1974:
1748:
813:
548:
1129:
1606:
1416:
942:
342:
1969:{\displaystyle Z(T):=\sum _{n=1}^{\infty }\exp \left({\frac {-E_{n}}{k_{\text{B}}T}}\right)=\sum _{n=1}^{\infty }\exp \left({\frac {-E\log n}{k_{\text{B}}T}}\right)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=\zeta (s)}
1426:
Let's suppose we would like to know the average time, suitably-normalised, that the
Riemann gas spends in a particular subspace. How might this frequency be related to the dimension of this subspace?
1229:
717:
1614:
What is even more remarkable is that although the Erdős-Kac theorem has the form of a statistical observation, it could not have been discovered using statistical methods. Indeed, for
410:
993:
207:
1488:
840:
626:
1721:
382:
1277:
1688:
1460:
579:
125:
1659:
2103:
987:, which views linear combinations of eigenstates as integer partitions. In fact, the reader may easily check that the successor function is not a linear function:
689:
1152:
965:
709:
1508:
1300:
985:
860:
648:
402:
227:
1516:
2165:
Bernard L. Julia, Statistical theory of numbers, in Number Theory and
Physics, eds. J. M. Luck, P. Moussa, and M. Waldschmidt, Springer Proceedings in
71:
because the particles are non-interacting. The idea of the primon gas was independently discovered by Donald
Spector. Later works by Ioannis Bakas and
1308:
868:
235:
2069:, field states with an even number of particles are bosons, while those with an odd number of particles are fermions. The fermion operator
630:
In short, the Fock space for primons has an orthonormal basis given by the positive natural numbers, but we think of each such number
154:
if we think of them as analogous to particles in quantum field theory. This Fock space has an orthonormal basis given by finite
1732:
2105:, in that the Möbius function is positive for bosons, negative for fermions, and zero on exclusion-principle-prohibited states.
2266:
2182:
D. Spector, Supersymmetry and the Möbius
Inversion Function, Communications in Mathematical Physics 127 (1990) pp. 239–252.
2261:
1433:
we may actually demonstrate that this frequency depends upon nothing more than the dimension of the subspace. In fact, if
2113:
The connections between number theory and quantum field theory can be somewhat further extended into connections between
2156:
D. J. G. Dueñas and N. F. Svaiter. Thermodynamics of the
Bosonic Randomized Riemann Gas. arXiv preprint arXiv:1401.8190.
2066:
1429:
If we characterize distinct linear subspaces as Erdős-Kac data which have the form of sparse binary vectors, using the
808:{\displaystyle \Phi \circ {\textbf {log}}\circ x_{n}={\textbf {log}}\circ F\circ x_{n}={\textbf {log}}\circ x_{n+1}}
2062:
1179:
2200:
D. Spector, Duality, Partial
Supersymmetry, and Arithmetic Number Theory, J. Math. Phys. 39 (1998) pp. 1919–1927
543:{\displaystyle n=2^{k_{2}}\cdot 3^{k_{3}}\cdot 5^{k_{5}}\cdot 7^{k_{7}}\cdot 11^{k_{11}}\cdots p^{k_{p}}\cdots }
2114:
1124:{\displaystyle \forall n\in \mathbb {N} ,F(n)=n+1\implies \forall x,y\in \mathbb {N} ^{*},F(x+y)\neq F(x)+F(y)}
2256:
2222:), Theorems that are essentially impossible to guess by empirical observation, URL (version: 2021-12-29):
1739:
161:
48:
1465:
2130:
2024:
1430:
821:
584:
44:
20:
1693:
350:
2138:
1240:
1163:
135:
2122:
2005:
2074:
1664:
1436:
556:
102:
1617:
52:
2079:
661:
56:
2191:
I. Bakas and M.J. Bowick, Curiosities of
Arithmetic Gases, J. Math. Phys. 32 (1991) p. 1881
1137:
950:
694:
158:
of primes. In other words, to specify one of these basis elements we can list the number
2134:
1601:{\displaystyle {\frac {\omega (n)-\ln \ln n}{\sqrt {\ln \ln n}}}\sim {\mathcal {N}}(0,1)}
1493:
1285:
970:
845:
633:
387:
212:
2209:
Steven L. Brunton. Notes on
Koopman Operator Theory. Cambridge University Press. 2019.
2250:
93:
76:
40:
32:
842:
is an algorithm for integer factorisation, analogous to the discrete logarithm, and
128:
2219:
2126:
2013:
72:
143:
2241:
1411:{\displaystyle E_{n}=\sum _{p}k_{p}E_{p}=E\cdot \sum _{p}k_{p}\log p=E\log n}
937:{\displaystyle {\textbf {log}}\circ x_{n}=\bigoplus _{k}a_{k}\cdot \ln p_{k}}
337:{\displaystyle |k_{2},k_{3},k_{5},k_{7},k_{11},\ldots ,k_{p},\ldots \rangle }
2237:
2023: = 1 corresponds to the divergence of the partition function at a
155:
650:
as a collection of primons: its prime factors, counted with multiplicity.
59:. It is a quantum field theory of a set of non-interacting particles, the
2118:
2065:
prohibits multi-particle states which include squares of primes. By the
711:
to lift dynamics from the space of states to the space of observables:
2070:
2058:
2223:
2054:
553:
we can also denote the basis elements of the Fock space as simply
150:, where states describe collections of primes - which we can call
36:
64:
1578:
2053:
The above second-quantized model takes the particles to be
2125:
takes the role of the spectrum of energy eigenvalues, the
75:, and Spector explored the connection of such systems to
947:
A precise motivation for defining the
Koopman operator
2082:
2073:
has a very concrete realization in this model as the
1751:
1696:
1667:
1620:
1519:
1496:
1468:
1439:
1311:
1288:
1243:
1182:
1140:
996:
973:
953:
871:
848:
824:
720:
697:
664:
636:
587:
559:
413:
390:
353:
238:
215:
164:
105:
654:
2242:
This Week's Finds in Mathematical Physics, Week 199
2097:
1968:
1715:
1682:
1653:
1600:
1502:
1482:
1454:
1410:
1294:
1271:
1223:
1169:to have eigenvalues proportional to log
1146:
1123:
979:
959:
936:
854:
834:
807:
703:
683:
642:
620:
573:
542:
396:
376:
336:
221:
201:
119:
2121:, where, corresponding to the example above, the
967:is that it represents a global linearisation of
1490:then the Erdős-Kac law tells us that for large
1462:counts the number of unique prime divisors of
384:is finite. Since any positive natural number
2220:https://mathoverflow.net/users/470546/bubblez
2173:, Springer-Verlag, Berlin, 1990, pp. 276–293.
8:
1218:
1194:
568:
331:
114:
1224:{\displaystyle H|p\rangle =E_{p}|p\rangle }
862:is the successor function. Thus, we have:
1042:
1038:
2081:
1943:
1934:
1928:
1917:
1894:
1870:
1854:
1843:
1820:
1808:
1798:
1782:
1771:
1750:
1707:
1695:
1666:
1619:
1577:
1576:
1520:
1518:
1495:
1476:
1475:
1467:
1438:
1378:
1368:
1349:
1339:
1329:
1316:
1310:
1287:
1248:
1242:
1210:
1204:
1186:
1181:
1139:
1064:
1060:
1059:
1007:
1006:
995:
972:
952:
928:
909:
899:
886:
873:
872:
870:
847:
826:
825:
823:
793:
780:
779:
770:
751:
750:
741:
728:
727:
719:
696:
669:
663:
635:
586:
560:
558:
529:
524:
509:
504:
489:
484:
469:
464:
449:
444:
429:
424:
412:
389:
368:
358:
352:
319:
300:
287:
274:
261:
248:
239:
237:
214:
169:
163:
106:
104:
2129:take the role of the prime numbers, the
404:has a unique factorization into primes:
2149:
2019:The divergence of the zeta function at
1611:has the standard normal distribution.
1422:Statistics of the phase-space dimension
39:illustrating correspondences between
7:
99:with an orthonormal basis of states
2057:. If the particles are taken to be
874:
827:
781:
752:
729:
1929:
1855:
1783:
1738:of the primon gas is given by the
1141:
1043:
997:
954:
721:
698:
691:, we may use the Koopman operator
202:{\displaystyle k_{p}=0,1,2,\dots }
14:
2224:https://mathoverflow.net/q/412762
1483:{\displaystyle n\in \mathbb {N} }
835:{\displaystyle {\textbf {log}}}
621:{\displaystyle n=1,2,3,\dots .}
16:Model from mathematical physics
2092:
2086:
1963:
1957:
1761:
1755:
1716:{\displaystyle N\geq 10^{100}}
1677:
1671:
1648:
1645:
1633:
1630:
1595:
1583:
1532:
1526:
1449:
1443:
1211:
1187:
1118:
1112:
1103:
1097:
1088:
1076:
1039:
1023:
1017:
561:
377:{\displaystyle \sum _{p}k_{p}}
240:
107:
1:
1272:{\displaystyle E_{p}=E\log p}
2133:take the role of integers,
1690:only begins to emerge for
1282:for some positive constant
2283:
1683:{\displaystyle \omega (X)}
1455:{\displaystyle \omega (n)}
1302:, we are naturally led to
574:{\displaystyle |n\rangle }
209:of primons for each prime
138:gives a new Hilbert space
120:{\displaystyle |p\rangle }
2063:Pauli exclusion principle
1654:{\displaystyle X\sim U()}
2115:topological field theory
2098:{\displaystyle \mu (n)}
2067:spin–statistics theorem
684:{\displaystyle x_{n}=n}
2099:
1970:
1933:
1859:
1787:
1717:
1684:
1655:
1602:
1504:
1484:
1456:
1412:
1296:
1273:
1225:
1148:
1125:
981:
961:
938:
856:
836:
809:
705:
685:
644:
622:
575:
544:
398:
378:
338:
223:
203:
121:
2267:Statistical mechanics
2137:taking the place the
2131:group representations
2100:
1971:
1913:
1839:
1767:
1740:Riemann zeta function
1727:Statistical mechanics
1718:
1685:
1656:
1603:
1505:
1485:
1457:
1413:
1297:
1274:
1226:
1149:
1147:{\displaystyle \Phi }
1126:
982:
962:
960:{\displaystyle \Phi }
939:
857:
837:
810:
706:
704:{\displaystyle \Phi }
686:
645:
623:
576:
545:
399:
379:
339:
224:
204:
122:
49:statistical mechanics
2262:Quantum field theory
2139:Dirichlet characters
2080:
2049:Supersymmetric model
2025:Hagedorn temperature
1749:
1694:
1665:
1661:the normal order of
1618:
1517:
1494:
1466:
1437:
1309:
1286:
1241:
1180:
1162:If we take a simple
1138:
994:
971:
951:
869:
846:
822:
718:
695:
662:
634:
585:
557:
411:
388:
351:
236:
213:
162:
103:
45:quantum field theory
21:mathematical physics
2109:More complex models
1164:quantum Hamiltonian
136:Second quantization
2123:spectrum of a ring
2095:
2006:Boltzmann constant
1966:
1733:partition function
1713:
1680:
1651:
1598:
1500:
1480:
1452:
1408:
1373:
1334:
1292:
1269:
1221:
1144:
1121:
977:
957:
934:
904:
852:
832:
805:
701:
681:
640:
618:
571:
540:
394:
374:
363:
334:
219:
199:
144:bosonic Fock space
117:
1949:
1904:
1897:
1830:
1823:
1571:
1570:
1503:{\displaystyle n}
1431:Erdős-Kac theorem
1364:
1325:
1295:{\displaystyle E}
980:{\displaystyle F}
895:
876:
855:{\displaystyle F}
829:
783:
754:
731:
643:{\displaystyle n}
397:{\displaystyle n}
354:
222:{\displaystyle p}
63:; it is called a
53:dynamical systems
2274:
2226:
2216:
2210:
2207:
2201:
2198:
2192:
2189:
2183:
2180:
2174:
2163:
2157:
2154:
2135:group characters
2104:
2102:
2101:
2096:
2012:is the absolute
1975:
1973:
1972:
1967:
1950:
1948:
1947:
1935:
1932:
1927:
1909:
1905:
1903:
1899:
1898:
1895:
1888:
1871:
1858:
1853:
1835:
1831:
1829:
1825:
1824:
1821:
1814:
1813:
1812:
1799:
1786:
1781:
1722:
1720:
1719:
1714:
1712:
1711:
1689:
1687:
1686:
1681:
1660:
1658:
1657:
1652:
1607:
1605:
1604:
1599:
1582:
1581:
1572:
1554:
1553:
1521:
1509:
1507:
1506:
1501:
1489:
1487:
1486:
1481:
1479:
1461:
1459:
1458:
1453:
1417:
1415:
1414:
1409:
1383:
1382:
1372:
1354:
1353:
1344:
1343:
1333:
1321:
1320:
1301:
1299:
1298:
1293:
1278:
1276:
1275:
1270:
1253:
1252:
1230:
1228:
1227:
1222:
1214:
1209:
1208:
1190:
1153:
1151:
1150:
1145:
1130:
1128:
1127:
1122:
1069:
1068:
1063:
1010:
986:
984:
983:
978:
966:
964:
963:
958:
943:
941:
940:
935:
933:
932:
914:
913:
903:
891:
890:
878:
877:
861:
859:
858:
853:
841:
839:
838:
833:
831:
830:
814:
812:
811:
806:
804:
803:
785:
784:
775:
774:
756:
755:
746:
745:
733:
732:
710:
708:
707:
702:
690:
688:
687:
682:
674:
673:
658:Given the state
649:
647:
646:
641:
627:
625:
624:
619:
580:
578:
577:
572:
564:
549:
547:
546:
541:
536:
535:
534:
533:
516:
515:
514:
513:
496:
495:
494:
493:
476:
475:
474:
473:
456:
455:
454:
453:
436:
435:
434:
433:
403:
401:
400:
395:
383:
381:
380:
375:
373:
372:
362:
347:where the total
343:
341:
340:
335:
324:
323:
305:
304:
292:
291:
279:
278:
266:
265:
253:
252:
243:
228:
226:
225:
220:
208:
206:
205:
200:
174:
173:
127:labelled by the
126:
124:
123:
118:
110:
57:Lee-Yang theorem
2282:
2281:
2277:
2276:
2275:
2273:
2272:
2271:
2247:
2246:
2234:
2229:
2217:
2213:
2208:
2204:
2199:
2195:
2190:
2186:
2181:
2177:
2164:
2160:
2155:
2151:
2147:
2111:
2078:
2077:
2075:Möbius function
2051:
2044:
2033:
2003:
1993:
1939:
1890:
1889:
1872:
1866:
1816:
1815:
1804:
1800:
1794:
1747:
1746:
1729:
1703:
1692:
1691:
1663:
1662:
1616:
1615:
1522:
1515:
1514:
1492:
1491:
1464:
1463:
1435:
1434:
1424:
1374:
1345:
1335:
1312:
1307:
1306:
1284:
1283:
1244:
1239:
1238:
1200:
1178:
1177:
1160:
1154:is canonical.
1136:
1135:
1058:
992:
991:
969:
968:
949:
948:
924:
905:
882:
867:
866:
844:
843:
820:
819:
789:
766:
737:
716:
715:
693:
692:
665:
660:
659:
656:
632:
631:
583:
582:
555:
554:
525:
520:
505:
500:
485:
480:
465:
460:
445:
440:
425:
420:
409:
408:
386:
385:
364:
349:
348:
315:
296:
283:
270:
257:
244:
234:
233:
211:
210:
165:
160:
159:
101:
100:
90:
85:
43:and methods in
17:
12:
11:
5:
2280:
2278:
2270:
2269:
2264:
2259:
2249:
2248:
2245:
2244:
2233:
2232:External links
2230:
2228:
2227:
2211:
2202:
2193:
2184:
2175:
2158:
2148:
2146:
2143:
2110:
2107:
2094:
2091:
2088:
2085:
2050:
2047:
2042:
2031:
2001:
1991:
1977:
1976:
1965:
1962:
1959:
1956:
1953:
1946:
1942:
1938:
1931:
1926:
1923:
1920:
1916:
1912:
1908:
1902:
1893:
1887:
1884:
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1120:
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1111:
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1057:
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1051:
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1045:
1041:
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1034:
1031:
1028:
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1019:
1016:
1013:
1009:
1005:
1002:
999:
976:
956:
945:
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927:
923:
920:
917:
912:
908:
902:
898:
894:
889:
885:
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851:
816:
815:
802:
799:
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792:
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723:
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652:
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614:
611:
608:
605:
602:
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567:
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488:
483:
479:
472:
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463:
459:
452:
448:
443:
439:
432:
428:
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419:
416:
393:
371:
367:
361:
357:
345:
344:
333:
330:
327:
322:
318:
314:
311:
308:
303:
299:
295:
290:
286:
282:
277:
273:
269:
264:
260:
256:
251:
247:
242:
218:
198:
195:
192:
189:
186:
183:
180:
177:
172:
168:
116:
113:
109:
89:
86:
84:
81:
31:discovered by
15:
13:
10:
9:
6:
4:
3:
2:
2279:
2268:
2265:
2263:
2260:
2258:
2257:Number theory
2255:
2254:
2252:
2243:
2239:
2236:
2235:
2231:
2225:
2221:
2215:
2212:
2206:
2203:
2197:
2194:
2188:
2185:
2179:
2176:
2172:
2168:
2162:
2159:
2153:
2150:
2144:
2142:
2141:, and so on.
2140:
2136:
2132:
2128:
2124:
2120:
2116:
2108:
2106:
2089:
2083:
2076:
2072:
2068:
2064:
2060:
2056:
2048:
2046:
2041:
2037:
2034: =
2030:
2026:
2022:
2017:
2015:
2011:
2007:
2000:
1996:
1990:
1986:
1983: =
1982:
1960:
1954:
1951:
1944:
1940:
1936:
1924:
1921:
1918:
1914:
1910:
1906:
1900:
1891:
1885:
1882:
1879:
1876:
1873:
1867:
1863:
1860:
1850:
1847:
1844:
1840:
1836:
1832:
1826:
1817:
1809:
1805:
1801:
1795:
1791:
1788:
1778:
1775:
1772:
1768:
1764:
1758:
1752:
1745:
1744:
1743:
1741:
1737:
1734:
1726:
1724:
1708:
1704:
1700:
1697:
1674:
1668:
1642:
1639:
1636:
1627:
1624:
1621:
1612:
1592:
1589:
1586:
1573:
1567:
1564:
1561:
1558:
1555:
1550:
1547:
1544:
1541:
1538:
1535:
1529:
1523:
1513:
1512:
1511:
1497:
1472:
1469:
1446:
1440:
1432:
1427:
1421:
1405:
1402:
1399:
1396:
1393:
1390:
1387:
1384:
1379:
1375:
1369:
1365:
1361:
1358:
1355:
1350:
1346:
1340:
1336:
1330:
1326:
1322:
1317:
1313:
1305:
1304:
1303:
1289:
1266:
1263:
1260:
1257:
1254:
1249:
1245:
1237:
1236:
1235:
1215:
1205:
1201:
1197:
1191:
1183:
1176:
1175:
1174:
1172:
1168:
1165:
1157:
1155:
1115:
1109:
1106:
1100:
1094:
1091:
1085:
1082:
1079:
1073:
1070:
1065:
1055:
1052:
1049:
1046:
1035:
1032:
1029:
1026:
1020:
1014:
1011:
1003:
1000:
990:
989:
988:
974:
929:
925:
921:
918:
915:
910:
906:
900:
896:
892:
887:
883:
879:
865:
864:
863:
849:
800:
797:
794:
790:
786:
776:
771:
767:
763:
760:
757:
747:
742:
738:
734:
724:
714:
713:
712:
678:
675:
670:
666:
653:
651:
637:
628:
615:
612:
609:
606:
603:
600:
597:
594:
591:
588:
565:
537:
530:
526:
521:
517:
510:
506:
501:
497:
490:
486:
481:
477:
470:
466:
461:
457:
450:
446:
441:
437:
430:
426:
421:
417:
414:
407:
406:
405:
391:
369:
365:
359:
355:
328:
325:
320:
316:
312:
309:
306:
301:
297:
293:
288:
284:
280:
275:
271:
267:
262:
258:
254:
249:
245:
232:
231:
230:
216:
196:
193:
190:
187:
184:
181:
178:
175:
170:
166:
157:
153:
149:
145:
141:
137:
133:
130:
129:prime numbers
111:
98:
95:
94:Hilbert space
87:
82:
80:
78:
77:string theory
74:
70:
66:
62:
58:
54:
50:
46:
42:
41:number theory
38:
34:
33:Bernard Julia
30:
26:
22:
2214:
2205:
2196:
2187:
2178:
2170:
2166:
2161:
2152:
2127:prime ideals
2112:
2052:
2039:
2035:
2028:
2020:
2018:
2009:
1998:
1994:
1988:
1984:
1980:
1978:
1735:
1730:
1613:
1610:
1428:
1425:
1281:
1233:
1170:
1166:
1161:
1133:
946:
817:
657:
629:
552:
346:
151:
147:
139:
131:
96:
91:
68:
60:
55:such as the
28:
24:
18:
2061:, then the
2014:temperature
1173:, that is,
92:Consider a
88:State space
73:Mark Bowick
29:Riemann gas
2251:Categories
2145:References
2071:(−1)
69:free model
25:primon gas
2238:John Baez
2218:BubbleZ (
2084:μ
1955:ζ
1930:∞
1915:∑
1883:
1874:−
1864:
1856:∞
1841:∑
1802:−
1792:
1784:∞
1769:∑
1701:≥
1669:ω
1625:∼
1574:∼
1565:
1559:
1548:
1542:
1536:−
1524:ω
1473:∈
1441:ω
1403:
1388:
1366:∑
1362:⋅
1327:∑
1264:
1219:⟩
1195:⟩
1142:Φ
1092:≠
1066:∗
1056:∈
1044:∀
1040:⟹
1004:∈
998:∀
955:Φ
922:
916:⋅
897:⨁
880:∘
787:∘
764:∘
758:∘
735:∘
725:∘
722:Φ
699:Φ
613:…
569:⟩
538:⋯
518:⋯
498:⋅
478:⋅
458:⋅
438:⋅
356:∑
332:⟩
329:…
310:…
197:…
156:multisets
115:⟩
83:The model
2119:K-theory
2059:fermions
2027:of
1158:Energies
2169:, Vol.
2167:Physics
2004:is the
1134:Hence,
152:primons
61:primons
2055:bosons
1997:where
818:where
581:where
142:, the
23:, the
1979:with
1234:with
67:or a
37:model
35:is a
2117:and
2008:and
1731:The
51:and
2016:.
1880:log
1861:exp
1789:exp
1723:.
1709:100
1510::
1400:log
1385:log
1261:log
875:log
828:log
782:log
753:log
730:log
146:on
65:gas
27:or
19:In
2253::
2240:,
2171:47
2045:.
1765::=
1742::
1705:10
1562:ln
1556:ln
1545:ln
1539:ln
919:ln
511:11
502:11
302:11
229::
134:.
79:.
47:,
2093:)
2090:n
2087:(
2043:B
2040:k
2038:/
2036:E
2032:H
2029:T
2021:s
2010:T
2002:B
1999:k
1995:T
1992:B
1989:k
1987:/
1985:E
1981:s
1964:)
1961:s
1958:(
1952:=
1945:s
1941:n
1937:1
1925:1
1922:=
1919:n
1911:=
1907:)
1901:T
1896:B
1892:k
1886:n
1877:E
1868:(
1851:1
1848:=
1845:n
1837:=
1833:)
1827:T
1822:B
1818:k
1810:n
1806:E
1796:(
1779:1
1776:=
1773:n
1762:)
1759:T
1756:(
1753:Z
1736:Z
1698:N
1678:)
1675:X
1672:(
1649:)
1646:]
1643:N
1640:,
1637:1
1634:[
1631:(
1628:U
1622:X
1596:)
1593:1
1590:,
1587:0
1584:(
1579:N
1568:n
1551:n
1533:)
1530:n
1527:(
1498:n
1477:N
1470:n
1450:)
1447:n
1444:(
1406:n
1397:E
1394:=
1391:p
1380:p
1376:k
1370:p
1359:E
1356:=
1351:p
1347:E
1341:p
1337:k
1331:p
1323:=
1318:n
1314:E
1290:E
1267:p
1258:E
1255:=
1250:p
1246:E
1216:p
1212:|
1206:p
1202:E
1198:=
1192:p
1188:|
1184:H
1171:p
1167:H
1119:)
1116:y
1113:(
1110:F
1107:+
1104:)
1101:x
1098:(
1095:F
1089:)
1086:y
1083:+
1080:x
1077:(
1074:F
1071:,
1061:N
1053:y
1050:,
1047:x
1036:1
1033:+
1030:n
1027:=
1024:)
1021:n
1018:(
1015:F
1012:,
1008:N
1001:n
975:F
930:k
926:p
911:k
907:a
901:k
893:=
888:n
884:x
850:F
801:1
798:+
795:n
791:x
777:=
772:n
768:x
761:F
748:=
743:n
739:x
679:n
676:=
671:n
667:x
638:n
616:.
610:,
607:3
604:,
601:2
598:,
595:1
592:=
589:n
566:n
562:|
531:p
527:k
522:p
507:k
491:7
487:k
482:7
471:5
467:k
462:5
451:3
447:k
442:3
431:2
427:k
422:2
418:=
415:n
392:n
370:p
366:k
360:p
326:,
321:p
317:k
313:,
307:,
298:k
294:,
289:7
285:k
281:,
276:5
272:k
268:,
263:3
259:k
255:,
250:2
246:k
241:|
217:p
194:,
191:2
188:,
185:1
182:,
179:0
176:=
171:p
167:k
148:H
140:K
132:p
112:p
108:|
97:H
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