324:. One fine point (of many): taking the distant stars as a fixed reference, the time for one complete spin is not the same as the time between sun facings (noons), and the latter varies depending on position in orbit. Again, in this case, the difference is small. Although Earth's orbit is nearly a perfect circle, so the distance to the Sun is nearly constant, the Earth's axis of spin is tilted with respect to the plane of the orbit; thus we have seasons.
1947:
153:
By "summer", I assume you're talking about "natural" or climatic summer, as opposed to official summer. In my country, we're now in winter, which is officially defined as the period 1 June-31 August. This roughly coincides with the natural winter - the times when it's cold, rainy, snowy, foggy etc
116:
every 400 years. If we didn't have them, today's date 11 July for example, which currently occurs in winter in the southern hemisphere, would progressively occur earlier and earlier. It would at some stage fall in autumn, then in summer, then in spring, then back to winter, and so on. It would
2048:
Ah, when we have an expression like this, which is only meaningful in a certain region, but can be analytically continued to form the function we really want, doesn't it feel like we (mankind) are missing something? It's like our concepts and notation can only reach a very small part of all the
1279:, and other tricks might be able to eliminate other terms as well, but you will almost always end up with an essentially difficult equation. In this particular example, you were lucky enough to have one of the roots equal to the average of the other two, simplifying the equation for
2271:
327:
Sunlight is a major factor in human activity; so is temperature, and so are other seasonal variations. In today's world, when so few people in the developed world engage in agriculture, it is hard to appreciate the critical importance of calendars to
226:
In the interest of historical accuracy, I would just like to point out that (a) the
Gregorian calendar was not adopted at a single date - Spain and Portugal adopted it first in 1582; Great Britain in 1752; Greece did not adopt it until 1923; (b) the
130:
As for daylight hours - those depend, of course, on the time of solar year and not on our calendars. So the longest day will still be in the beginning of summer, but it won't be the middle of June but a different calendar day every time. --
1746:
Many different sources, including wikipedia, state that the riemann zeta function have trivial zeroes at negative integers. However, this argument states that at the negative even integers, (in the example, s=-2,) the function diverges.
1750:
2049:
mathematical object out there, and some only if we bend over backward like in this case. Like we are a one-dimensional creature, living on a string, beginning to suspect that that is not all, there must be more dimensions out there! β
331:
We empirically discover that the length of a day and the length of a year and the length of a month exhibit no exact simple numeric relationships. However, a year (suitably defined) contains 365 days (suitably defined)
2079:
1469:
The short version is: Every rational root of a polynomial (with integer coefficients, of course) can be expressed as a divisor of the free term divided by a divisor of the leading term (the coefficient of
308:, again completing each orbit with almost no timing variation from the one before. In this case, these two periods are β for most practical purposes β not only constant but independent. Not so with the
1146:
202:
That's easy. From 1582 to today there have been 106 years which are multiples of 4. Among these, 1700, 1800 and 1900 are not leap years. So we would be 103 days ahead of today, which is
October 18. --
162:
it would still be mid-winter. To correct this, the official start and end dates of the seasons would need to be moved one further day ahead every year that would otherwise have been a leap year. --
781:
108:
I have been wondering for quite some time- what would happen if we did not observe leap years? Would our whole calendar just get one day ahead every four years? Would it affect our daylight hours?
2007:
66:
45:
59:
554:
2319:
1200:
55:
51:
480:
266:
Indeed I was being Mr. Sad & Pedantic - I just didn't want anyone to walk away from the RDs with the misguided impression that there were no leap years before the 16th century.
896:
1551:
in your above explanation you mentioned 'other tricks' for eliminating other terms of a polynomial. Could you expand on 'other tricks' or show me where to find out about them?
599:
1679:
830:
1620:
1724:
1397:
1326:
672:
633:
1243:
1942:{\displaystyle \zeta {(-2)}=\sum _{n=1}^{\infty }{\frac {1}{n^{-2}}}=\sum _{n=1}^{\infty }n^{2}=1+4+9+16+25+36...={\frac {\infty (\infty +1)(2\infty +1)}{6}}=\infty }
1495:
984:
1352:
1277:
952:
932:
336:. That fraction is our problem. What exactly is it, and how do we accomodate it? And, as requested, what happens if we ignore it? The fraction is almost exactly β
25:
367:
to be in the Spring, but over hundreds of years the drift in the calendar became too much to ignore. So the church dictated a new calendar. Not surprisingly,
85:
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the
283:
Note that the Muslim calendar is purely lunar with no adjustments to bring it into sync with the solar year, so their dates rotate through the seasons.
252:
For the record, I was thinking about mentioning those points, but ultimately concluded that they aren't relevant for the question or the answer. --
1204:
When I say this I suppose I really mean could you solve something relatively large, say n=9 by constantly applying the appropriate substitution?
2321:
factor in there plus the fact that no other factors are unbounded (which is why you don't have trivial zeroes at nonnegative even integers).
2283:
But it's much easier to see that there are trivial zeros of the zeta function at negative even values from the functional equation, with that
179:
Now someone who REALLY likes math figure out exactly what time of the year it would be if there were no leap years starting from the time the
37:
2266:{\displaystyle \zeta (s)={\frac {1}{1-2^{1-s}}}\sum _{n=0}^{\infty }{\frac {1}{2^{n+1}}}\sum _{k=0}^{n}(-1)^{k}{n \choose k}(k+1)^{-s}.}
1534:
21:
239:, which had a leap year every 4 years with no exceptions; (c) if we had stayed with the Julian calendar we would now be 13 days
901:
This last stage obviously confirms that the three solutions are consecutive but gives them as -1,0, and 1, which is incorrect.
1047:
2361:
2352:
2338:
2325:
2278:
2053:
2035:
2017:
1953:
1735:
1555:
1541:
1528:
1510:
1462:
1453:
1429:
1412:
1032:
1023:
1014:
1000:
990:
908:
390:
287:
270:
261:
247:
211:
189:
166:
140:
121:
682:
154:- which is a good thing. However, if the 103-day scenario you worked out below applied, and seasonal designations were
1962:
158:
officially changed, today would now be called 18 October and we'd be calling the season "spring" (officially), but
340:, but a little less. If we ignore it, then a calendar based on days bears no fixed relationship with the seasons.
1216:
2348:
1731:
1524:
1506:
1449:
1408:
257:
207:
136:
86:
17:
1501:
time. In your example, all roots are rational, and they are all a divisor of 24 divided by a divisor of 1. --
491:
2334:
that global definition, I just wanted to point out it's there for anyone who dislikes analytic continuation.
2286:
1155:
413:
1626:, which describes some (relatively complicated) transformations which can be used to transform a general
2071:
2028:
1516:
356:
1574:
This is indeed an example, but not a particularly useful one as it doesn't give something simpler than
374:
All timekeeping, not just calendars, must juggle science and commerce and politics and so on. Consider
841:
185:
375:
565:
2344:
2010:
1727:
1548:
1520:
1502:
1445:
1404:
253:
203:
132:
1633:
792:
1577:
401:
I have just discovered
Cardano's method for solving cubics and have tested it out, to no avail.
368:
348:
228:
180:
1684:
1357:
1286:
304:
spins around its axis, and the rate at which is does so is fairly constant. It also orbits the
297:
are a fun topic, involving mathematics, astronomy, agronomy, religion, and politics β at least!
2322:
2014:
1020:
641:
607:
2343:
Is this expression not a relatively efficient way of evaluating the function numerically? --
1627:
1552:
1459:
1458:
If you're offering to explain or point me in the right direction then yes, I am interested.
1426:
1222:
1205:
1029:
997:
905:
344:
2358:
2335:
2275:
2050:
2032:
1538:
1473:
1011:
987:
957:
352:
236:
1561:
Here's a substitution to get rid of the linear term in a cubic ax^3 + bx^2 + cx + d = 0
1331:
1256:
1950:
1422:
267:
244:
1533:
In case you think this is a major result, it isn't: it's a trivial application of the
937:
917:
1623:
1497:). Since there are only finitely many such divisors, you can find them all in finite
485:
Hence my solutions are obviously 2, 3 and 4 respectively. Expanding this results in
284:
163:
118:
386:
times? With contributors around the world, we certainly couldn't use local time! --
404:
I would appreciate someone telling me where I have gone wrong or what to do next.
379:
387:
382:
were recently a cause for internation debate. Ever notice that
Knowledge uses
117:
take approximately 400*365/97 = 1505 years for the whole cycle to occur. --
2026:
Indeed, that series only defines the zeta function for z with real part : -->
294:
113:
74:
347:, which makes life complicated indeed!) The common Western calendar, the
1253:(roots). The substitution you suggest can eliminate the term of degree
1250:
1019:
That's a big mistake my students made whenever they did substitutions.
378:, with it's shifting of noon. And although few people know about them,
317:
1444:
roots of any polynomial, if for any reason those are of interest. --
360:
313:
301:
309:
2027:
1. The function is extended to the complex plane (except 1) by
1249:
th degree cannot be found using only arithmetic operations and
1959:
The sum form of the function is not valid for all values. For
383:
321:
305:
1567:
this should produce a cubic of the form My^3 + Ny^2 + S = 0
79:
Welcome to the
Knowledge Mathematics Reference Desk Archives
2357:
According to the article, Borwein's method is much better.
1141:{\displaystyle x^{n}+a_{1}x^{n-1}+a_{2}x^{n-2}...=0\,\!}
351:, was motivated by an inaccurate use of leap years (the
343:
We cannot ignore religion and politics. (Consider the
2289:
2082:
1965:
1753:
1687:
1636:
1580:
1570:
dunno if that's a "trick" you'd want to see or not..
1476:
1440:
Of course, there is a general method to find all the
1360:
1334:
1289:
1259:
1225:
1158:
1050:
960:
940:
920:
844:
795:
776:{\displaystyle (t+3)^{3}-9(t+3)^{2}+26(t+3)-24=0\,\!}
685:
644:
610:
568:
494:
416:
676:
Substituting this into the original equation leaves
1283:. Otherwise you would have to solve something like
2313:
2265:
2001:
1941:
1718:
1673:
1614:
1489:
1391:
1346:
1320:
1271:
1237:
1194:
1140:
978:
946:
926:
890:
824:
775:
666:
627:
593:
548:
474:
235:3 leap years in every 400 years from the previous
2229:
2216:
1245:, a solution to a general polynomial equation of
2002:{\displaystyle \zeta (s),s\in \mathbb {Z} ^{-}}
1190:
1136:
886:
820:
771:
662:
623:
544:
470:
8:
1622:. A better example I can suggest appears in
1354:case you might have to solve something like
1006:Hehe, you're welcome. I was a little afraid
231:did not introduce leap years - in fact, it
104:Leap Years - What Would We Do Without Them?
2296:
2288:
2251:
2228:
2215:
2213:
2207:
2188:
2177:
2159:
2150:
2144:
2133:
2114:
2098:
2081:
1993:
1989:
1988:
1964:
1888:
1843:
1833:
1822:
1804:
1795:
1789:
1778:
1757:
1752:
1692:
1686:
1641:
1635:
1585:
1579:
1481:
1475:
1365:
1359:
1333:
1294:
1288:
1258:
1224:
1177:
1171:
1157:
1107:
1097:
1078:
1068:
1055:
1049:
1042:In theory can you solve any polynomial
959:
939:
919:
843:
800:
794:
730:
702:
684:
643:
615:
609:
581:
567:
549:{\displaystyle x^{3}-9x^{2}+26x-24=0\,\!}
515:
499:
493:
415:
49:
36:
2314:{\displaystyle \sin {\frac {\pi s}{2}}}
1195:{\displaystyle x=t-{a_{1} \over n}\,\!}
1188:
1134:
884:
818:
769:
660:
621:
559:Then I have to use the substitution of
542:
468:
65:
359:problems with its yearly major event:
43:
1564:Let x = y + (β(b^2 - 3ac) - 6ac)/3a
475:{\displaystyle (x-2)(x-3)(x-4)=0\,\!}
7:
954:takes the values 2, 3 and 4 because
2330:Sure, only a madman would actually
2220:
2145:
1936:
1915:
1897:
1891:
1834:
1790:
1010:was missing some difficult part. β
635:and in this case is therefore -9.
32:
1949:What's wrong with this argument?
1535:fundamental theorem of arithmetic
1328:which is a little harder. In the
996:Wow, is my face red. Many thanks
891:{\displaystyle t(t+1)(t-1)=0\,\!}
1515:Apparently, this is called the
594:{\displaystyle x=t-{a \over 3}}
112:Ultimately, yes. There are 97
2248:
2235:
2204:
2194:
2092:
2086:
1975:
1969:
1924:
1909:
1906:
1894:
1767:
1758:
934:takes the values -1, 0 and 1,
875:
863:
860:
848:
754:
742:
727:
714:
699:
686:
604:where a is the co-efficent of
459:
447:
444:
432:
429:
417:
1:
1674:{\displaystyle x^{5}-5x-4t=0}
825:{\displaystyle t^{3}-t=0\,\!}
33:
1615:{\displaystyle x^{3}+ax+b=0}
1028:Nice to know I'm not alone!
2074:gives a global definition:
1719:{\displaystyle x^{5}+x+a=0}
1392:{\displaystyle t^{5}+t+3=0}
1321:{\displaystyle t^{3}+t+3=0}
1150:by making the substitution
835:Which can then simplify to
407:I start with the equation
2394:
2362:13:42, 12 July 2007 (UTC)
2353:12:55, 12 July 2007 (UTC)
2339:11:36, 12 July 2007 (UTC)
2326:22:11, 11 July 2007 (UTC)
2279:21:02, 11 July 2007 (UTC)
2054:20:36, 11 July 2007 (UTC)
2036:20:17, 11 July 2007 (UTC)
2018:19:02, 11 July 2007 (UTC)
1954:18:51, 11 July 2007 (UTC)
1736:12:37, 12 July 2007 (UTC)
1556:00:09, 12 July 2007 (UTC)
1542:23:10, 11 July 2007 (UTC)
1529:22:09, 11 July 2007 (UTC)
1511:21:51, 11 July 2007 (UTC)
1463:21:47, 11 July 2007 (UTC)
1454:21:43, 11 July 2007 (UTC)
1430:23:22, 11 July 2007 (UTC)
1413:21:40, 11 July 2007 (UTC)
1033:17:55, 11 July 2007 (UTC)
1024:17:51, 11 July 2007 (UTC)
1015:16:22, 11 July 2007 (UTC)
1001:16:20, 11 July 2007 (UTC)
991:16:18, 11 July 2007 (UTC)
909:15:49, 11 July 2007 (UTC)
667:{\displaystyle x=t+3\,\!}
628:{\displaystyle x^{2}\,\!}
391:10:13, 12 July 2007 (UTC)
288:15:29, 11 July 2007 (UTC)
271:14:46, 11 July 2007 (UTC)
262:14:25, 11 July 2007 (UTC)
248:14:17, 11 July 2007 (UTC)
212:13:55, 11 July 2007 (UTC)
190:13:44, 11 July 2007 (UTC)
167:10:43, 12 July 2007 (UTC)
141:13:43, 11 July 2007 (UTC)
122:13:31, 11 July 2007 (UTC)
18:Knowledge:Reference desk
1499:(and I believe polylog)
1421:For quintics there are
1238:{\displaystyle n\geq 5}
2315:
2267:
2193:
2149:
2009:, you need to use the
2003:
1943:
1838:
1794:
1720:
1675:
1616:
1491:
1393:
1348:
1322:
1273:
1239:
1196:
1142:
980:
948:
928:
892:
826:
777:
668:
629:
595:
550:
476:
87:current reference desk
2316:
2268:
2173:
2129:
2072:Riemann zeta function
2029:analytic continuation
2004:
1944:
1818:
1774:
1742:Riemann Zeta Function
1721:
1676:
1617:
1517:rational root theorem
1492:
1490:{\displaystyle x^{n}}
1394:
1349:
1323:
1274:
1240:
1197:
1143:
981:
979:{\displaystyle x=t+3}
949:
929:
893:
827:
778:
669:
638:Substiuting in gives
630:
596:
551:
477:
357:Roman Catholic Church
2287:
2080:
1963:
1751:
1685:
1634:
1578:
1474:
1358:
1332:
1287:
1257:
1223:
1217:AbelβRuffini theorem
1156:
1048:
958:
938:
918:
842:
793:
786:This simplifies to
683:
642:
608:
566:
492:
414:
376:daylight saving time
371:had their own ideas.
2011:functional equation
1681:, or equivalently,
1630:to one of the form
1347:{\displaystyle n=5}
1272:{\displaystyle n-1}
369:various governments
2311:
2263:
1999:
1939:
1716:
1671:
1612:
1487:
1389:
1344:
1318:
1269:
1235:
1192:
1191:
1189:
1138:
1137:
1135:
976:
944:
924:
888:
887:
885:
822:
821:
819:
773:
772:
770:
664:
663:
661:
625:
624:
622:
591:
546:
545:
543:
472:
471:
469:
349:Gregorian calendar
243:the current date.
229:Gregorian calendar
183:was established --
181:Gregorian calendar
2309:
2227:
2171:
2127:
1931:
1813:
1186:
947:{\displaystyle x}
927:{\displaystyle t}
589:
334:and a little more
93:
92:
73:
72:
2385:
2320:
2318:
2317:
2312:
2310:
2305:
2297:
2272:
2270:
2269:
2264:
2259:
2258:
2234:
2233:
2232:
2219:
2212:
2211:
2192:
2187:
2172:
2170:
2169:
2151:
2148:
2143:
2128:
2126:
2125:
2124:
2099:
2008:
2006:
2005:
2000:
1998:
1997:
1992:
1948:
1946:
1945:
1940:
1932:
1927:
1889:
1848:
1847:
1837:
1832:
1814:
1812:
1811:
1796:
1793:
1788:
1770:
1725:
1723:
1722:
1717:
1697:
1696:
1680:
1678:
1677:
1672:
1646:
1645:
1628:quintic equation
1621:
1619:
1618:
1613:
1590:
1589:
1496:
1494:
1493:
1488:
1486:
1485:
1398:
1396:
1395:
1390:
1370:
1369:
1353:
1351:
1350:
1345:
1327:
1325:
1324:
1319:
1299:
1298:
1278:
1276:
1275:
1270:
1244:
1242:
1241:
1236:
1219:states that for
1201:
1199:
1198:
1193:
1187:
1182:
1181:
1172:
1147:
1145:
1144:
1139:
1118:
1117:
1102:
1101:
1089:
1088:
1073:
1072:
1060:
1059:
985:
983:
982:
977:
953:
951:
950:
945:
933:
931:
930:
925:
897:
895:
894:
889:
831:
829:
828:
823:
805:
804:
782:
780:
779:
774:
735:
734:
707:
706:
673:
671:
670:
665:
634:
632:
631:
626:
620:
619:
600:
598:
597:
592:
590:
582:
555:
553:
552:
547:
520:
519:
504:
503:
481:
479:
478:
473:
345:Islamic calendar
75:
38:Mathematics desk
34:
2393:
2392:
2388:
2387:
2386:
2384:
2383:
2382:
2298:
2285:
2284:
2247:
2214:
2203:
2155:
2110:
2103:
2078:
2077:
2070:If you prefer,
1987:
1961:
1960:
1890:
1839:
1800:
1749:
1748:
1744:
1688:
1683:
1682:
1637:
1632:
1631:
1581:
1576:
1575:
1477:
1472:
1471:
1361:
1356:
1355:
1330:
1329:
1290:
1285:
1284:
1255:
1254:
1221:
1220:
1173:
1154:
1153:
1103:
1093:
1074:
1064:
1051:
1046:
1045:
956:
955:
936:
935:
916:
915:
840:
839:
796:
791:
790:
726:
698:
681:
680:
640:
639:
611:
606:
605:
564:
563:
511:
495:
490:
489:
412:
411:
399:
353:Julian calendar
339:
237:Julian calendar
188:
106:
101:
30:
29:
28:
12:
11:
5:
2391:
2389:
2381:
2380:
2379:
2378:
2377:
2376:
2375:
2374:
2373:
2372:
2371:
2370:
2369:
2368:
2367:
2366:
2365:
2364:
2345:Meni Rosenfeld
2308:
2304:
2301:
2295:
2292:
2273:
2262:
2257:
2254:
2250:
2246:
2243:
2240:
2237:
2231:
2226:
2223:
2218:
2210:
2206:
2202:
2199:
2196:
2191:
2186:
2183:
2180:
2176:
2168:
2165:
2162:
2158:
2154:
2147:
2142:
2139:
2136:
2132:
2123:
2120:
2117:
2113:
2109:
2106:
2102:
2097:
2094:
2091:
2088:
2085:
2061:
2060:
2059:
2058:
2057:
2056:
2041:
2040:
2039:
2038:
2021:
2020:
1996:
1991:
1986:
1983:
1980:
1977:
1974:
1971:
1968:
1938:
1935:
1930:
1926:
1923:
1920:
1917:
1914:
1911:
1908:
1905:
1902:
1899:
1896:
1893:
1887:
1884:
1881:
1878:
1875:
1872:
1869:
1866:
1863:
1860:
1857:
1854:
1851:
1846:
1842:
1836:
1831:
1828:
1825:
1821:
1817:
1810:
1807:
1803:
1799:
1792:
1787:
1784:
1781:
1777:
1773:
1769:
1766:
1763:
1760:
1756:
1743:
1740:
1739:
1738:
1728:Meni Rosenfeld
1715:
1712:
1709:
1706:
1703:
1700:
1695:
1691:
1670:
1667:
1664:
1661:
1658:
1655:
1652:
1649:
1644:
1640:
1611:
1608:
1605:
1602:
1599:
1596:
1593:
1588:
1584:
1559:
1558:
1549:Meni Rosenfeld
1546:
1545:
1544:
1521:Meni Rosenfeld
1513:
1503:Meni Rosenfeld
1484:
1480:
1467:
1466:
1465:
1446:Meni Rosenfeld
1438:
1437:
1436:
1435:
1434:
1433:
1432:
1423:Bring Radicals
1405:Meni Rosenfeld
1388:
1385:
1382:
1379:
1376:
1373:
1368:
1364:
1343:
1340:
1337:
1317:
1314:
1311:
1308:
1305:
1302:
1297:
1293:
1268:
1265:
1262:
1234:
1231:
1228:
1185:
1180:
1176:
1170:
1167:
1164:
1161:
1133:
1130:
1127:
1124:
1121:
1116:
1113:
1110:
1106:
1100:
1096:
1092:
1087:
1084:
1081:
1077:
1071:
1067:
1063:
1058:
1054:
1040:
1039:
1038:
1037:
1036:
1035:
994:
993:
975:
972:
969:
966:
963:
943:
923:
899:
898:
883:
880:
877:
874:
871:
868:
865:
862:
859:
856:
853:
850:
847:
833:
832:
817:
814:
811:
808:
803:
799:
784:
783:
768:
765:
762:
759:
756:
753:
750:
747:
744:
741:
738:
733:
729:
725:
722:
719:
716:
713:
710:
705:
701:
697:
694:
691:
688:
659:
656:
653:
650:
647:
618:
614:
602:
601:
588:
585:
580:
577:
574:
571:
557:
556:
541:
538:
535:
532:
529:
526:
523:
518:
514:
510:
507:
502:
498:
483:
482:
467:
464:
461:
458:
455:
452:
449:
446:
443:
440:
437:
434:
431:
428:
425:
422:
419:
398:
397:Cubic equation
395:
394:
393:
372:
355:) causing the
341:
337:
329:
325:
316:; not so with
298:
291:
290:
281:
280:
279:
278:
277:
276:
275:
274:
273:
254:Meni Rosenfeld
219:
218:
217:
216:
215:
214:
204:Meni Rosenfeld
195:
194:
193:
192:
184:
174:
173:
172:
171:
170:
169:
146:
145:
144:
143:
133:Meni Rosenfeld
125:
124:
105:
102:
100:
97:
95:
91:
90:
82:
81:
71:
70:
64:
48:
41:
40:
31:
15:
14:
13:
10:
9:
6:
4:
3:
2:
2390:
2363:
2360:
2356:
2355:
2354:
2350:
2346:
2342:
2341:
2340:
2337:
2333:
2329:
2328:
2327:
2324:
2306:
2302:
2299:
2293:
2290:
2282:
2281:
2280:
2277:
2274:
2260:
2255:
2252:
2244:
2241:
2238:
2224:
2221:
2208:
2200:
2197:
2189:
2184:
2181:
2178:
2174:
2166:
2163:
2160:
2156:
2152:
2140:
2137:
2134:
2130:
2121:
2118:
2115:
2111:
2107:
2104:
2100:
2095:
2089:
2083:
2076:
2075:
2073:
2069:
2068:
2067:
2066:
2065:
2064:
2063:
2062:
2055:
2052:
2047:
2046:
2045:
2044:
2043:
2042:
2037:
2034:
2030:
2025:
2024:
2023:
2022:
2019:
2016:
2012:
1994:
1984:
1981:
1978:
1972:
1966:
1958:
1957:
1956:
1955:
1952:
1933:
1928:
1921:
1918:
1912:
1903:
1900:
1885:
1882:
1879:
1876:
1873:
1870:
1867:
1864:
1861:
1858:
1855:
1852:
1849:
1844:
1840:
1829:
1826:
1823:
1819:
1815:
1808:
1805:
1801:
1797:
1785:
1782:
1779:
1775:
1771:
1764:
1761:
1754:
1741:
1737:
1733:
1729:
1713:
1710:
1707:
1704:
1701:
1698:
1693:
1689:
1668:
1665:
1662:
1659:
1656:
1653:
1650:
1647:
1642:
1638:
1629:
1625:
1624:Bring radical
1609:
1606:
1603:
1600:
1597:
1594:
1591:
1586:
1582:
1573:
1572:
1571:
1568:
1565:
1562:
1557:
1554:
1550:
1547:
1543:
1540:
1536:
1532:
1531:
1530:
1526:
1522:
1518:
1514:
1512:
1508:
1504:
1500:
1482:
1478:
1468:
1464:
1461:
1457:
1456:
1455:
1451:
1447:
1443:
1439:
1431:
1428:
1424:
1420:
1419:
1418:
1417:
1416:
1415:
1414:
1410:
1406:
1402:
1386:
1383:
1380:
1377:
1374:
1371:
1366:
1362:
1341:
1338:
1335:
1315:
1312:
1309:
1306:
1303:
1300:
1295:
1291:
1282:
1266:
1263:
1260:
1252:
1248:
1232:
1229:
1226:
1218:
1214:
1210:
1209:
1208:
1207:
1202:
1183:
1178:
1174:
1168:
1165:
1162:
1159:
1151:
1148:
1131:
1128:
1125:
1122:
1119:
1114:
1111:
1108:
1104:
1098:
1094:
1090:
1085:
1082:
1079:
1075:
1069:
1065:
1061:
1056:
1052:
1043:
1034:
1031:
1027:
1026:
1025:
1022:
1018:
1017:
1016:
1013:
1009:
1005:
1004:
1003:
1002:
999:
992:
989:
973:
970:
967:
964:
961:
941:
921:
913:
912:
911:
910:
907:
902:
881:
878:
872:
869:
866:
857:
854:
851:
845:
838:
837:
836:
815:
812:
809:
806:
801:
797:
789:
788:
787:
766:
763:
760:
757:
751:
748:
745:
739:
736:
731:
723:
720:
717:
711:
708:
703:
695:
692:
689:
679:
678:
677:
674:
657:
654:
651:
648:
645:
636:
616:
612:
586:
583:
578:
575:
572:
569:
562:
561:
560:
539:
536:
533:
530:
527:
524:
521:
516:
512:
508:
505:
500:
496:
488:
487:
486:
465:
462:
456:
453:
450:
441:
438:
435:
426:
423:
420:
410:
409:
408:
405:
402:
396:
392:
389:
385:
381:
377:
373:
370:
366:
362:
358:
354:
350:
346:
342:
335:
330:
326:
323:
320:orbiting the
319:
315:
312:orbiting the
311:
307:
303:
299:
296:
293:
292:
289:
286:
282:
272:
269:
265:
264:
263:
259:
255:
251:
250:
249:
246:
242:
238:
234:
230:
225:
224:
223:
222:
221:
220:
213:
209:
205:
201:
200:
199:
198:
197:
196:
191:
187:
182:
178:
177:
176:
175:
168:
165:
161:
157:
152:
151:
150:
149:
148:
147:
142:
138:
134:
129:
128:
127:
126:
123:
120:
115:
111:
110:
109:
103:
98:
96:
88:
84:
83:
80:
77:
76:
68:
61:
57:
53:
47:
42:
39:
35:
27:
23:
19:
2331:
2323:Donald Hosek
2015:Donald Hosek
1745:
1569:
1566:
1563:
1560:
1498:
1441:
1400:
1280:
1246:
1212:
1203:
1152:
1149:
1044:
1041:
1021:Donald Hosek
1007:
995:
903:
900:
834:
785:
675:
637:
603:
558:
484:
406:
403:
400:
380:leap seconds
364:
333:
240:
232:
160:climatically
159:
155:
107:
94:
78:
1553:Algebra man
1460:Algebra man
1403:harder. --
1206:Algebra man
1030:Algebra man
998:Algebra man
906:Algebra man
26:Mathematics
2359:Algebraist
2336:Algebraist
2276:Algebraist
2051:Bromskloss
2033:Algebraist
1539:Algebraist
1215:fast. The
1012:Bromskloss
988:Bromskloss
986:, right? β
295:Leap years
1951:Indeed123
1399:which is
363:. It was
328:survival.
268:Gandalf61
245:Gandalf61
114:leap days
50:<<
1442:rational
1427:Cronholm
1251:radicals
904:Thanks.
365:supposed
285:Gzuckier
164:JackofOz
119:JackofOz
24: |
22:Archives
20: |
318:Mercury
233:removed
99:July 11
89:pages.
67:July 12
46:July 10
361:Easter
241:behind
1883:36...
1726:. --
1519:. --
388:KSmrq
314:Earth
302:Earth
69:: -->
63:: -->
62:: -->
44:<
16:<
2349:talk
1732:talk
1525:talk
1507:talk
1450:talk
1409:talk
1401:much
1211:Not
310:Moon
300:The
258:talk
208:talk
137:talk
56:July
2332:use
2291:sin
914:If
384:UTC
322:Sun
306:Sun
156:not
60:Aug
52:Jun
2351:)
2300:Ο
2294:β‘
2253:β
2198:β
2175:β
2146:β
2131:β
2119:β
2108:β
2084:ΞΆ
2031:.
2013:.
1995:β
1985:β
1967:ΞΆ
1937:β
1916:β
1898:β
1892:β
1877:25
1871:16
1835:β
1820:β
1806:β
1791:β
1776:β
1762:β
1755:ΞΆ
1734:)
1657:β
1648:β
1537:.
1527:)
1509:)
1452:)
1425:--
1411:)
1264:β
1230:β₯
1213:so
1169:β
1112:β
1083:β
870:β
807:β
761:24
758:β
740:26
709:β
579:β
534:24
531:β
525:26
506:β
454:β
439:β
424:β
260:)
210:)
139:)
58:|
54:|
2347:(
2307:2
2303:s
2261:.
2256:s
2249:)
2245:1
2242:+
2239:k
2236:(
2230:)
2225:k
2222:n
2217:(
2209:k
2205:)
2201:1
2195:(
2190:n
2185:0
2182:=
2179:k
2167:1
2164:+
2161:n
2157:2
2153:1
2141:0
2138:=
2135:n
2122:s
2116:1
2112:2
2105:1
2101:1
2096:=
2093:)
2090:s
2087:(
1990:Z
1982:s
1979:,
1976:)
1973:s
1970:(
1934:=
1929:6
1925:)
1922:1
1919:+
1913:2
1910:(
1907:)
1904:1
1901:+
1895:(
1886:=
1880:+
1874:+
1868:+
1865:9
1862:+
1859:4
1856:+
1853:1
1850:=
1845:2
1841:n
1830:1
1827:=
1824:n
1816:=
1809:2
1802:n
1798:1
1786:1
1783:=
1780:n
1772:=
1768:)
1765:2
1759:(
1730:(
1714:0
1711:=
1708:a
1705:+
1702:x
1699:+
1694:5
1690:x
1669:0
1666:=
1663:t
1660:4
1654:x
1651:5
1643:5
1639:x
1610:0
1607:=
1604:b
1601:+
1598:x
1595:a
1592:+
1587:3
1583:x
1523:(
1505:(
1483:n
1479:x
1448:(
1407:(
1387:0
1384:=
1381:3
1378:+
1375:t
1372:+
1367:5
1363:t
1342:5
1339:=
1336:n
1316:0
1313:=
1310:3
1307:+
1304:t
1301:+
1296:3
1292:t
1281:t
1267:1
1261:n
1247:n
1233:5
1227:n
1184:n
1179:1
1175:a
1166:t
1163:=
1160:x
1132:0
1129:=
1126:.
1123:.
1120:.
1115:2
1109:n
1105:x
1099:2
1095:a
1091:+
1086:1
1080:n
1076:x
1070:1
1066:a
1062:+
1057:n
1053:x
1008:I
974:3
971:+
968:t
965:=
962:x
942:x
922:t
882:0
879:=
876:)
873:1
867:t
864:(
861:)
858:1
855:+
852:t
849:(
846:t
816:0
813:=
810:t
802:3
798:t
767:0
764:=
755:)
752:3
749:+
746:t
743:(
737:+
732:2
728:)
724:3
721:+
718:t
715:(
712:9
704:3
700:)
696:3
693:+
690:t
687:(
658:3
655:+
652:t
649:=
646:x
617:2
613:x
587:3
584:a
576:t
573:=
570:x
540:0
537:=
528:x
522:+
517:2
513:x
509:9
501:3
497:x
466:0
463:=
460:)
457:4
451:x
448:(
445:)
442:3
436:x
433:(
430:)
427:2
421:x
418:(
338:4
256:(
206:(
186:L
135:(
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