6₅₂₆ ₄₄₁ ₁₃₂ 69 119 *0₀₂₅ 9₀₅₉ 8₀₉₄ 8₁₂₈ 7₁₆₃ 6₁₉₇ 6₂₃₂ 5₂₆₆ 5₃₀₁ 4₃₃₅ ₄₂₈ ₁₃₀ 71 126 *0₀₂₅ 9₀₆₀ 8₀₉₆ 8₁₃₁ 7₁₆₇ 7₂₀₂ 6₂₃₈ 5₂₇₃ 5₃₀₉ 4₃₄₄ ₄₁₅ ₁₂₈ 73 133 *2₂₂₅ *1₂₆₁ *0₂₉₈ *0₃₃₄ 9₃₇₁ 9₄₀₇ 8₄₄₄ 7₄₈₀ 7₅₁₇ 6₅₅₃ ₄₀₂ ₁₂₆ 75 140 6₆₂₅ 5₆₆₂ 5₇₀₀ 4₇₃₇ 3₇₇₅ 3₈₁₂ 2₈₅₀ 1₈₈₇ 1₉₂₅ 0₉₆₂ ₃₉₀ ₁₂₄ 77 148 *2₂₂₅ *1₂₆₃ *1₃₀₂ *0₃₄₀ 9₃₇₉ 9₄₁₇ 8₄₅₆ 7₄₉₄ 7₅₃₃ 6₅₇₁ ₃₇₇ ₁₂₂ 79 156 *0₀₂₅ 9₀₆₄ 9₁₀₄ 8₁₄₃ 7₁₈₃ 7₂₂₂ 6₂₆₂ 6₃₀₁ 5₃₄₁ 4₃₈₀ ₃₆₅ ₁₂₀ 81 164 *0₀₂₅ 9₀₆₅ 9₁₀₆ 8₁₄₆ 7₁₈₇ 7₂₂₇ 6₂₆₈ 6₃₀₈ 5₃₄₉ 4₃₈₉ ₃₅₃ ₁₁₈ 83 172 *2₂₂₅ *1₂₆₆ *1₃₀₈ *0₃₄₉ 9₃₉₁ 9₄₃₂ 8₄₇₄ 8₅₁₅ 7₅₅₇ 6₅₉₈ ₃₄₁ ₁₁₆ 85 180 6₆₂₅ 5₆₆₇ 5₇₁₀ 4₇₅₂ 3₇₉₅ 3₈₃₇ 2₈₈₀ 2₉₂₂ 1₉₆₅ 1*₀₀₇ ₃₃₀ ₁₁₄ 87 189 *2₂₂₅ *1₂₆₈ *1₃₁₂ *0₃₅₅ 9₃₉₉ 9₄₄₂ 8₄₈₆ 8₅₂₉ 7₅₇₃ 7₆₁₆ ₃₁₈ ₁₁₂ 89 198 *0₀₂₅ 9₀₆₉ 9₁₁₄ 8₁₅₈ 8₂₀₃ 7₂₄₇ 6₂₉₂ 6₃₃₆ 5₃₈₁ 5₄₂₅ ₃₀₇ ₁₁₀ 91 207 *0₀₂₅ 9₀₇₀ 9₁₁₆ 8₁₆₁ 8₂₀₇ 7₂₅₂ 6₂₉₈ 6₃₄₃ 5₃₈₉ 5₄₃₄ ₂₉₆ ₁₀₈ 93 216 *2₂₂₅ *1₂₇₁ *1₃₁₈ *0₃₆₄ *0₄₁₁ 9₄₅₇ 9₅₀₄ 8₅₅₀ 7₅₉₇ 7₆₄₃ ₂₈₅ ₁₀₆ 95 225 6₆₂₅ 5₆₇₂ 5₇₂₀ 4₇₆₇ 4₈₁₅ 3₈₆₂ 3₉₁₀ 2₉₅₇ 2*₀₀₅ 1*₀₅₂ ₂₇₅ ₁₀₄ 97 235 *2₂₂₅ *1₂₇₃ *1₃₂₂ *0₃₇₀ *0₄₁₉ 9₄₆₇ 9₅₁₆ 8₅₆₄ 8₆₁₃ 7₆₆₁ ₂₆₄ ₁₀₂ 99 245 *0₀₂₅ 9₀₇₄ 9₁₂₄ 8₁₇₃ 8₂₂₃ 7₂₇₂ 7₃₂₂ 6₃₇₁ 6₄₂₁ 5₄₇₀ ₂₅₄ ₁₀₀ B1 A1 100 99 98 97 96 95 94 93 92 91 n1 N1 C 00 50 01 52 04 56 09 62 16 70

(15)

Bojko’s table of quarter-squares (reconstruction, D. Roegel, 2013)

N n 10 11 12 13 14 15 16 17 18 19 A B 0 0 *0₀₀₀ 9₀₀₀ 8₀₀₀ 7₀₀₀ 6₀₀₀ 5₀₀₀ 4₀₀₀ 3₀₀₀ 2₀₀₀ 1₀₀₀ ₉₉₈ ₁₉₉ 2 0 *1₁₁₀ *0₁₁₁ 9₁₁₂ 8₁₁₃ 7₁₁₄ 6₁₁₅ 5₁₁₆ 4₁₁₇ 3₁₁₈ 2₁₁₉ ₉₇₈ ₁₉₇ 4 0 *4₄₂₀ *3₄₂₂ *2₄₂₄ *1₄₂₆ *0₄₂₈ 9₄₃₀ 8₄₃₂ 7₄₃₄ 6₄₃₆ 5₄₃₈ ₉₅₈ ₁₉₅ 6 0 9₉₃₀ 8₉₃₃ 7₉₃₆ 6₉₃₉ 5₉₄₂ 4₉₄₅ 3₉₄₈ 2₉₅₁ 1₉₅₄ 0₉₅₇ ₉₃₉ ₁₉₃ 8 1 *6₆₄₀ *5₆₄₄ *4₆₄₈ *3₆₅₂ *2₆₅₆ *1₆₆₀ *0₆₆₄ 9₆₆₈ 8₆₇₂ 7₆₇₆ ₉₁₉ ₁₉₁ 10 2 *5₅₅₀ *4₅₅₅ *3₅₆₀ *2₅₆₅ *1₅₇₀ *0₅₇₅ 9₅₈₀ 8₅₈₅ 7₅₉₀ 6₅₉₅ ₉₀₀ ₁₈₉ 12 3 *6₆₆₀ *5₆₆₆ *4₆₇₂ *3₆₇₈ *2₆₈₄ *1₆₉₀ *0₆₉₆ *0₇₀₂ 9₇₀₈ 8₇₁₄ ₈₈₁ ₁₈₇ 14 4 9₉₇₀ 8₉₇₇ 7₉₈₄ 6₉₉₁ 5₉₉₈ 5*₀₀₅ 4*₀₁₂ 3*₀₁₉ 2*₀₂₆ 1*₀₃₃ ₈₆₃ ₁₈₅ 16 6 *4₄₈₀ *3₄₈₈ *2₄₉₆ *2₅₀₄ *1₅₁₂ *0₅₂₀ 9₅₂₈ 8₅₃₆ 7₅₄₄ 6₅₅₂ ₈₄₄ ₁₈₃ 18 8 *1₁₉₀ *0₁₉₉ *0₂₀₈ 9₂₁₇ 8₂₂₆ 7₂₃₅ 6₂₄₄ 5₂₅₃ 4₂₆₂ 3₂₇₁ ₈₂₆ ₁₈₁ 20 10 *1₁₀₀ *0₁₁₀ 9₁₂₀ 8₁₃₀ 7₁₄₀ 6₁₅₀ 5₁₆₀ 4₁₇₀ 3₁₈₀ 2₁₉₀ ₈₀₈ ₁₇₉ 22 12 *2₂₁₀ *1₂₂₁ *0₂₃₂ 9₂₄₃ 8₂₅₄ 7₂₆₅ 6₂₇₆ 5₂₈₇ 4₂₉₈ 4₃₀₉ ₇₉₀ ₁₇₇ 24 14 *5₅₂₀ *4₅₃₂ *3₅₄₄ *2₅₅₆ *1₅₆₈ *0₅₈₀ 9₅₉₂ 9₆₀₄ 8₆₁₆ 7₆₂₈ ₇₇₂ ₁₇₅ 26 17 *0₀₃₀ 9₀₄₃ 8₀₅₆ 7₀₆₉ 6₀₈₂ 5₀₉₅ 5₁₀₈ 4₁₂₁ 3₁₃₄ 2₁₄₇ ₇₅₅ ₁₇₃ 28 19 *7₇₄₀ *6₇₅₄ *5₇₆₈ *4₇₈₂ *3₇₉₆ *3₈₁₀ *2₈₂₄ *1₈₃₈ *0₈₅₂ 9₈₆₆ ₇₃₇ ₁₇₁ 30 22 *6₆₅₀ *5₆₆₅ *4₆₈₀ *3₆₉₅ *3₇₁₀ *2₇₂₅ *1₇₄₀ *0₇₅₅ 9₇₇₀ 8₇₈₅ ₇₂₀ ₁₆₉ 32 25 7₇₆₀ 6₇₇₆ 5₇₉₂ 5₈₀₈ 4₈₂₄ 3₈₄₀ 2₈₅₆ 1₈₇₂ 0₈₈₈ 0₉₀₄ ₇₀₄ ₁₆₇ 34 29 *0₀₇₀ 9₀₈₇ 9₁₀₄ 8₁₂₁ 7₁₃₈ 6₁₅₅ 5₁₇₂ 4₁₈₉ 4₂₀₆ 3₂₂₃ ₆₈₇ ₁₆₅ 36 32 *5₅₈₀ *4₅₉₈ *4₆₁₆ *3₆₃₄ *2₆₅₂ *1₆₇₀ *0₆₈₈ *0₇₀₆ 9₇₂₄ 8₇₄₂ ₆₇₀ ₁₆₃ 38 36 *2₂₉₀ *2₃₀₉ *1₃₂₈ *0₃₄₇ 9₃₆₆ 8₃₈₅ 8₄₀₄ 7₄₂₃ 6₄₄₂ 5₄₆₁ ₆₅₄ ₁₆₁ 40 40 *2₂₀₀ *1₂₂₀ *0₂₄₀ 9₂₆₀ 8₂₈₀ 8₃₀₀ 7₃₂₀ 6₃₄₀ 5₃₆₀ 4₃₈₀ ₆₃₈ ₁₅₉ 42 44 *3₃₁₀ *2₃₃₁ *1₃₅₂ *0₃₇₃ 9₃₉₄ 9₄₁₅ 8₄₃₆ 7₄₅₇ 6₄₇₈ 5₄₉₉ ₆₂₂ ₁₅₇ 44 48 *6₆₂₀ *5₆₄₂ *4₆₆₄ *3₆₈₆ *3₇₀₈ *2₇₃₀ *1₇₅₂ *0₇₇₄ 9₇₉₆ 9₈₁₈ ₆₀₆ ₁₅₅ 46 53 *1₁₃₀ *0₁₅₃ 9₁₇₆ 8₁₉₉ 8₂₂₂ 7₂₄₅ 6₂₆₈ 5₂₉₁ 5₃₁₄ 4₃₃₇ ₅₉₁ ₁₅₃ 48 57 8₈₄₀ 7₈₆₄ 6₈₈₈ 6₉₁₂ 5₉₃₆ 4₉₆₀ 3₉₈₄ 3*₀₀₈ 2*₀₃₂ 1*₀₅₆ ₅₇₆ ₁₅₁ 50 62 7₇₅₀ 6₇₇₅ 6₈₀₀ 5₈₂₅ 4₈₅₀ 3₈₇₅ 3₉₀₀ 2₉₂₅ 1₉₅₀ 0₉₇₅ ₅₆₁ ₁₄₉ 52 67 8₈₆₀ 7₈₈₆ 7₉₁₂ 6₉₃₈ 5₉₆₄ 4₉₉₀ 4*₀₁₆ 3*₀₄₂ 2*₀₆₈ 1*₀₉₄ ₅₄₆ ₁₄₇ 54 73 *1₁₇₀ *0₁₉₇ *0₂₂₄ 9₂₅₁ 8₂₇₈ 8₃₀₅ 7₃₃₂ 6₃₅₉ 5₃₈₆ 5₄₁₃ ₅₃₁ ₁₄₅ 56 78 6₆₈₀ 6₇₀₈ 5₇₃₆ 4₇₆₄ 3₇₉₂ 3₈₂₀ 2₈₄₈ 1₈₇₆ 1₉₀₄ 0₉₃₂ ₅₁₇ ₁₄₃ 58 84 *3₃₉₀ *3₄₁₉ *2₄₄₈ *1₄₇₇ *1₅₀₆ *0₅₃₅ 9₅₆₄ 8₅₉₃ 8₆₂₂ 7₆₅₁ ₅₀₂ ₁₄₁ 60 90 *3₃₀₀ *2₃₃₀ *1₃₆₀ *0₃₉₀ *0₄₂₀ 9₄₅₀ 8₄₈₀ 8₅₁₀ 7₅₄₀ 6₅₇₀ ₄₈₈ ₁₃₉ 62 96 *4₄₁₀ *3₄₄₁ *2₄₇₂ *2₅₀₃ *1₅₃₄ *0₅₆₅ 9₅₉₆ 9₆₂₇ 8₆₅₈ 7₆₈₉ ₄₇₄ ₁₃₇ 64 102 7₇₂₀ 6₇₅₂ 5₇₈₄ 5₈₁₆ 4₈₄₈ 3₈₈₀ 3₉₁₂ 2₉₄₄ 1₉₇₆ 1*₀₀₈ ₄₆₁ ₁₃₅ 66 109 *2₂₃₀ *1₂₆₃ *0₂₉₆ *0₃₂₉ 9₃₆₂ 8₃₉₅ 8₄₂₈ 7₄₆₁ 6₄₉₄ 6₅₂₇ ₄₄₇ ₁₃₃ 68 115 9₉₄₀ 8₉₇₄ 8*₀₀₈ 7*₀₄₂ 6*₀₇₆ 6*₁₁₀ 5*₁₄₄ 4*₁₇₈ 4*₂₁₂ 3*₂₄₆ ₄₃₄ ₁₃₁ 70 122 8₈₅₀ 7₈₈₅ 7₉₂₀ 6₉₅₅ 5₉₉₀ 5*₀₂₅ 4*₀₆₀ 3*₀₉₅ 3*₁₃₀ 2*₁₆₅ ₄₂₁ ₁₂₉ 72 129 9₉₆₀ 8₉₉₆ 8*₀₃₂ 7*₀₆₈ 7*₁₀₄ 6*₁₄₀ 5*₁₇₆ 5*₂₁₂ 4*₂₄₈ 3*₂₈₄ ₄₀₈ ₁₂₇ 74 137 *2₂₇₀ *2₃₀₇ *1₃₄₄ *0₃₈₁ *0₄₁₈ 9₄₅₅ 8₄₉₂ 8₅₂₉ 7₅₆₆ 7₆₀₃ ₃₉₅ ₁₂₅ 76 144 7₇₈₀ 7₈₁₈ 6₈₅₆ 5₈₉₄ 5₉₃₂ 4₉₇₀ 4*₀₀₈ 3*₀₄₆ 2*₀₈₄ 2*₁₂₂ ₃₈₃ ₁₂₃ 78 152 *4₄₉₀ *4₅₂₉ *3₅₆₈ *3₆₀₇ *2₆₄₆ *1₆₈₅ *1₇₂₄ *0₇₆₃ *0₈₀₂ 9₈₄₁ ₃₇₀ ₁₂₁ 80 160 *4₄₀₀ *3₄₄₀ *2₄₈₀ *2₅₂₀ *1₅₆₀ *1₆₀₀ *0₆₄₀ 9₆₈₀ 9₇₂₀ 8₇₆₀ ₃₅₈ ₁₁₉ 82 168 *5₅₁₀ *4₅₅₁ *3₅₉₂ *3₆₃₃ *2₆₇₄ *2₇₁₅ *1₇₅₆ *0₇₉₇ *0₈₃₈ 9₈₇₉ ₃₄₆ ₁₁₇ 84 176 8₈₂₀ 7₈₆₂ 7₉₀₄ 6₉₄₆ 5₉₈₈ 5*₀₃₀ 4*₀₇₂ 4*₁₁₄ 3*₁₅₆ 2*₁₉₈ ₃₃₅ ₁₁₅ 86 185 *3₃₃₀ *2₃₇₃ *2₄₁₆ *1₄₅₉ *1₅₀₂ *0₅₄₅ 9₅₈₈ 9₆₃₁ 8₆₇₄ 8₇₁₇ ₃₂₃ ₁₁₃ 88 194 *0₀₄₀ 9₀₈₄ 9₁₂₈ 8₁₇₂ 8₂₁₆ 7₂₆₀ 7₃₀₄ 6₃₄₈ 5₃₉₂ 5₄₃₆ ₃₁₂ ₁₁₁ 90 202 9₉₅₀ 8₉₉₅ 8*₀₄₀ 7*₀₈₅ 7*₁₃₀ 6*₁₇₅ 6*₂₂₀ 5*₂₆₅ 5*₃₁₀ 4*₃₅₅ ₃₀₁ ₁₀₉ 92 212 *0₀₆₀ *0₁₀₆ 9₁₅₂ 8₁₉₈ 8₂₄₄ 7₂₉₀ 7₃₃₆ 6₃₈₂ 6₄₂₈ 5₄₇₄ ₂₉₀ ₁₀₇ 94 221 *3₃₇₀ *3₄₁₇ *2₄₆₄ *2₅₁₁ *1₅₅₈ *1₆₀₅ *0₆₅₂ 9₆₉₉ 9₇₄₆ 8₇₉₃ ₂₇₉ ₁₀₅ 96 230 8₈₈₀ 8₉₂₈ 7₉₇₆ 7*₀₂₄ 6*₀₇₂ 6*₁₂₀ 5*₁₆₈ 5*₂₁₆ 4*₂₆₄ 4*₃₁₂ ₂₆₉ ₁₀₃ 98 240 5₅₉₀ 5₆₃₉ 4₆₈₈ 4₇₃₇ 3₇₈₆ 3₈₃₅ 2₈₈₄ 2₉₃₃ 1₉₈₂ 1*₀₃₁ ₂₅₉ ₁₀₁ B1 A1 90 89 88 87 86 85 84 83 82 81 n1 N1 C 25 30 36 42 49 56 64 72 81 90

(16)

Bojko’s table of quarter-squares (reconstruction, D. Roegel, 2013)

N n 10 11 12 13 14 15 16 17 18 19 A B 1 0 *0₀₃₀ 9₀₃₀ 8₀₃₁ 7₀₃₁ 6₀₃₂ 5₀₃₃ 4₀₃₃ 3₀₃₄ 2₀₃₄ 1₀₃₅ ₉₈₈ ₁₉₈ 3 0 *2₂₄₀ *1₂₄₁ *0₂₄₃ 9₂₄₄ 8₂₄₆ 7₂₄₈ 6₂₄₉ 5₂₅₁ 4₂₅₂ 3₂₅₄ ₉₆₈ ₁₉₆ 5 0 *6₆₅₀ *5₆₅₂ *4₆₅₅ *3₆₅₇ *2₆₆₀ *1₆₆₃ *0₆₆₅ 9₆₆₈ 8₆₇₀ 7₆₇₃ ₉₄₈ ₁₉₄ 7 1 *2₂₆₀ *1₂₆₃ *0₂₆₇ 9₂₇₀ 8₂₇₄ 7₂₇₈ 6₂₈₁ 5₂₈₅ 4₂₈₈ 3₂₉₂ ₉₂₉ ₁₉₂ 9 2 *0₀₇₀ 9₀₇₄ 8₀₇₉ 7₀₈₃ 6₀₈₈ 5₀₉₃ 4₀₉₇ 4₁₀₂ 3₁₀₆ 2₁₁₁ ₉₁₀ ₁₉₀ 11 3 *0₀₈₀ 9₀₈₅ 8₀₉₁ 7₀₉₆ 7₁₀₂ 6₁₀₈ 5₁₁₃ 4₁₁₉ 3₁₂₄ 2₁₃₀ ₈₉₁ ₁₈₈ 13 4 *2₂₉₀ *1₂₉₆ *1₃₀₃ *0₃₀₉ 9₃₁₆ 8₃₂₃ 7₃₂₉ 6₃₃₆ 5₃₄₂ 4₃₄₉ ₈₇₂ ₁₈₆ 15 5 *7₇₀₀ *6₇₀₇ *5₇₁₅ *4₇₂₂ *3₇₃₀ *2₇₃₈ *1₇₄₅ *0₇₅₃ 9₇₆₀ 8₇₆₈ ₈₅₃ ₁₈₄ 17 7 *3₃₁₀ *2₃₁₈ *1₃₂₇ *0₃₃₅ 9₃₄₄ 8₃₅₃ 7₃₆₁ 6₃₇₀ 5₃₇₈ 4₃₈₇ ₈₃₅ ₁₈₂ 19 9 *1₁₂₀ *0₁₂₉ 9₁₃₉ 8₁₄₈ 7₁₅₈ 6₁₆₈ 5₁₇₇ 4₁₈₇ 3₁₉₆ 3₂₀₆ ₈₁₇ ₁₈₀ 21 11 *1₁₃₀ *0₁₄₀ 9₁₅₁ 8₁₆₁ 7₁₇₂ 6₁₈₃ 5₁₉₃ 5₂₀₄ 4₂₁₄ 3₂₂₅ ₇₉₉ ₁₇₈ 23 13 *3₃₄₀ *2₃₅₁ *1₃₆₃ *0₃₇₄ 9₃₈₆ 8₃₉₈ 8₄₀₉ 7₄₂₁ 6₄₃₂ 5₄₄₄ ₇₈₁ ₁₇₆ 25 15 *7₇₅₀ *6₇₆₂ *5₇₇₅ *4₇₈₇ *4₈₀₀ *3₈₁₃ *2₈₂₅ *1₈₃₈ *0₈₅₀ 9₈₆₃ ₇₆₃ ₁₇₄ 27 18 *3₃₆₀ *2₃₇₃ *1₃₈₇ *1₄₀₀ *0₄₁₄ 9₄₂₈ 8₄₄₁ 7₄₅₅ 6₄₆₈ 5₄₈₂ ₇₄₆ ₁₇₂ 29 21 *1₁₇₀ *0₁₈₄ 9₁₉₉ 9₂₁₃ 8₂₂₈ 7₂₄₃ 6₂₅₇ 5₂₇₂ 4₂₈₆ 4₃₀₁ ₇₂₉ ₁₇₀ 31 24 *1₁₈₀ *0₁₉₅ *0₂₁₁ 9₂₂₆ 8₂₄₂ 7₂₅₈ 6₂₇₃ 5₂₈₉ 5₃₀₄ 4₃₂₀ ₇₁₂ ₁₆₈ 33 27 *3₃₉₀ *3₄₀₆ *2₄₂₃ *1₄₃₉ *0₄₅₆ 9₄₇₃ 8₄₈₉ 8₅₀₆ 7₅₂₂ 6₅₃₉ ₆₉₅ ₁₆₆ 35 30 8₈₀₀ 7₈₁₇ 6₈₃₅ 5₈₅₂ 4₈₇₀ 3₈₈₈ 3₉₀₅ 2₉₂₃ 1₉₄₀ 0₉₅₈ ₆₇₉ ₁₆₄ 37 34 *4₄₁₀ *3₄₂₈ *2₄₄₇ *1₄₆₅ *0₄₈₄ *0₅₀₃ 9₅₂₁ 8₅₄₀ 7₅₅₈ 6₅₇₇ ₆₆₂ ₁₆₂ 39 38 *2₂₂₀ *1₂₃₉ *0₂₅₉ 9₂₇₈ 8₂₉₈ 8₃₁₈ 7₃₃₇ 6₃₅₇ 5₃₇₆ 4₃₉₆ ₆₄₆ ₁₆₀ 41 42 *2₂₃₀ *1₂₅₀ *0₂₇₁ 9₂₉₁ 9₃₁₂ 8₃₃₃ 7₃₅₃ 6₃₇₄ 5₃₉₄ 5₄₁₅ ₆₃₀ ₁₅₈ 43 46 *4₄₄₀ *3₄₆₁ *2₄₈₃ *2₅₀₄ *1₅₂₆ *0₅₄₈ 9₅₆₉ 8₅₉₁ 8₆₁₂ 7₆₃₄ ₆₁₄ ₁₅₆ 45 50 8₈₅₀ 7₈₇₂ 6₈₉₅ 6₉₁₇ 5₉₄₀ 4₉₆₃ 3₉₈₅ 3*₀₀₈ 2*₀₃₀ 1*₀₅₃ ₅₉₉ ₁₅₄ 47 55 *4₄₆₀ *3₄₈₃ *3₅₀₇ *2₅₃₀ *1₅₅₄ *0₅₇₈ *0₆₀₁ 9₆₂₅ 8₆₄₈ 7₆₇₂ ₅₈₃ ₁₅₂ 49 60 *2₂₇₀ *1₂₉₄ *1₃₁₉ *0₃₄₃ 9₃₆₈ 8₃₉₃ 8₄₁₇ 7₄₄₂ 6₄₆₆ 5₄₉₁ ₅₆₈ ₁₅₀ 51 65 *2₂₈₀ *2₃₀₅ *1₃₃₁ *0₃₅₆ 9₃₈₂ 9₄₀₈ 8₄₃₃ 7₄₅₉ 6₄₈₄ 6₅₁₀ ₅₅₃ ₁₄₈ 53 70 *4₄₉₀ *4₅₁₆ *3₅₄₃ *2₅₆₉ *1₅₉₆ *1₆₂₃ *0₆₄₉ 9₆₇₆ 9₇₀₂ 8₇₂₉ ₅₃₈ ₁₄₆ 55 75 9₉₀₀ 8₉₂₇ 7₉₅₅ 6₉₈₂ 6*₀₁₀ 5*₀₃₈ 4*₀₆₅ 3*₀₉₃ 3*₁₂₀ 2*₁₄₈ ₅₂₄ ₁₄₄ 57 81 *5₅₁₀ *4₅₃₈ *3₅₆₇ *2₅₉₅ *2₆₂₄ *1₆₅₃ *0₆₈₁ *0₇₁₀ 9₇₃₈ 8₇₆₇ ₅₀₉ ₁₄₂ 59 87 *3₃₂₀ *2₃₄₉ *1₃₇₉ *1₄₀₈ *0₄₃₈ 9₄₆₈ 8₄₉₇ 8₅₂₇ 7₅₅₆ 6₅₈₆ ₄₉₅ ₁₄₀ 61 93 *3₃₃₀ *2₃₆₀ *1₃₉₁ *1₄₂₁ *0₄₅₂ 9₄₈₃ 9₅₁₃ 8₅₄₄ 7₅₇₄ 7₆₀₅ ₄₈₁ ₁₃₈ 63 99 *5₅₄₀ *4₅₇₁ *4₆₀₃ *3₆₃₄ *2₆₆₆ *1₆₉₈ *1₇₂₉ *0₇₆₁ 9₇₉₂ 9₈₂₄ ₄₆₇ ₁₃₆ 65 105 9₉₅₀ 8₉₈₂ 8*₀₁₅ 7*₀₄₇ 6*₀₈₀ 6*₁₁₃ 5*₁₄₅ 4*₁₇₈ 4*₂₁₀ 3*₂₄₃ ₄₅₄ ₁₃₄ 67 112 *5₅₆₀ *4₅₉₃ *4₆₂₇ *3₆₆₀ *2₆₉₄ *2₇₂₈ *1₇₆₁ *0₇₉₅ *0₈₂₈ 9₈₆₂ ₄₄₀ ₁₃₂ 69 119 *3₃₇₀ *3₄₀₄ *2₄₃₉ *1₄₇₃ *1₅₀₈ *0₅₄₃ 9₅₇₇ 9₆₁₂ 8₆₄₆ 7₆₈₁ ₄₂₇ ₁₃₀ 71 126 *3₃₈₀ *3₄₁₅ *2₄₅₁ *1₄₈₆ *1₅₂₂ *0₅₅₈ 9₅₉₃ 9₆₂₉ 8₆₆₄ 8₇₀₀ ₄₁₄ ₁₂₈ 73 133 5₅₉₀ 5₆₂₆ 4₆₆₃ 3₆₉₉ 3₇₃₆ 2₇₇₃ 2₈₀₉ 1₈₄₆ 0₈₈₂ 0₉₁₉ ₄₀₂ ₁₂₆ 75 141 *0₀₀₀ 9₀₃₇ 8₀₇₅ 8₁₁₂ 7₁₅₀ 6₁₈₈ 6₂₂₅ 5₂₆₃ 5₃₀₀ 4₃₃₈ ₃₈₉ ₁₂₄ 77 148 6₆₁₀ 5₆₄₈ 4₆₈₇ 4₇₂₅ 3₇₆₄ 3₈₀₃ 2₈₄₁ 1₈₈₀ 1₉₁₈ 0₉₅₇ ₃₇₇ ₁₂₂ 79 156 *4₄₂₀ *3₄₅₉ *2₄₉₉ *2₅₃₈ *1₅₇₈ *1₆₁₈ *0₆₅₇ 9₆₉₇ 9₇₃₆ 8₇₇₆ ₃₆₄ ₁₂₀ 81 164 *4₄₃₀ *3₄₇₀ *3₅₁₁ *2₅₅₁ *1₅₉₂ *1₆₃₃ *0₆₇₃ *0₇₁₄ 9₇₅₄ 8₇₉₅ ₃₅₂ ₁₁₈ 83 172 6₆₄₀ 5₆₈₁ 5₇₂₃ 4₇₆₄ 4₈₀₆ 3₈₄₈ 2₈₈₉ 2₉₃₁ 1₉₇₂ 1*₀₁₄ ₃₄₁ ₁₁₆ 85 181 *0₀₅₀ 9₀₉₂ 9₁₃₅ 8₁₇₇ 8₂₂₀ 7₂₆₃ 7₃₀₅ 6₃₄₈ 5₃₉₀ 5₄₃₃ ₃₂₉ ₁₁₄ 87 189 6₆₆₀ 6₇₀₃ 5₇₄₇ 4₇₉₀ 4₈₃₄ 3₈₇₈ 3₉₂₁ 2₉₆₅ 2*₀₀₈ 1*₀₅₂ ₃₁₈ ₁₁₂ 89 198 *4₄₇₀ *4₅₁₄ *3₅₅₉ *3₆₀₃ *2₆₄₈ *1₆₉₃ *1₇₃₇ *0₇₈₂ *0₈₂₆ 9₈₇₁ ₃₀₆ ₁₁₀ 91 207 *4₄₈₀ *4₅₂₅ *3₅₇₁ *3₆₁₆ *2₆₆₂ *2₇₀₈ *1₇₅₃ *0₇₉₉ *0₈₄₄ 9₈₉₀ ₂₉₅ ₁₀₈ 93 216 6₆₉₀ 6₇₃₆ 5₇₈₃ 5₈₂₉ 4₈₇₆ 4₉₂₃ 3₉₆₉ 3*₀₁₆ 2*₀₆₂ 2*₁₀₉ ₂₈₅ ₁₀₆ 95 226 *1₁₀₀ *0₁₄₇ 9₁₉₅ 9₂₄₂ 8₂₉₀ 8₃₃₈ 7₃₈₅ 7₄₃₃ 6₄₈₀ 6₅₂₈ ₂₇₄ ₁₀₄ 97 235 7₇₁₀ 6₇₅₈ 6₈₀₇ 5₈₅₅ 5₉₀₄ 4₉₅₃ 4*₀₀₁ 3*₀₅₀ 2*₀₉₈ 2*₁₄₇ ₂₆₄ ₁₀₂ 99 245 5₅₂₀ 4₅₆₉ 4₆₁₉ 3₆₆₈ 3₇₁₈ 2₇₆₈ 2₈₁₇ 1₈₆₇ 1₉₁₆ 0₉₆₆ ₂₅₄ ₁₀₀ B1 A1 90 89 88 87 86 85 84 83 82 81 n1 N1 C 25 80 36 92 49 06 64 22 81 40

(17)

Bojko’s table of quarter-squares (reconstruction, D. Roegel, 2013)

N n 20 21 22 23 24 25 26 27 28 29 A B 0 0 *0₀₀₁ 9₀₀₁ 8₀₀₁ 7₀₀₁ 6₀₀₁ 5₀₀₁ 4₀₀₁ 3₀₀₁ 2₀₀₁ 1₀₀₂ ₉₉₇ ₁₉₉ 2 0 *1₁₂₁ *0₁₂₂ 9₁₂₃ 8₁₂₄ 7₁₂₅ 6₁₂₆ 5₁₂₇ 4₁₂₈ 3₁₂₉ 2₁₃₁ ₉₇₇ ₁₉₇ 4 0 *4₄₄₁ *3₄₄₃ *2₄₄₅ *1₄₄₇ *0₄₄₉ 9₄₅₁ 8₄₅₃ 7₄₅₅ 6₄₅₇ 5₄₆₀ ₉₅₇ ₁₉₅ 6 0 9₉₆₁ 8₉₆₄ 7₉₆₇ 6₉₇₀ 5₉₇₃ 4₉₇₆ 3₉₇₉ 2₉₈₂ 1₉₈₅ 0₉₈₉ ₉₃₈ ₁₉₃ 8 1 *6₆₈₁ *5₆₈₅ *4₆₈₉ *3₆₉₃ *2₆₉₇ *2₇₀₁ *1₇₀₅ *0₇₀₉ 9₇₁₃ 8₇₁₈ ₉₁₈ ₁₉₁ 10 2 *6₆₀₁ *5₆₀₆ *4₆₁₁ *3₆₁₆ *2₆₂₁ *1₆₂₆ *0₆₃₁ 9₆₃₆ 8₆₄₁ 7₆₄₇ ₈₉₉ ₁₈₉ 12 3 *7₇₂₁ *6₇₂₇ *5₇₃₃ *4₇₃₉ *3₇₄₅ *2₇₅₁ *1₇₅₇ *0₇₆₃ 9₇₆₉ 8₇₇₆ ₈₈₀ ₁₈₇ 14 5 *0₀₄₁ 9₀₄₈ 8₀₅₅ 7₀₆₂ 6₀₆₉ 5₀₇₆ 4₀₈₃ 3₀₉₀ 2₀₉₇ 2₁₀₅ ₈₆₂ ₁₈₅ 16 6 *5₅₆₁ *4₅₆₉ *3₅₇₇ *2₅₈₅ *1₅₉₃ *1₆₀₁ *0₆₀₉ 9₆₁₇ 8₆₂₅ 7₆₃₄ ₈₄₃ ₁₈₃ 18 8 *2₂₈₁ *1₂₉₀ *0₂₉₉ *0₃₀₈ 9₃₁₇ 8₃₂₆ 7₃₃₅ 6₃₄₄ 5₃₅₃ 4₃₆₃ ₈₂₅ ₁₈₁ 20 10 *2₂₀₁ *1₂₁₁ *0₂₂₁ 9₂₃₁ 8₂₄₁ 7₂₅₁ 6₂₆₁ 5₂₇₁ 4₂₈₁ 3₂₉₂ ₈₀₇ ₁₇₉ 22 12 *3₃₂₁ *2₃₃₂ *1₃₄₃ *0₃₅₄ 9₃₆₅ 8₃₇₆ 7₃₈₇ 6₃₉₈ 6₄₀₉ 5₄₂₁ ₇₈₉ ₁₇₇ 24 14 *6₆₄₁ *5₆₅₃ *4₆₆₅ *3₆₇₇ *2₆₈₉ *2₇₀₁ *1₇₁₃ *0₇₂₅ 9₇₃₇ 8₇₅₀ ₇₇₁ ₁₇₅ 26 17 *1₁₆₁ *0₁₇₄ 9₁₈₇ 9₂₀₀ 8₂₁₃ 7₂₂₆ 6₂₃₉ 5₂₅₂ 4₂₆₅ 3₂₇₉ ₇₅₄ ₁₇₃ 28 19 8₈₈₁ 7₈₉₅ 7₉₀₉ 6₉₂₃ 5₉₃₇ 4₉₅₁ 3₉₆₅ 2₉₇₉ 1₉₉₃ 1*₀₀₈ ₇₃₇ ₁₇₁ 30 22 8₈₀₁ 7₈₁₆ 6₈₃₁ 5₈₄₆ 4₈₆₁ 3₈₇₆ 2₈₉₁ 2₉₀₆ 1₉₂₁ 0₉₃₇ ₇₂₀ ₁₆₉ 32 25 9₉₂₁ 8₉₃₇ 7₉₅₃ 6₉₆₉ 5₉₈₅ 5*₀₀₁ 4*₀₁₇ 3*₀₃₃ 2*₀₄₉ 1*₀₆₆ ₇₀₃ ₁₆₇ 34 29 *2₂₄₁ *1₂₅₈ *0₂₇₅ 9₂₉₂ 9₃₀₉ 8₃₂₆ 7₃₄₃ 6₃₆₀ 5₃₇₇ 4₃₉₅ ₆₈₆ ₁₆₅ 36 32 7₇₆₁ 6₇₇₉ 5₇₉₇ 5₈₁₅ 4₈₃₃ 3₈₅₁ 2₈₆₉ 1₈₈₇ 1₉₀₅ 0₉₂₄ ₆₇₀ ₁₆₃ 38 36 *4₄₈₁ *4₅₀₀ *3₅₁₉ *2₅₃₈ *1₅₅₇ *0₅₇₆ 9₅₉₅ 9₆₁₄ 8₆₃₃ 7₆₅₃ ₆₅₃ ₁₆₁ 40 40 *4₄₀₁ *3₄₂₁ *2₄₄₁ *1₄₆₁ *0₄₈₁ *0₅₀₁ 9₅₂₁ 8₅₄₁ 7₅₆₁ 6₅₈₂ ₆₃₇ ₁₅₉ 42 44 *5₅₂₁ *4₅₄₂ *3₅₆₃ *2₅₈₄ *2₆₀₅ *1₆₂₆ *0₆₄₇ 9₆₆₈ 8₆₈₉ 8₇₁₁ ₆₂₁ ₁₅₇ 44 48 8₈₄₁ 7₈₆₃ 6₈₈₅ 6₉₀₇ 5₉₂₉ 4₉₅₁ 3₉₇₃ 2₉₉₅ 2*₀₁₇ 1*₀₄₀ ₆₀₆ ₁₅₅ 46 53 *3₃₆₁ *2₃₈₄ *2₄₀₇ *1₄₃₀ *0₄₅₃ 9₄₇₆ 8₄₉₉ 8₅₂₂ 7₅₄₅ 6₅₆₉ ₅₉₀ ₁₅₃ 48 58 *0₀₈₁ *0₁₀₅ 9₁₂₉ 8₁₅₃ 7₁₇₇ 7₂₀₁ 6₂₂₅ 5₂₄₉ 4₂₇₃ 3₂₉₈ ₅₇₅ ₁₅₁ 50 63 *0₀₀₁ 9₀₂₆ 8₀₅₁ 7₀₇₆ 7₁₀₁ 6₁₂₆ 5₁₅₁ 4₁₇₆ 4₂₀₁ 3₂₂₇ ₅₆₀ ₁₄₉ 52 68 *1₁₂₁ *0₁₄₇ 9₁₇₃ 8₁₉₉ 8₂₂₅ 7₂₅₁ 6₂₇₇ 6₃₀₃ 5₃₂₉ 4₃₅₆ ₅₄₅ ₁₄₇ 54 73 *4₄₄₁ *3₄₆₈ *2₄₉₅ *2₅₂₂ *1₅₄₉ *0₅₇₆ *0₆₀₃ 9₆₃₀ 8₆₅₇ 7₆₈₅ ₅₃₀ ₁₄₅ 56 78 9₉₆₁ 8₉₈₉ 8*₀₁₇ 7*₀₄₅ 6*₀₇₃ 6*₁₀₁ 5*₁₂₉ 4*₁₅₇ 3*₁₈₅ 3*₂₁₄ ₅₁₆ ₁₄₃ 58 84 6₆₈₁ 6₇₁₀ 5₇₃₉ 4₇₆₈ 3₇₉₇ 3₈₂₆ 2₈₅₅ 1₈₈₄ 1₉₁₃ 0₉₄₃ ₅₀₂ ₁₄₁ 60 90 *6₆₀₁ *5₆₃₁ *4₆₆₁ *3₆₉₁ *3₇₂₁ *2₇₅₁ *1₇₈₁ *1₈₁₁ *0₈₄₁ 9₈₇₂ ₄₈₇ ₁₃₉ 62 96 7₇₂₁ 6₇₅₂ 5₇₈₃ 5₈₁₄ 4₈₄₅ 3₈₇₆ 3₉₀₇ 2₉₃₈ 1₉₆₉ 1*₀₀₁ ₄₇₄ ₁₃₇ 64 103 *0₀₄₁ 9₀₇₃ 9₁₀₅ 8₁₃₇ 7₁₆₉ 7₂₀₁ 6₂₃₃ 5₂₆₅ 4₂₉₇ 4₃₃₀ ₄₆₀ ₁₃₅ 66 109 *5₅₆₁ *4₅₉₄ *4₆₂₇ *3₆₆₀ *2₆₉₃ *2₇₂₆ *1₇₅₉ *0₇₉₂ *0₈₂₅ 9₈₅₉ ₄₄₆ ₁₃₃ 68 116 *2₂₈₁ *2₃₁₅ *1₃₄₉ *0₃₈₃ *0₄₁₇ 9₄₅₁ 8₄₈₅ 8₅₁₉ 7₅₅₃ 6₅₈₈ ₄₃₃ ₁₃₁ 70 123 *2₂₀₁ *1₂₃₆ *0₂₇₁ *0₃₀₆ 9₃₄₁ 8₃₇₆ 8₄₁₁ 7₄₄₆ 6₄₈₁ 6₅₁₇ ₄₂₀ ₁₂₉ 72 130 *3₃₂₁ *2₃₅₇ *1₃₉₃ *1₄₂₉ *0₄₆₅ *0₅₀₁ 9₅₃₇ 8₅₇₃ 8₆₀₉ 7₆₄₆ ₄₀₇ ₁₂₇ 74 137 6₆₄₁ 5₆₇₈ 5₇₁₅ 4₇₅₂ 3₇₈₉ 3₈₂₆ 2₈₆₃ 2₉₀₀ 1₉₃₇ 0₉₇₅ ₃₉₅ ₁₂₅ 76 145 *1₁₆₁ *0₁₉₉ *0₂₃₇ 9₂₇₅ 9₃₁₃ 8₃₅₁ 7₃₈₉ 7₄₂₇ 6₄₆₅ 6₅₀₄ ₃₈₂ ₁₂₃ 78 152 8₈₈₁ 8₉₂₀ 7₉₅₉ 6₉₉₈ 6*₀₃₇ 5*₀₇₆ 5*₁₁₅ 4*₁₅₄ 3*₁₉₃ 3*₂₃₃ ₃₇₀ ₁₂₁ 80 160 8₈₀₁ 7₈₄₁ 6₈₈₁ 6₉₂₁ 5₉₆₁ 5*₀₀₁ 4*₀₄₁ 3*₀₈₁ 3*₁₂₁ 2*₁₆₂ ₃₅₈ ₁₁₉ 82 168 9₉₂₁ 8₉₆₂ 8*₀₀₃ 7*₀₄₄ 6*₀₈₅ 6*₁₂₆ 5*₁₆₇ 5*₂₀₈ 4*₂₄₉ 3*₂₉₁ ₃₄₆ ₁₁₇ 84 177 *2₂₄₁ *1₂₈₃ *1₃₂₅ *0₃₆₇ *0₄₀₉ 9₄₅₁ 8₄₉₃ 8₅₃₅ 7₅₇₇ 7₆₂₀ ₃₃₄ ₁₁₅ 86 185 7₇₆₁ 7₈₀₄ 6₈₄₇ 5₈₉₀ 5₉₃₃ 4₉₇₆ 4*₀₁₉ 3*₀₆₂ 3*₁₀₅ 2*₁₄₉ ₃₂₃ ₁₁₃ 88 194 *4₄₈₁ *4₅₂₅ *3₅₆₉ *3₆₁₃ *2₆₅₇ *2₇₀₁ *1₇₄₅ *0₇₈₉ *0₈₃₃ 9₈₇₈ ₃₁₁ ₁₁₁ 90 203 *4₄₀₁ *3₄₄₆ *2₄₉₁ *2₅₃₆ *1₅₈₁ *1₆₂₆ *0₆₇₁ *0₇₁₆ 9₇₆₁ 9₈₀₇ ₃₀₀ ₁₀₉ 92 212 5₅₂₁ 4₅₆₇ 4₆₁₃ 3₆₅₉ 3₇₀₅ 2₇₅₁ 1₇₉₇ 1₈₄₃ 0₈₈₉ 0₉₃₆ ₂₉₀ ₁₀₇ 94 221 8₈₄₁ 7₈₈₈ 7₉₃₅ 6₉₈₂ 6*₀₂₉ 5*₀₇₆ 5*₁₂₃ 4*₁₇₀ 4*₂₁₇ 3*₂₆₅ ₂₇₉ ₁₀₅ 96 231 *3₃₆₁ *3₄₀₉ *2₄₅₇ *2₅₀₅ *1₅₅₃ *1₆₀₁ *0₆₄₉ 9₆₉₇ 9₇₄₅ 8₇₉₄ ₂₆₈ ₁₀₃ 98 241 *0₀₈₁ *0₁₃₀ 9₁₇₉ 9₂₂₈ 8₂₇₇ 8₃₂₆ 7₃₇₅ 7₄₂₄ 6₄₇₃ 6₅₂₃ ₂₅₈ ₁₀₁ B1 A1 80 79 78 77 76 75 74 73 72 71 n1 N1 C 00 10 21 32 44 56 69 82 96 10