164:, who once upon a time grazed on the fields of the Thrinacian isle of Sicily, divided into four herds of different colours, one milk white, another a glossy black, a third yellow and the last dappled. In each herd were bulls, mighty in number according to these proportions: Understand, stranger, that the white bulls were equal to a half and a third of the black together with the whole of the yellow, while the black were equal to the fourth part of the dappled and a fifth, together with, once more, the whole of the yellow. Observe further that the remaining bulls, the dappled, were equal to a sixth part of the white and a seventh, together with all of the yellow. These were the proportions of the cows: The white were precisely equal to the third part and a fourth of the whole herd of the black; while the black were equal to the fourth part once more of the dappled and with it a fifth part, when all, including the bulls, went to pasture together. Now the dappled in four parts were equal in number to a fifth part and a sixth of the yellow herd. Finally the yellow were in number equal to a sixth part and a seventh of the white herd. If thou canst accurately tell, O stranger, the number of cattle of the Sun, giving separately the number of well-fed bulls and again the number of females according to each colour, thou wouldst not be called unskilled or ignorant of numbers, but not yet shalt thou be numbered among the wise.
168:
the dappled bulls were gathered into one herd they stood in such a manner that their number, beginning from one, grew slowly greater till it completed a triangular figure, there being no bulls of other colours in their midst nor none of them lacking. If thou art able, O stranger, to find out all these things and gather them together in your mind, giving all the relations, thou shalt depart crowned with glory and knowing that thou hast been adjudged perfect in this species of wisdom.
2672:
20:
892:
2788:
589:
604:
297:
167:
But come, understand also all these conditions regarding the cattle of the Sun. When the white bulls mingled their number with the black, they stood firm, equal in depth and breadth, and the plains of
Thrinacia, stretching far in all ways, were filled with their multitude. Again, when the yellow and
144:, Germany, which contained many Greek and Latin manuscripts. A few years later, Lessing published translations of some of the manuscripts with commentaries. Among them was a Greek poem of forty-four lines, containing an arithmetical problem which asks the reader to find the number of cattle in
887:{\displaystyle {\begin{aligned}B&=7\,460\,514=4657\times 1602,\\W&=10\,366\,482=4657\times 2226,\\D&=7\,358\,060=4657\times 1580,\\Y&=4\,149\,387=4657\times 891,\\b&=4\,893\,246,\\w&=7\,206\,360,\\d&=3\,515\,820,\\y&=5\,439\,213,\end{aligned}}}
1982:" (For, as said , the problem shall, if it had not been composed by Archimedes himself, yet have been recognized by him worthy that he would have sent it to Eratosthenes , in order to submit it to the surveyor at Alexandria for solution. The title says this;
1973:
Denn, wie gesagt, das
Problem soll, wenn es nicht von dem Archimedes selbst abgefaßt worden, doch von ihm für werth erkannt seyn, daß er es den Eratosthenes geschicket hätte, um es den Meßkünstern zu Alexandria zur Auflösung vorzulegen. Dieses besagt die
1202:
584:{\displaystyle {\begin{aligned}W&={\frac {5}{6}}B+Y,\\B&={\frac {9}{20}}D+Y,\\D&={\frac {13}{42}}W+Y,\\w&={\frac {7}{12}}(B+b),\\b&={\frac {9}{20}}(D+d),\\d&={\frac {11}{30}}(Y+y),\\y&={\frac {13}{42}}(W+w),\end{aligned}}}
1591:
1095:
1889:
1760:
609:
302:
1276:
1678:
1381:
1425:
1110:
85:
The problem remained unsolved for a number of years, due partly to the difficulty of computing the huge numbers involved in the solution. The general solution was found in 1880 by
1938:
269:
214:
1311:
983:
953:
1912:
289:
234:
2154:
1473:
2181:
2732:
1003:
2565:
989:. The general solution to this part of the problem was first found by A. Amthor in 1880. The following version of it was described by
2320:
2652:
1997:
2063:
The problem was solved independently in 1895 by Adam Henry Bell, a surveyor and civil engineer of
Hillsboro, Illinois, USA. See:
1798:
2276:
2114:
1693:
2812:
2469:
125:
2827:
2405:
2822:
2603:
1427:(first solved by Amthor). Modern computers can easily print out all digits of the answer. This was first done at the
1217:
2617:
2610:
178:
2503:
sequence A096151 (Decimal expansion of the 206545-digit integer solution to
Archimedes's cattle problem)
2206:
1386:
The size of the smallest herd that could satisfy both the first and second parts of the problem is then given by
1197:{\displaystyle w=300\,426\,607\,914\,281\,713\,365{\sqrt {609}}+84\,129\,507\,677\,858\,393\,258{\sqrt {7766}},}
2631:
2558:
2081:
1626:
1322:
137:
67:
2696:
1393:
2460:
Benson, G. (2014). "Archimedes the Poet: Generic
Innovation and Mathematical Fantasy in the Cattle Problem".
2817:
2722:
2589:
2214:
1960:
Zur
Geschichte und Litteratur: aus den Schätzen der Herzoglichen Bibliothek zu Wolfenbüttel, Zweyter Beytrag
2316:
1428:
2301:, Historisch-literarische Abteilung der Zeitschrift für Mathematik und Physik 25 (1880) 121–136, 153–171.
2727:
1958:
71:
2701:
2185:
2250:
595:
145:
47:
2791:
2773:
2763:
2551:
121:
86:
51:
2671:
1965:
On
History and Literature: from the treasures of the ducal library at WolfenbĂĽttel, second article
1917:
598:
and has infinitely many solutions. The least positive integers satisfying the seven equations are
2768:
2706:
2638:
2528:
2481:
2448:
2419:
2384:
2364:
105:
994:
2099:
1967:] (in German and Greek). Braunschweig, (Germany): Fürstlicher Waysenhaus. pp. 421–425.
2758:
2737:
2686:
2596:
2051:. Vol. 1. New York, New York, USA: American Encyclopedic Library Association. p. 18.
1617:
986:
109:
2312:
2044:
2028:
23:
Smallest solution to
Archimedes's cattle problem with each icon representing around 10 cattle
2473:
2440:
2409:
2356:
1432:
239:
184:
161:
2167:
1284:
19:
2163:
2142:
1894:
Amthor's approach discussed in the previous section was essentially to find the smallest
962:
932:
141:
75:
2659:
1897:
274:
219:
2806:
2485:
2236:
2228:
2146:
1790:
1464:
1452:
1451:
The constraints of the second part of the problem are straightforward and the actual
1314:
997:: the solution given above for the first part of the problem should be multiplied by
990:
956:
907:
cattle, and the other solutions are integral multiples of these. Note that given the
112:, he calculated the first digits of the smallest solution, showing that it is about
1782:
908:
91:
1940:. The fundamental solution of this equation has more than 100,000 decimal digits.
2106:. Vol. 2. Chicago, Illinois, USA: Munsell Publishing Co. pp. 1049–1050.
2645:
2524:
2509:
1435:, R. A. German, and Charles Robert Zarnke. They used a combination of the
2624:
2574:
2513:
59:
1586:{\displaystyle B+W=7\,460\,514\,k+10\,366\,482\,k=(2^{2})(3)(11)(29)(4657)k,}
2477:
236:, and the number of white, black, dappled, and yellow cows are written as
181:. If the number of white, black, dappled, and yellow bulls are written as
2004:
Zeitschrift fĂĽr
Mathematik und Physik: Historisch-literarische Abtheilung
1440:
1436:
2452:
2423:
2368:
2067:
Bell, A.H. (1895). "On the celebrated 'Cattle
Problem' of Archimedes".
2037:] (in German). Dresden, Germany: K. Blochmann und Sohn. p. 31.
79:
55:
2347:
Bell, A. H. (1895), "The "Cattle Problem." By Archimedies 251 B. C.",
1090:{\displaystyle n={\frac {(w^{4658j}-w^{-4658j})^{2}}{(4657)(79072)}},}
2753:
1455:
that needs to be solved can easily be given. First, it one asks that
63:
2444:
2414:
2393:
2360:
1608:. That solves the first condition. For the second, it requires that
177:
The first part of the problem can be solved readily by setting up a
124:. The decimal form is too long for humans to calculate exactly, but
70:
in a Greek manuscript containing a poem of forty-four lines, in the
62:, the problem involves computing the number of cattle in a herd of
2499:
160:
If thou art diligent and wise, O stranger, compute the number of
594:
which is a system of seven equations with eight unknowns. It is
156:
The problem, as translated into English by Ivor Thomas, states:
66:
from a given set of restrictions. The problem was discovered by
2547:
2543:
2502:
1884:{\displaystyle p^{2}-(2^{2})(609)(7766)(4657^{2})q^{2}=1.}
1755:{\displaystyle t={\frac {-1\pm {\sqrt {1+8(D+Y)}}}{2}}.}
140:
was appointed librarian of the Herzog August Library in
2392:
Williams, H. C.; German, R. A.; Zarnke, C. R. (1965).
2043:
Singer, Isadore; de Leon, Edward Warren, eds. (1910).
2035:
Program of the Gymnasium of the Holy Cross in Dresden
2030:
Programm des Gymnasiums zum Heiligen Kreuz in Dresden
1920:
1900:
1801:
1696:
1629:
1476:
1396:
1325:
1287:
1220:
1113:
1006:
965:
935:
607:
300:
277:
242:
222:
187:
2313:"Unbundling Computing at The University of Waterloo"
914:= 4657 then the first four numbers are multiples of
2746:
2715:
2679:
2581:
2439:(4). Mathematical Association of America: 305–319.
2355:(5), Mathematical Association of America: 140–141,
128:packages on computers can write it out explicitly.
1932:
1906:
1883:
1754:
1672:
1585:
1419:
1375:
1305:
1270:
1196:
1089:
977:
947:
886:
583:
283:
263:
228:
208:
2505:—Complete decimal solution to the second problem
2431:Vardi, I. (1998). "Archimedes' Cattle Problem".
1271:{\displaystyle w^{2}=u+v{\sqrt {(609)(7766)}},}
2394:"Solution of the Cattle Problem of Archimedes"
2022:Biographical information about August Amthor:
291:, the problem is simply to find a solution to
148:. It is now generally credited to Archimedes.
108:of the Holy Cross) in Dresden, Germany. Using
2559:
2311:Harold Alkema and Kenneth McLaughlin (2007).
2137:
2135:
8:
2381:100 Great Problems of Elementary Mathematics
2155:Notices of the American Mathematical Society
2082:"The 'Cattle Problem' by Archimedes 251 B.C"
2002:
2000:[The cattle problem of Archimedes].
1971:
98:
2098:Bateman, Newton; Selby, Paul, eds. (1918).
929:The second part of the problem states that
40:
32:
2566:
2552:
2544:
2233:Number Theory, an Approach Through History
2413:
2041:Brief biography about Amthor appears in:
1919:
1899:
1869:
1856:
1822:
1806:
1800:
1715:
1703:
1695:
1673:{\displaystyle D+Y={\frac {t^{2}+t}{2}}.}
1649:
1642:
1628:
1532:
1518:
1514:
1510:
1500:
1496:
1492:
1475:
1411:
1407:
1395:
1376:{\displaystyle u^{2}-(609)(7766)v^{2}=1.}
1361:
1330:
1324:
1286:
1243:
1225:
1219:
1184:
1180:
1176:
1172:
1168:
1164:
1160:
1147:
1143:
1139:
1135:
1131:
1127:
1123:
1112:
1055:
1039:
1023:
1013:
1005:
964:
934:
873:
869:
845:
841:
817:
813:
789:
785:
749:
745:
709:
705:
669:
665:
629:
625:
608:
606:
549:
507:
465:
423:
387:
351:
315:
301:
299:
276:
241:
221:
186:
2251:"The Cattle Problem Statement (English)"
2182:"Archimedes' Cattle Problem (Statement)"
1914:such that it is integrally divisible by
1420:{\displaystyle 7.76\times 10^{206\,544}}
18:
1949:
120:cattle, far more than could fit in the
2375:Dörrie, Heinrich (1965). "Archimedes'
1785:of this quadratic is a perfect square
2733:List of things named after Archimedes
2275:Merriman, Mansfield (November 1905).
2113:Merriman, Mansfield (November 1905).
1998:"Das Problema bovinum des Archimedes"
16:Mathematical problem in number theory
7:
2049:International Insurance Encyclopedia
2299:Das Problema Bovinum des Archimedes
2104:Historical Encyclopedia of Illinois
1996:Krumbiegel, B.; Amthor, A. (1880).
1467:, or using the values given above,
1313:is the fundamental solution of the
2277:"The cattle problem of Archimedes"
2115:"The cattle problem of Archimedes"
1986:...) See pages 423–424 (in Greek).
1957:Lessing, Gotthold Ephraim (1773).
14:
2653:The Method of Mechanical Theorems
2349:The American Mathematical Monthly
2323:from the original on 4 April 2011
2787:
2786:
2670:
2027:(School administration) (1876).
2531:from the original on 2021-12-19
2025:Amthor's full name appears in:
2008:(in German, Greek, and Latin).
96:(1845–1916), headmaster of the
2514:"Holy Cow that's a Big Number"
2470:Johns Hopkins University Press
2111:Bell's occupations appear in:
1862:
1849:
1846:
1840:
1837:
1831:
1828:
1815:
1738:
1726:
1574:
1568:
1565:
1559:
1556:
1550:
1547:
1541:
1538:
1525:
1354:
1348:
1345:
1339:
1300:
1288:
1260:
1254:
1251:
1245:
1078:
1072:
1069:
1063:
1052:
1016:
926:will appear repeatedly below.
571:
559:
529:
517:
487:
475:
445:
433:
146:the herd of the god of the sun
1:
2433:American Mathematical Monthly
2406:American Mathematical Society
2096:Bell's full name appears in:
2086:American Mathematical Monthly
126:multiple-precision arithmetic
2604:On the Equilibrium of Planes
1933:{\displaystyle 2\times 4657}
1104:is any positive integer and
100:Gymnasium zum Heiligen Kreuz
2692:Archimedes's cattle problem
2147:"Solving the Pell Equation"
28:Archimedes's cattle problem
2844:
2680:Discoveries and inventions
2618:On the Sphere and Cylinder
2611:Quadrature of the Parabola
2398:Mathematics of Computation
2297:B. Krumbiegel, A. Amthor,
1765:Substituting the value of
2782:
2668:
2069:The Mathematical Magazine
58:solutions. Attributed to
2632:On Conoids and Spheroids
2045:"Amthor, August (Ph.D.)"
138:Gotthold Ephraim Lessing
87:Carl Ernst August Amthor
68:Gotthold Ephraim Lessing
2590:Measurement of a Circle
2281:Popular Science Monthly
2215:Oxford University Press
2119:Popular Science Monthly
1777:and finding a value of
1207:Equivalently, squaring
2317:University of Waterloo
2003:
1972:
1934:
1908:
1885:
1756:
1674:
1587:
1429:University of Waterloo
1421:
1377:
1307:
1272:
1198:
1091:
979:
949:
888:
585:
285:
265:
264:{\displaystyle w,b,d,}
230:
210:
209:{\displaystyle W,B,D,}
170:
99:
41:
33:
24:
2813:Diophantine equations
2728:Archimedes Palimpsest
2697:Archimedes' principle
2478:10.1353/are.2014.0008
1935:
1909:
1886:
1757:
1675:
1588:
1433:Hugh C. Williams
1422:
1378:
1308:
1306:{\displaystyle (u,v)}
1273:
1199:
1092:
980:
950:
889:
586:
286:
266:
231:
211:
158:
72:Herzog August Library
22:
2828:Cattle in literature
2723:Archimedes' heat ray
2211:Ptolemaic Alexandria
1918:
1898:
1799:
1789:entails solving the
1694:
1627:
1596:thus one should set
1474:
1394:
1323:
1285:
1218:
1111:
1004:
963:
933:
897:which is a total of
605:
298:
275:
240:
220:
185:
52:polynomial equations
48:Diophantine analysis
2823:Works by Archimedes
2774:Eutocius of Ascalon
2764:Apollonius of Perga
2333:(includes pictures)
2255:New York University
2080:Bell, A.H. (1895).
2012:: 121–136, 153–171.
1969:From pp. 422–423: "
1600:= (3)(11)(29)(4657)
978:{\displaystyle Y+D}
948:{\displaystyle W+B}
179:system of equations
122:observable universe
42:problema Archimedis
2769:Hero of Alexandria
2707:Claw of Archimedes
2702:Archimedes's screw
2639:On Floating Bodies
2385:Dover Publications
2188:on 24 January 2007
2143:Lenstra, H. W. Jr.
1930:
1904:
1881:
1752:
1670:
1583:
1417:
1373:
1303:
1268:
1194:
1087:
991:H. W. Lenstra
975:
945:
884:
882:
581:
579:
281:
261:
226:
206:
110:logarithmic tables
46:) is a problem in
25:
2800:
2799:
2759:Eudoxus of Cnidus
2738:Pseudo-Archimedes
2687:Archimedean solid
2597:The Sand Reckoner
2100:"Fish, Albert E."
1907:{\displaystyle v}
1747:
1741:
1665:
1618:triangular number
1604:for some integer
1390:= 1 and is about
1263:
1189:
1152:
1082:
987:triangular number
557:
515:
473:
431:
395:
359:
323:
284:{\displaystyle y}
229:{\displaystyle Y}
162:cattle of the Sun
2835:
2790:
2789:
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2554:
2545:
2540:
2538:
2536:
2518:
2501:
2489:
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2427:
2417:
2388:
2377:Problema Bovinum
2371:
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2247:
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2240:
2225:
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2218:
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2197:
2196:
2194:
2193:
2184:. Archived from
2177:
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2151:
2139:
2130:
2126:
2107:
2093:
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2055:
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1061:
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1049:
1031:
1030:
1014:
984:
982:
981:
976:
954:
952:
951:
946:
906:
905:
902:
893:
891:
890:
885:
883:
590:
588:
587:
582:
580:
558:
550:
516:
508:
474:
466:
432:
424:
396:
388:
360:
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324:
316:
290:
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235:
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215:
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207:
119:
117:
102:
95:
44:
36:
34:problema bovinum
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2842:
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2834:
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2803:
2802:
2801:
2796:
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2742:
2711:
2675:
2666:
2577:
2572:
2534:
2532:
2516:
2508:
2496:
2459:
2445:10.2307/2589706
2430:
2415:10.2307/2003954
2391:
2387:. pp. 3–7.
2374:
2361:10.2307/2968125
2346:
2343:
2341:Further reading
2338:
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2191:
2189:
2180:Rorres, Chris.
2179:
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2017:
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1001:
995:Pell's equation
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182:
175:
154:
134:
115:
113:
89:
50:, the study of
17:
12:
11:
5:
2841:
2839:
2831:
2830:
2825:
2820:
2818:Large integers
2815:
2805:
2804:
2798:
2797:
2795:
2794:
2783:
2780:
2779:
2777:
2776:
2771:
2766:
2761:
2756:
2750:
2748:
2747:Related people
2744:
2743:
2741:
2740:
2735:
2730:
2725:
2719:
2717:
2713:
2712:
2710:
2709:
2704:
2699:
2694:
2689:
2683:
2681:
2677:
2676:
2669:
2667:
2665:
2664:
2660:Book of Lemmas
2656:
2649:
2642:
2635:
2628:
2621:
2614:
2607:
2600:
2593:
2585:
2583:
2579:
2578:
2573:
2571:
2570:
2563:
2556:
2548:
2542:
2541:
2506:
2495:
2494:External links
2492:
2491:
2490:
2457:
2428:
2389:
2372:
2342:
2339:
2336:
2335:
2303:
2290:
2267:
2242:
2220:
2198:
2172:
2162:(2): 182–192,
2131:
2129:
2128:
2109:
2108:; see p. 1050.
2094:
2077:
2056:
2054:
2053:
2039:
2015:
1988:
1948:
1947:
1945:
1942:
1929:
1926:
1923:
1903:
1892:
1891:
1880:
1877:
1872:
1868:
1864:
1859:
1855:
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1842:
1839:
1836:
1833:
1830:
1825:
1821:
1817:
1814:
1809:
1805:
1781:such that the
1763:
1762:
1751:
1746:
1740:
1737:
1734:
1731:
1728:
1725:
1722:
1719:
1714:
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1594:
1593:
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1506:
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1356:
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1347:
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1329:
1302:
1299:
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1290:
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1262:
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1247:
1242:
1239:
1236:
1233:
1228:
1224:
1205:
1204:
1193:
1188:
1183:
1179:
1175:
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1167:
1163:
1159:
1156:
1151:
1146:
1142:
1138:
1134:
1130:
1126:
1122:
1119:
1116:
1098:
1097:
1086:
1080:
1077:
1074:
1071:
1068:
1065:
1058:
1054:
1048:
1045:
1042:
1038:
1034:
1029:
1026:
1022:
1018:
1012:
1009:
974:
971:
968:
944:
941:
938:
895:
894:
879:
876:
872:
868:
865:
862:
860:
858:
855:
854:
851:
848:
844:
840:
837:
834:
832:
830:
827:
826:
823:
820:
816:
812:
809:
806:
804:
802:
799:
798:
795:
792:
788:
784:
781:
778:
776:
774:
771:
770:
767:
764:
761:
758:
755:
752:
748:
744:
741:
738:
736:
734:
731:
730:
727:
724:
721:
718:
715:
712:
708:
704:
701:
698:
696:
694:
691:
690:
687:
684:
681:
678:
675:
672:
668:
664:
661:
658:
656:
654:
651:
650:
647:
644:
641:
638:
635:
632:
628:
624:
621:
618:
616:
614:
611:
610:
592:
591:
576:
573:
570:
567:
564:
561:
556:
553:
548:
545:
543:
541:
538:
537:
534:
531:
528:
525:
522:
519:
514:
511:
506:
503:
501:
499:
496:
495:
492:
489:
486:
483:
480:
477:
472:
469:
464:
461:
459:
457:
454:
453:
450:
447:
444:
441:
438:
435:
430:
427:
422:
419:
417:
415:
412:
411:
408:
405:
402:
399:
394:
391:
386:
383:
381:
379:
376:
375:
372:
369:
366:
363:
358:
355:
350:
347:
345:
343:
340:
339:
336:
333:
330:
327:
322:
319:
314:
311:
309:
307:
304:
303:
280:
260:
257:
254:
251:
248:
245:
225:
205:
202:
199:
196:
193:
190:
174:
171:
153:
150:
133:
130:
15:
13:
10:
9:
6:
4:
3:
2:
2840:
2829:
2826:
2824:
2821:
2819:
2816:
2814:
2811:
2810:
2808:
2793:
2785:
2784:
2781:
2775:
2772:
2770:
2767:
2765:
2762:
2760:
2757:
2755:
2752:
2751:
2749:
2745:
2739:
2736:
2734:
2731:
2729:
2726:
2724:
2721:
2720:
2718:
2716:Miscellaneous
2714:
2708:
2705:
2703:
2700:
2698:
2695:
2693:
2690:
2688:
2685:
2684:
2682:
2678:
2673:
2662:
2661:
2657:
2655:
2654:
2650:
2648:
2647:
2643:
2641:
2640:
2636:
2634:
2633:
2629:
2627:
2626:
2622:
2620:
2619:
2615:
2613:
2612:
2608:
2606:
2605:
2601:
2599:
2598:
2594:
2592:
2591:
2587:
2586:
2584:
2582:Written works
2580:
2576:
2569:
2564:
2562:
2557:
2555:
2550:
2549:
2546:
2530:
2526:
2522:
2515:
2511:
2507:
2504:
2498:
2497:
2493:
2487:
2483:
2479:
2475:
2471:
2467:
2463:
2458:
2454:
2450:
2446:
2442:
2438:
2434:
2429:
2425:
2421:
2416:
2411:
2407:
2403:
2399:
2395:
2390:
2386:
2382:
2378:
2373:
2370:
2366:
2362:
2358:
2354:
2350:
2345:
2344:
2340:
2322:
2318:
2314:
2307:
2304:
2300:
2294:
2291:
2286:
2282:
2278:
2271:
2268:
2256:
2252:
2246:
2243:
2238:
2234:
2230:
2224:
2221:
2216:
2212:
2208:
2202:
2199:
2187:
2183:
2176:
2173:
2169:
2165:
2161:
2157:
2156:
2148:
2144:
2138:
2136:
2132:
2127:; see p. 664.
2124:
2120:
2116:
2110:
2105:
2101:
2095:
2091:
2087:
2083:
2078:
2074:
2070:
2065:
2064:
2060:
2057:
2050:
2046:
2040:
2036:
2032:
2031:
2024:
2023:
2019:
2016:
2011:
2007:
2005:
1999:
1992:
1989:
1981:
1979:
1966:
1962:
1961:
1953:
1950:
1943:
1941:
1927:
1924:
1921:
1901:
1878:
1875:
1870:
1866:
1857:
1853:
1843:
1834:
1823:
1819:
1812:
1807:
1803:
1795:
1794:
1793:
1792:
1791:Pell equation
1788:
1784:
1780:
1776:
1772:
1768:
1749:
1744:
1735:
1732:
1729:
1723:
1720:
1717:
1712:
1709:
1706:
1700:
1697:
1690:
1689:
1688:
1686:
1667:
1662:
1658:
1655:
1650:
1646:
1639:
1636:
1633:
1630:
1623:
1622:
1621:
1619:
1615:
1611:
1607:
1603:
1599:
1580:
1577:
1571:
1562:
1553:
1544:
1533:
1529:
1522:
1519:
1515:
1511:
1507:
1504:
1501:
1497:
1493:
1489:
1486:
1483:
1480:
1477:
1470:
1469:
1468:
1466:
1462:
1458:
1454:
1453:Pell equation
1447:Pell equation
1446:
1444:
1442:
1441:IBM 1620
1438:
1437:IBM 7040
1434:
1431:, in 1965 by
1430:
1412:
1408:
1404:
1400:
1397:
1389:
1370:
1367:
1362:
1358:
1351:
1342:
1336:
1331:
1327:
1319:
1318:
1317:
1316:
1315:Pell equation
1297:
1294:
1291:
1265:
1257:
1248:
1240:
1237:
1234:
1231:
1226:
1222:
1214:
1213:
1212:
1210:
1191:
1186:
1181:
1177:
1173:
1169:
1165:
1161:
1157:
1154:
1149:
1144:
1140:
1136:
1132:
1128:
1124:
1120:
1117:
1114:
1107:
1106:
1105:
1103:
1084:
1075:
1066:
1056:
1046:
1043:
1040:
1036:
1032:
1027:
1024:
1020:
1010:
1007:
1000:
999:
998:
996:
992:
988:
972:
969:
966:
958:
957:square number
942:
939:
936:
927:
925:
921:
917:
913:
910:
877:
874:
870:
866:
863:
861:
856:
849:
846:
842:
838:
835:
833:
828:
821:
818:
814:
810:
807:
805:
800:
793:
790:
786:
782:
779:
777:
772:
765:
762:
759:
756:
753:
750:
746:
742:
739:
737:
732:
725:
722:
719:
716:
713:
710:
706:
702:
699:
697:
692:
685:
682:
679:
676:
673:
670:
666:
662:
659:
657:
652:
645:
642:
639:
636:
633:
630:
626:
622:
619:
617:
612:
601:
600:
599:
597:
596:indeterminate
574:
568:
565:
562:
554:
551:
546:
544:
539:
532:
526:
523:
520:
512:
509:
504:
502:
497:
490:
484:
481:
478:
470:
467:
462:
460:
455:
448:
442:
439:
436:
428:
425:
420:
418:
413:
406:
403:
400:
397:
392:
389:
384:
382:
377:
370:
367:
364:
361:
356:
353:
348:
346:
341:
334:
331:
328:
325:
320:
317:
312:
310:
305:
294:
293:
292:
278:
258:
255:
252:
249:
246:
243:
223:
203:
200:
197:
194:
191:
188:
180:
172:
169:
165:
163:
157:
151:
149:
147:
143:
139:
131:
129:
127:
123:
111:
107:
103:
101:
93:
88:
83:
81:
77:
73:
69:
65:
61:
57:
53:
49:
45:
43:
37:
35:
29:
21:
2691:
2663:(apocryphal)
2658:
2651:
2644:
2637:
2630:
2623:
2616:
2609:
2602:
2595:
2588:
2533:. Retrieved
2520:
2465:
2461:
2436:
2432:
2401:
2397:
2380:
2376:
2352:
2348:
2325:. Retrieved
2306:
2298:
2293:
2284:
2280:
2270:
2258:. Retrieved
2254:
2245:
2232:
2223:
2210:
2207:Fraser, P.M.
2201:
2190:. Retrieved
2186:the original
2175:
2159:
2153:
2122:
2118:
2103:
2089:
2085:
2072:
2068:
2059:
2048:
2034:
2029:
2018:
2009:
2001:
1991:
1970:
1964:
1959:
1952:
1893:
1786:
1783:discriminant
1778:
1774:
1770:
1766:
1764:
1684:
1683:Solving for
1682:
1616:should be a
1613:
1609:
1605:
1601:
1597:
1595:
1463:should be a
1460:
1456:
1450:
1387:
1385:
1280:
1208:
1206:
1101:
1099:
928:
923:
919:
915:
911:
909:prime number
896:
593:
176:
166:
159:
155:
142:WolfenbĂĽttel
135:
97:
84:
76:WolfenbĂĽttel
39:
31:
27:
26:
2646:Ostomachion
2535:25 November
2525:Brady Haran
2510:Alex Bellos
2472:: 169–196.
2408:: 671–674.
1974:Aufschrift;
1443:computers.
1211:results in
993:, based on
918:, and both
90: [
64:the sun god
2807:Categories
2625:On Spirals
2575:Archimedes
2287:: 660–665.
2237:Birkhäuser
2192:2007-01-24
2125:: 660–665.
2092:: 140–141.
2075:: 163–164.
1944:References
60:Archimedes
2486:162393743
1925:×
1813:−
1713:±
1707:−
1401:×
1337:−
1041:−
1033:−
760:×
720:×
680:×
640:×
136:In 1769,
106:Gymnasium
82:in 1773.
2792:Category
2529:Archived
2462:Arethusa
2327:April 5,
2321:Archived
2231:(1972).
2229:Weil, A.
2209:(1972).
2145:(2002),
173:Solution
30:(or the
2521:YouTube
2517:(video)
2453:2589706
2424:2003954
2369:2968125
2260:11 July
2168:1875156
152:Problem
132:History
80:Germany
56:integer
2754:Euclid
2484:
2451:
2422:
2404:(92).
2367:
2166:
1984:
1976:
1465:square
1281:where
1100:where
959:, and
2482:S2CID
2468:(2).
2449:JSTOR
2420:JSTOR
2365:JSTOR
2150:(PDF)
2033:[
1963:[
1076:79072
985:is a
955:is a
94:]
54:with
2537:2019
2500:OEIS
2329:2011
2262:2022
1928:4657
1854:4657
1844:7766
1773:and
1572:4657
1439:and
1398:7.76
1352:7766
1258:7766
1187:7766
1067:4657
1044:4658
1025:4658
922:and
757:4657
723:1580
717:4657
683:2226
677:4657
643:1602
637:4657
271:and
216:and
114:7.76
2474:doi
2441:doi
2437:105
2410:doi
2379:".
2357:doi
1978:...
1835:609
1516:482
1512:366
1498:514
1494:460
1413:544
1409:206
1343:609
1249:609
1182:258
1178:393
1174:858
1170:677
1166:507
1162:129
1150:609
1145:365
1141:713
1137:281
1133:914
1129:607
1125:426
1121:300
924:p+1
904:082
901:389
875:213
871:439
847:820
843:515
819:360
815:206
791:246
787:893
763:891
751:387
747:149
711:060
707:358
671:482
667:366
631:514
627:460
74:in
38:or
2809::
2527:.
2523:.
2519:.
2512:.
2480:.
2466:47
2464:.
2447:.
2435:.
2418:.
2402:19
2400:.
2396:.
2383:.
2363:,
2351:,
2319:.
2315:.
2285:67
2283:.
2279:.
2253:.
2235:.
2213:.
2164:MR
2160:49
2158:,
2152:,
2134:^
2123:67
2121:.
2117:.
2102:.
2088:.
2084:.
2071:.
2047:.
2010:25
1879:1.
1769:+
1687:,
1620::
1612:+
1563:29
1554:11
1508:10
1459:+
1405:10
1371:1.
1158:84
899:50
663:10
555:42
552:13
513:30
510:11
471:20
429:12
393:42
390:13
357:20
118:10
92:de
78:,
2567:e
2560:t
2553:v
2539:.
2488:.
2476::
2455:.
2443::
2426:.
2412::
2359::
2353:2
2331:.
2264:.
2239:.
2217:.
2195:.
2090:2
2073:2
1922:2
1902:v
1876:=
1871:2
1867:q
1863:)
1858:2
1850:(
1847:)
1841:(
1838:)
1832:(
1829:)
1824:2
1820:2
1816:(
1808:2
1804:p
1787:p
1779:q
1775:k
1771:Y
1767:D
1750:.
1745:2
1739:)
1736:Y
1733:+
1730:D
1727:(
1724:8
1721:+
1718:1
1710:1
1701:=
1698:t
1685:t
1668:.
1663:2
1659:t
1656:+
1651:2
1647:t
1640:=
1637:Y
1634:+
1631:D
1614:Y
1610:D
1606:q
1602:q
1598:k
1581:,
1578:k
1575:)
1569:(
1566:)
1560:(
1557:)
1551:(
1548:)
1545:3
1542:(
1539:)
1534:2
1530:2
1526:(
1523:=
1520:k
1505:+
1502:k
1490:7
1487:=
1484:W
1481:+
1478:B
1461:W
1457:B
1388:j
1368:=
1363:2
1359:v
1355:)
1349:(
1346:)
1340:(
1332:2
1328:u
1301:)
1298:v
1295:,
1292:u
1289:(
1266:,
1261:)
1255:(
1252:)
1246:(
1241:v
1238:+
1235:u
1232:=
1227:2
1223:w
1209:w
1192:,
1155:+
1118:=
1115:w
1102:j
1085:,
1079:)
1073:(
1070:)
1064:(
1057:2
1053:)
1047:j
1037:w
1028:j
1021:w
1017:(
1011:=
1008:n
973:D
970:+
967:Y
943:B
940:+
937:W
920:p
916:p
912:p
878:,
867:5
864:=
857:y
850:,
839:3
836:=
829:d
822:,
811:7
808:=
801:w
794:,
783:4
780:=
773:b
766:,
754:=
743:4
740:=
733:Y
726:,
714:=
703:7
700:=
693:D
686:,
674:=
660:=
653:W
646:,
634:=
623:7
620:=
613:B
575:,
572:)
569:w
566:+
563:W
560:(
547:=
540:y
533:,
530:)
527:y
524:+
521:Y
518:(
505:=
498:d
491:,
488:)
485:d
482:+
479:D
476:(
468:9
463:=
456:b
449:,
446:)
443:b
440:+
437:B
434:(
426:7
421:=
414:w
407:,
404:Y
401:+
398:W
385:=
378:D
371:,
368:Y
365:+
362:D
354:9
349:=
342:B
335:,
332:Y
329:+
326:B
321:6
318:5
313:=
306:W
279:y
259:,
256:d
253:,
250:b
247:,
244:w
224:Y
204:,
201:D
198:,
195:B
192:,
189:W
116:Ă—
104:(
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