515:
hence certainly not prime, while in the sequence of values taken by the first polynomial, none are divisible by 3. Thus it seems plausible that the first polynomial will produce values with a higher density of primes than will the second. At the very least, this observation gives little reason to believe that the corresponding diagonals will be equally dense with primes. One should, of course, consider divisibility by primes other than 3. Examining divisibility by 5 as well, remainders upon division by 15 repeat with pattern 1, 11, 14, 10, 14, 11, 1, 14, 5, 4, 11, 11, 4, 5, 14 for the first polynomial, and with pattern 5, 0, 3, 14, 3, 0, 5, 3, 9, 8, 0, 0, 8, 9, 3 for the second, implying that only three out of 15 values in the second sequence are potentially prime (being divisible by neither 3 nor 5), while 12 out of 15 values in the first sequence are potentially prime (since only three are divisible by 5 and none are divisible by 3).
1471:
1495:
139:
1820:
151:
1427:
1459:
2047:
1483:
158:
often, the number spiral is started with the number 1 at the center, but it is possible to start with any number, and the same concentration of primes along diagonal, horizontal, and vertical lines is observed. Starting with 41 at the center gives a diagonal containing an unbroken string of 40 primes (starting from 1523 southwest of the origin, decreasing to 41 at the origin, and increasing to 1601 northeast of the origin), the longest example of its kind.
28:
1447:
20:
1149:
514:
takes successive values 0, 1, 2, .... For the first of these polynomials, the sequence of remainders is 1, 2, 2, 1, 2, 2, ..., while for the second, it is 2, 0, 0, 2, 0, 0, .... This implies that in the sequence of values taken by the second polynomial, two out of every three are divisible by 3, and
1416:
are also included in the Ulam spiral. The number 1 has only a single factor, itself; each prime number has two factors, itself and 1; composite numbers are divisible by at least three different factors. Using the size of the dot representing an integer to indicate the number of factors and coloring
191:
column, Gardner mentioned the earlier paper of
Klauber. Klauber describes his construction as follows, "The integers are arranged in triangular order with 1 at the apex, the second line containing numbers 2 to 4, the third 5 to 9, and so forth. When the primes have been indicated, it is found that
175:
to extend the calculation to about 100,000 points. The group also computed the density of primes among numbers up to 10,000,000 along some of the prime-rich lines as well as along some of the prime-poor lines. Images of the spiral up to 65,000 points were displayed on "a scope attached to the
157:
In the figure, primes appear to concentrate along certain diagonal lines. In the 201×201 Ulam spiral shown above, diagonal lines are clearly visible, confirming the pattern to that point. Horizontal and vertical lines with a high density of primes, while less prominent, are also evident. Most
166:
According to
Gardner, Ulam discovered the spiral in 1963 while doodling during the presentation of "a long and very boring paper" at a scientific meeting. These hand calculations amounted to "a few hundred points". Shortly afterwards, Ulam, with collaborators Myron Stein and Mark Wells, used
74:
Ulam and
Gardner emphasized the striking appearance in the spiral of prominent diagonal, horizontal, and vertical lines containing large numbers of primes. Both Ulam and Gardner noted that the existence of such prominent lines is not unexpected, as lines in the spiral correspond to
593:
1013:
576:
odd. Conjecture F provides a formula that can be used to estimate the density of primes along such rays. It implies that there will be considerable variability in the density along different rays. In particular, the density is highly sensitive to the
1400:
takes the values 0, 1, 2, ... This curve asymptotically approaches a horizontal line in the left half of the figure. (In the Ulam spiral, Euler's polynomial forms two diagonal lines, one in the top half of the figure, corresponding to even values of
518:
While rigorously-proved results about primes in quadratic sequences are scarce, considerations like those above give rise to a plausible conjecture on the asymptotic density of primes in such sequences, which is described in the next section.
865:
for this polynomial is approximately 6.6, meaning that the numbers it generates are almost seven times as likely to be prime as random numbers of comparable size, according to the conjecture. This particular polynomial is related to Euler's
116:
constructed a triangular, non-spiral array containing vertical and diagonal lines exhibiting a similar concentration of prime numbers. Like Ulam, Klauber noted the connection with prime-generating polynomials, such as Euler's.
800:
539:
stated a series of conjectures, one of which, if true, would explain some of the striking features of the Ulam spiral. This conjecture, which Hardy and
Littlewood called "Conjecture F", is a special case of the
965:
1470:
1320:
102:. In particular, no quadratic polynomial has ever been proved to generate infinitely many primes, much less to have a high asymptotic density of them, although there is a well-supported
853:
can take values bigger or smaller than 1, some polynomials, according to the conjecture, will be especially rich in primes, and others especially poor. An unusually rich polynomial is 4
415:
1144:{\displaystyle A=\varepsilon \prod _{p}{\biggl (}{\frac {p}{p-1}}{\biggr )}\,\prod _{\varpi }{\biggl (}1-{\frac {1}{\varpi -1}}{\Bigl (}{\frac {\Delta }{\varpi }}{\Bigr )}{\biggr )}}
688:
are both even, the polynomial produces only even values, and is therefore composite except possibly for the value 2. Hardy and
Littlewood assert that, apart from these situations,
1426:
261:
1216:
508:
463:
325:
1277:
998:
1244:
192:
there are concentrations in certain vertical and diagonal lines, and amongst these the so-called Euler sequences with high concentrations of primes are discovered."
23:
Ulam spiral of size 201×201. Black dots represent prime numbers. Diagonal, vertical, and horizontal lines with a high density of prime numbers are clearly visible.
417:
and are therefore never prime except possibly when one of the factors equals 1. Such examples correspond to diagonals that are devoid of primes or nearly so.
556:. Rays emanating from the central region of the Ulam spiral making angles of 45° with the horizontal and vertical correspond to numbers of the form 4
1458:
738:
1365:. As in the Ulam spiral, quadratic polynomials generate numbers that lie in straight lines. Vertical lines correspond to numbers of the form
94:+ 41, are believed to produce a high density of prime numbers. Nevertheless, the Ulam spiral is connected with major unsolved problems in
1796:
Hardy, G. H.; Littlewood, J. E. (1923), "Some
Problems of 'Partitio Numerorum'; III: On the Expression of a Number as a Sum of Primes",
282:. It is therefore no surprise that all primes other than 2 lie in alternate diagonals of the Ulam spiral. Some polynomials, such as
899:
138:
1482:
184:
and featured on the front cover of that issue. Some of the photographs of Stein, Ulam, and Wells were reproduced in the column.
1494:
867:
84:
1787:
1700:
1388:
occurs in each full rotation. (In the Ulam spiral, two squares occur in each rotation.) Euler's prime-generating polynomial,
1290:
420:
To gain insight into why some of the remaining odd diagonals may have a higher concentration of primes than others, consider
1883:
1380:
Robert Sacks devised a variant of the Ulam spiral in 1994. In the Sacks spiral, the non-negative integers are plotted on an
608:= 0, 1, 2, ... have been highlighted in purple. The prominent parallel line in the lower half of the figure corresponds to 4
1420:
Spirals following other tilings of the plane also generate lines rich in prime numbers, for example hexagonal spirals.
1645:
1985:
1902:
Stein, M. L.; Ulam, S. M.; Wells, M. B. (1964), "A Visual
Display of Some Properties of the Distribution of Primes",
1839:
31:
For comparison, a spiral with random odd numbers colored black (at the same density of primes in a 200x200 spiral).
1007:
and therefore takes one of the values 0, 1, or 2. Hardy and
Littlewood break the product into three factors as
330:
1476:
Hexagonal number spiral with prime numbers in green and more highly composite numbers in darker shades of blue.
172:
1904:
1827:
1722:
278:
is even, the lines are diagonal, and either all numbers are odd, or all are even, depending on the value of
1850:"Quadratic polynomials producing consecutive, distinct primes and class groups of complex quadratic fields"
150:
541:
1279:
equals 1, 2, or 0 depending on whether the discriminant is 0, a non-zero square, or a non-square modulo
2071:
1653:
206:
2120:
1819:
200:
Diagonal, horizontal, and vertical lines in the number spiral correspond to polynomials of the form
652:
is positive. If the coefficients contain a common factor greater than 1 or if the discriminant Δ =
528:
57:
1186:
2243:
536:
468:
423:
285:
113:
1253:
974:
2279:
705:
1377:. Vertical and diagonal lines with a high density of prime numbers are evident in the figure.
837:
expected in a random set of numbers having the same density as the set of numbers of the form
176:
machine" and then photographed. The Ulam spiral was described in Martin
Gardner's March 1964
2238:
2036:
1978:
1229:
2187:
1662:
1405:
in the sequence, the other in the bottom half of the figure corresponding to odd values of
826:
99:
1627:
8:
2248:
1755:
1511:
1345:≈ 11.3, currently the highest known value, has been discovered by Jacobson and Williams.
63:
1666:
2219:
2204:
2167:
1949:
1921:
1768:
1739:
1381:
1001:
708:
and remains open. Hardy and
Littlewood further assert that, asymptotically, the number
76:
1849:
2182:
2172:
2130:
2026:
1835:
1783:
1696:
1446:
665:
48:
2284:
2081:
1971:
1941:
1913:
1866:
1857:
1807:
1798:
1764:
1735:
1731:
1670:
1413:
1675:
2214:
2066:
2016:
1690:
1284:
16:
Visualization of the prime numbers formed by arranging the integers into a spiral
2145:
2031:
1932:
Stein, M.; Ulam, S. M. (1967), "An Observation on the Distribution of Primes",
1780:
Martin Gardner's Sixth Book of Mathematical Diversions from Scientific American
1775:
1750:
572:
even; horizontal and vertical rays correspond to numbers of the same form with
130:
52:
1963:
2273:
2224:
1753:(March 1964), "Mathematical Games: The Remarkable Lore of the Prime Number",
1516:
1385:
704:
takes the values 0, 1, 2, ... This statement is a special case of an earlier
661:
110:
95:
27:
1720:
Daus, P. H. (1932), "The March Meeting of the Southern California Section",
67:
a short time later. It is constructed by writing the positive integers in a
2199:
2194:
2140:
2086:
1879:
795:{\displaystyle P(n)\sim A{\frac {1}{\sqrt {a}}}{\frac {\sqrt {n}}{\log n}}}
578:
44:
1871:
2110:
2061:
532:
1417:
prime numbers red and composite numbers blue produces the figure shown.
672:
takes the values 0, 1, 2, ... (except possibly for one or two values of
2232:
2135:
2115:
1953:
1925:
1812:
1743:
1384:
rather than the square spiral used by Ulam, and are spaced so that one
544:
and asserts an asymptotic formula for the number of primes of the form
2253:
2177:
2102:
2076:
2021:
1464:
Ulam spiral of size 150×150 showing both prime and composite numbers.
168:
125:
The Ulam spiral is constructed by writing the positive integers in a
68:
1945:
1917:
1432:
Klauber triangle with prime numbers generated by Euler's polynomial
2209:
2125:
19:
861:+ 41 which forms a visible line in the Ulam spiral. The constant
960:{\displaystyle A=\prod \limits _{p}{\frac {p-\omega (p)}{p-1}}~}
1994:
1646:"New quadratic polynomials with high densities of prime values"
1338:. Consequently, such primes do not contribute to the product.
327:, while producing only odd values, factorize over the integers
126:
833:
set equal to one is the asymptotic number of primes less than
2002:
1998:
80:
1595:
1544:
103:
1488:
Number spiral with 7503 primes visible on regular triangle.
1155:
Here the factor ε, corresponding to the prime 2, is 1 if
592:
1534:
1532:
1832:
Archimedes' Revenge: The Joys and Perils of Mathematics
1353:
Klauber's 1932 paper describes a triangle in which row
1315:{\displaystyle \left({\frac {\Delta }{\varpi }}\right)}
624:
Conjecture F is concerned with polynomials of the form
1246:
runs over the infinitely-many odd primes not dividing
893:
is given by a product running over all prime numbers,
522:
1573:
1571:
1556:
1529:
1293:
1256:
1232:
1189:
1175:
runs over the finitely-many odd primes dividing both
1016:
977:
902:
741:
471:
426:
333:
288:
209:
1643:
1607:
1912:(5), Mathematical Association of America: 516–520,
1730:(7), Mathematical Association of America: 373–374,
1583:
664:, the polynomial factorizes and therefore produces
1568:
1314:
1271:
1238:
1210:
1143:
992:
959:
794:
502:
457:
409:
319:
255:
1940:(1), Mathematical Association of America: 43–44,
1136:
1129:
1112:
1081:
1063:
1038:
676:where one of the factors equals 1). Moreover, if
109:In 1932, 31 years prior to Ulam's discovery, the
2271:
1993:
1795:
1632:, The Online Archive of California, p. 117
1000:is number of zeros of the quadratic polynomial
1901:
1601:
1550:
106:as to what that asymptotic density should be.
1979:
616:+ 41 or, equivalently, to negative values of
1644:Jacobson Jr., M. J.; Williams, H. C (2003),
2095:
510:. Compute remainders upon division by 3 as
1986:
1972:
1283:. This is accounted for by the use of the
410:{\displaystyle (4n^{2}+8n+3)=(2n+1)(2n+3)}
1931:
1878:
1870:
1811:
1674:
1068:
71:and specially marking the prime numbers.
591:
79:, and certain such polynomials, such as
26:
18:
1826:
1774:
1749:
1625:
1613:
1562:
1538:
1396:+ 41, now appears as a single curve as
700:takes prime values infinitely often as
43:is a graphical depiction of the set of
2272:
1847:
1589:
1412:Additional structure may be seen when
1967:
1890:, Mathematical Association of America
1695:(3rd ed.), Springer, p. 8,
1719:
1577:
145:and then marking the prime numbers:
1688:
1500:Ulam spiral with 10 million primes.
910:
523:Hardy and Littlewood's Conjecture F
13:
1769:10.1038/scientificamerican0364-120
1692:Unsolved problems in number theory
1629:Guide to the Martin Gardner papers
1300:
1119:
889:to the even numbers. The constant
885:, or equivalently, by restricting
14:
2296:
1171:is even. The first product index
2045:
1818:
1493:
1481:
1469:
1457:
1445:
1425:
256:{\displaystyle f(n)=4n^{2}+bn+c}
173:Los Alamos Scientific Laboratory
149:
137:
1834:, New York: Fawcett Colombine,
1782:, University of Chicago Press,
1712:
1682:
120:
1884:"Prime generating polynomials"
1736:10.1080/00029890.1932.11987331
1637:
1619:
1266:
1260:
1199:
1193:
987:
981:
937:
931:
751:
745:
404:
389:
386:
371:
365:
334:
219:
213:
195:
1:
1934:American Mathematical Monthly
1905:American Mathematical Monthly
1723:American Mathematical Monthly
1676:10.1090/S0025-5718-02-01418-7
1522:
274:are integer constants. When
1602:Stein, Ulam & Wells 1964
1551:Stein, Ulam & Wells 1964
1341:A quadratic polynomial with
1211:{\displaystyle \omega (p)=0}
7:
1505:
1348:
1226:. The second product index
868:prime-generating polynomial
527:In their 1923 paper on the
503:{\displaystyle 4n^{2}+6n+5}
458:{\displaystyle 4n^{2}+6n+1}
320:{\displaystyle 4n^{2}+8n+3}
85:prime-generating polynomial
51:in 1963 and popularized in
47:, devised by mathematician
10:
2301:
1654:Mathematics of Computation
1272:{\displaystyle \omega (p)}
993:{\displaystyle \omega (p)}
161:
2160:
2054:
2043:
2009:
1334:there is one root modulo
706:conjecture of Bunyakovsky
1689:Guy, Richard K. (2004),
1626:Hartwig, Daniel (2013),
716:) of primes of the form
596:The primes of the form 4
1239:{\displaystyle \varpi }
542:Bateman–Horn conjecture
1357:contains the numbers (
1316:
1273:
1240:
1212:
1145:
994:
961:
796:
621:
504:
459:
411:
321:
257:
187:In an addendum to the
32:
24:
1872:10.4064/aa-74-1-17-30
1848:Mollin, R.A. (1996),
1317:
1274:
1241:
1213:
1146:
995:
962:
797:
595:
505:
460:
412:
322:
258:
77:quadratic polynomials
30:
22:
1291:
1254:
1230:
1187:
1183:. For these primes
1014:
975:
900:
829:, this formula with
827:prime number theorem
739:
469:
424:
331:
286:
207:
1756:Scientific American
1667:2003MaCom..72..499J
1512:Pattern recognition
1250:. For these primes
1222:then cannot divide
581:of the polynomial,
529:Goldbach Conjecture
189:Scientific American
182:Scientific American
64:Scientific American
1813:10.1007/BF02403921
1440:+ 41 highlighted.
1409:in the sequence.)
1382:Archimedean spiral
1361:− 1) + 1 through
1312:
1269:
1236:
1208:
1141:
1078:
1035:
990:
957:
918:
877:+ 41 by replacing
792:
622:
500:
455:
407:
317:
253:
178:Mathematical Games
58:Mathematical Games
33:
25:
2267:
2266:
2156:
2155:
1882:(July 17, 2006),
1789:978-0-226-28250-3
1702:978-0-387-20860-2
1414:composite numbers
1306:
1125:
1108:
1069:
1059:
1026:
956:
952:
909:
790:
778:
770:
769:
666:composite numbers
648:are integers and
129:arrangement on a
100:Landau's problems
2292:
2093:
2092:
2072:Boerdijk–Coxeter
2049:
2048:
1988:
1981:
1974:
1965:
1964:
1956:
1928:
1898:
1897:
1895:
1875:
1874:
1858:Acta Arithmetica
1854:
1844:
1823:
1822:
1816:
1815:
1799:Acta Mathematica
1792:
1771:
1746:
1706:
1705:
1686:
1680:
1679:
1678:
1661:(241): 499–519,
1650:
1641:
1635:
1633:
1623:
1617:
1611:
1605:
1599:
1593:
1587:
1581:
1575:
1566:
1560:
1554:
1548:
1542:
1536:
1497:
1485:
1473:
1461:
1449:
1429:
1322:. When a prime
1321:
1319:
1318:
1313:
1311:
1307:
1299:
1278:
1276:
1275:
1270:
1245:
1243:
1242:
1237:
1217:
1215:
1214:
1209:
1163:is odd and 2 if
1150:
1148:
1147:
1142:
1140:
1139:
1133:
1132:
1126:
1118:
1116:
1115:
1109:
1107:
1093:
1085:
1084:
1077:
1067:
1066:
1060:
1058:
1044:
1042:
1041:
1034:
999:
997:
996:
991:
966:
964:
963:
958:
954:
953:
951:
940:
920:
917:
801:
799:
798:
793:
791:
789:
774:
773:
771:
765:
761:
509:
507:
506:
501:
484:
483:
464:
462:
461:
456:
439:
438:
416:
414:
413:
408:
349:
348:
326:
324:
323:
318:
301:
300:
262:
260:
259:
254:
237:
236:
153:
141:
114:Laurence Klauber
2300:
2299:
2295:
2294:
2293:
2291:
2290:
2289:
2270:
2269:
2268:
2263:
2152:
2106:
2091:
2050:
2046:
2041:
2005:
1992:
1961:
1959:
1946:10.2307/2314055
1918:10.2307/2312588
1893:
1891:
1852:
1842:
1817:
1790:
1715:
1710:
1709:
1703:
1687:
1683:
1648:
1642:
1638:
1624:
1620:
1612:
1608:
1600:
1596:
1588:
1584:
1576:
1569:
1561:
1557:
1549:
1545:
1537:
1530:
1525:
1508:
1501:
1498:
1489:
1486:
1477:
1474:
1465:
1462:
1453:
1450:
1441:
1430:
1351:
1298:
1294:
1292:
1289:
1288:
1285:Legendre symbol
1255:
1252:
1251:
1231:
1228:
1227:
1188:
1185:
1184:
1135:
1134:
1128:
1127:
1117:
1111:
1110:
1097:
1092:
1080:
1079:
1073:
1062:
1061:
1048:
1043:
1037:
1036:
1030:
1015:
1012:
1011:
976:
973:
972:
941:
921:
919:
913:
901:
898:
897:
779:
772:
760:
740:
737:
736:
525:
479:
475:
470:
467:
466:
434:
430:
425:
422:
421:
344:
340:
332:
329:
328:
296:
292:
287:
284:
283:
232:
228:
208:
205:
204:
198:
164:
123:
17:
12:
11:
5:
2298:
2288:
2287:
2282:
2265:
2264:
2262:
2261:
2256:
2251:
2246:
2241:
2236:
2229:
2228:
2227:
2217:
2212:
2207:
2202:
2197:
2192:
2191:
2190:
2185:
2180:
2170:
2164:
2162:
2158:
2157:
2154:
2153:
2151:
2150:
2149:
2148:
2138:
2133:
2128:
2123:
2118:
2113:
2108:
2104:
2099:
2097:
2090:
2089:
2084:
2079:
2074:
2069:
2064:
2058:
2056:
2052:
2051:
2044:
2042:
2040:
2039:
2034:
2029:
2024:
2019:
2013:
2011:
2007:
2006:
1991:
1990:
1983:
1976:
1968:
1958:
1957:
1929:
1899:
1876:
1845:
1840:
1824:
1793:
1788:
1772:
1747:
1716:
1714:
1711:
1708:
1707:
1701:
1681:
1636:
1618:
1606:
1604:, p. 520.
1594:
1582:
1580:, p. 373.
1567:
1565:, p. 124.
1555:
1553:, p. 517.
1543:
1541:, p. 122.
1527:
1526:
1524:
1521:
1520:
1519:
1514:
1507:
1504:
1503:
1502:
1499:
1492:
1490:
1487:
1480:
1478:
1475:
1468:
1466:
1463:
1456:
1454:
1451:
1444:
1442:
1431:
1424:
1386:perfect square
1350:
1347:
1310:
1305:
1302:
1297:
1268:
1265:
1262:
1259:
1235:
1207:
1204:
1201:
1198:
1195:
1192:
1153:
1152:
1138:
1131:
1124:
1121:
1114:
1106:
1103:
1100:
1096:
1091:
1088:
1083:
1076:
1072:
1065:
1057:
1054:
1051:
1047:
1040:
1033:
1029:
1025:
1022:
1019:
989:
986:
983:
980:
969:
968:
950:
947:
944:
939:
936:
933:
930:
927:
924:
916:
912:
908:
905:
803:
802:
788:
785:
782:
777:
768:
764:
759:
756:
753:
750:
747:
744:
728:and less than
662:perfect square
524:
521:
499:
496:
493:
490:
487:
482:
478:
474:
454:
451:
448:
445:
442:
437:
433:
429:
406:
403:
400:
397:
394:
391:
388:
385:
382:
379:
376:
373:
370:
367:
364:
361:
358:
355:
352:
347:
343:
339:
336:
316:
313:
310:
307:
304:
299:
295:
291:
264:
263:
252:
249:
246:
243:
240:
235:
231:
227:
224:
221:
218:
215:
212:
197:
194:
163:
160:
155:
154:
143:
142:
131:square lattice
122:
119:
53:Martin Gardner
49:Stanisław Ulam
15:
9:
6:
4:
3:
2:
2297:
2286:
2283:
2281:
2280:Prime numbers
2278:
2277:
2275:
2260:
2257:
2255:
2252:
2250:
2247:
2245:
2242:
2240:
2237:
2235:
2234:
2230:
2226:
2223:
2222:
2221:
2218:
2216:
2213:
2211:
2208:
2206:
2203:
2201:
2198:
2196:
2193:
2189:
2186:
2184:
2181:
2179:
2176:
2175:
2174:
2171:
2169:
2166:
2165:
2163:
2159:
2147:
2144:
2143:
2142:
2139:
2137:
2134:
2132:
2129:
2127:
2124:
2122:
2119:
2117:
2114:
2112:
2109:
2107:
2101:
2100:
2098:
2094:
2088:
2085:
2083:
2080:
2078:
2075:
2073:
2070:
2068:
2065:
2063:
2060:
2059:
2057:
2053:
2038:
2035:
2033:
2030:
2028:
2025:
2023:
2020:
2018:
2015:
2014:
2012:
2008:
2004:
2000:
1996:
1989:
1984:
1982:
1977:
1975:
1970:
1969:
1966:
1962:
1955:
1951:
1947:
1943:
1939:
1935:
1930:
1927:
1923:
1919:
1915:
1911:
1907:
1906:
1900:
1889:
1885:
1881:
1880:Pegg, Jr., Ed
1877:
1873:
1868:
1864:
1860:
1859:
1851:
1846:
1843:
1841:0-449-00089-3
1837:
1833:
1829:
1828:Hoffman, Paul
1825:
1821:
1814:
1809:
1805:
1801:
1800:
1794:
1791:
1785:
1781:
1777:
1773:
1770:
1766:
1762:
1758:
1757:
1752:
1748:
1745:
1741:
1737:
1733:
1729:
1725:
1724:
1718:
1717:
1704:
1698:
1694:
1693:
1685:
1677:
1672:
1668:
1664:
1660:
1656:
1655:
1647:
1640:
1631:
1630:
1622:
1616:, p. 88.
1615:
1610:
1603:
1598:
1592:, p. 21.
1591:
1586:
1579:
1574:
1572:
1564:
1559:
1552:
1547:
1540:
1535:
1533:
1528:
1518:
1517:Prime k-tuple
1515:
1513:
1510:
1509:
1496:
1491:
1484:
1479:
1472:
1467:
1460:
1455:
1452:Sacks spiral.
1448:
1443:
1439:
1435:
1428:
1423:
1422:
1421:
1418:
1415:
1410:
1408:
1404:
1399:
1395:
1391:
1387:
1383:
1378:
1376:
1372:
1368:
1364:
1360:
1356:
1346:
1344:
1339:
1337:
1333:
1329:
1325:
1308:
1303:
1295:
1286:
1282:
1263:
1257:
1249:
1233:
1225:
1221:
1205:
1202:
1196:
1190:
1182:
1178:
1174:
1170:
1166:
1162:
1158:
1122:
1104:
1101:
1098:
1094:
1089:
1086:
1074:
1070:
1055:
1052:
1049:
1045:
1031:
1027:
1023:
1020:
1017:
1010:
1009:
1008:
1006:
1003:
984:
978:
948:
945:
942:
934:
928:
925:
922:
914:
906:
903:
896:
895:
894:
892:
888:
884:
880:
876:
872:
869:
864:
860:
856:
852:
848:
844:
840:
836:
832:
828:
824:
820:
816:
812:
808:
786:
783:
780:
775:
766:
762:
757:
754:
748:
742:
735:
734:
733:
731:
727:
723:
719:
715:
711:
707:
703:
699:
695:
691:
687:
683:
679:
675:
671:
667:
663:
659:
655:
651:
647:
643:
639:
635:
631:
627:
619:
615:
611:
607:
603:
599:
594:
590:
588:
584:
580:
575:
571:
567:
563:
559:
555:
551:
547:
543:
538:
534:
530:
520:
516:
513:
497:
494:
491:
488:
485:
480:
476:
472:
452:
449:
446:
443:
440:
435:
431:
427:
418:
401:
398:
395:
392:
383:
380:
377:
374:
368:
362:
359:
356:
353:
350:
345:
341:
337:
314:
311:
308:
305:
302:
297:
293:
289:
281:
277:
273:
269:
250:
247:
244:
241:
238:
233:
229:
225:
222:
216:
210:
203:
202:
201:
193:
190:
185:
183:
179:
174:
170:
159:
152:
148:
147:
146:
140:
136:
135:
134:
132:
128:
118:
115:
112:
111:herpetologist
107:
105:
101:
97:
96:number theory
93:
89:
86:
82:
78:
72:
70:
69:square spiral
66:
65:
60:
59:
54:
50:
46:
45:prime numbers
42:
38:
29:
21:
2258:
2231:
2096:Biochemistry
1960:
1937:
1933:
1909:
1903:
1892:, retrieved
1887:
1862:
1856:
1831:
1803:
1797:
1779:
1760:
1754:
1727:
1721:
1713:Bibliography
1691:
1684:
1658:
1652:
1639:
1628:
1621:
1614:Gardner 1971
1609:
1597:
1585:
1563:Gardner 1964
1558:
1546:
1539:Gardner 1964
1437:
1433:
1419:
1411:
1406:
1402:
1397:
1393:
1389:
1379:
1374:
1370:
1366:
1362:
1358:
1354:
1352:
1342:
1340:
1335:
1331:
1327:
1323:
1280:
1247:
1223:
1219:
1180:
1176:
1172:
1168:
1164:
1160:
1156:
1154:
1004:
970:
890:
886:
882:
878:
874:
870:
862:
858:
854:
850:
849:. But since
846:
842:
838:
834:
830:
822:
818:
814:
810:
806:
804:
732:is given by
729:
725:
721:
717:
713:
709:
701:
697:
693:
689:
685:
681:
677:
673:
669:
657:
653:
649:
645:
641:
637:
633:
629:
625:
623:
617:
613:
609:
605:
601:
597:
586:
582:
579:discriminant
573:
569:
565:
561:
557:
553:
549:
545:
526:
517:
511:
419:
279:
275:
271:
267:
265:
199:
188:
186:
181:
177:
165:
156:
144:
124:
121:Construction
108:
91:
87:
73:
62:
56:
41:prime spiral
40:
36:
34:
2244:Pitch angle
2220:Logarithmic
2168:Archimedean
2131:Polyproline
1776:Gardner, M.
1763:: 120–128,
1751:Gardner, M.
1590:Mollin 1996
821:but not on
809:depends on
196:Explanation
37:Ulam spiral
2274:Categories
2233:On Spirals
2183:Hyperbolic
1888:Math Games
1523:References
604:+ 41 with
537:Littlewood
180:column in
104:conjecture
61:column in
2254:Spirangle
2249:Theodorus
2188:Poinsot's
2178:Epispiral
2022:Curvature
2017:Algebraic
1894:1 January
1865:: 17–30,
1578:Daus 1932
1304:ϖ
1301:Δ
1258:ω
1234:ϖ
1191:ω
1123:ϖ
1120:Δ
1102:−
1099:ϖ
1090:−
1075:ϖ
1071:∏
1053:−
1028:∏
1024:ε
979:ω
971:in which
946:−
929:ω
926:−
911:∏
825:. By the
784:
755:∼
169:MANIAC II
2210:Involute
2205:Fermat's
2146:Collagen
2082:Symmetry
1830:(1988),
1806:: 1–70,
1778:(1971),
1506:See also
1349:Variants
1330:but not
1326:divides
98:such as
2285:Spirals
2239:Padovan
2173:Cotes's
2161:Spirals
2067:Antenna
2055:Helices
2027:Gallery
2003:helices
1995:Spirals
1954:2314055
1926:2312588
1744:2300380
1663:Bibcode
1218:since
162:History
2225:Golden
2141:Triple
2121:Double
2087:Triple
2037:Topics
2010:Curves
1999:curves
1952:
1924:
1838:
1786:
1742:
1699:
1002:modulo
955:
881:with 2
817:, and
805:where
644:, and
636:where
266:where
127:spiral
2200:Euler
2195:Doyle
2136:Super
2111:Alpha
2062:Angle
1950:JSTOR
1922:JSTOR
1853:(PDF)
1740:JSTOR
1649:(PDF)
660:is a
568:with
533:Hardy
81:Euler
2259:Ulam
2215:List
2116:Beta
2077:Hemi
2032:List
2001:and
1896:2019
1836:ISBN
1784:ISBN
1697:ISBN
1179:and
684:and
585:− 16
535:and
465:and
270:and
35:The
1942:doi
1914:doi
1867:doi
1808:doi
1765:doi
1761:210
1732:doi
1671:doi
1436:−
857:− 2
781:log
668:as
656:− 4
612:+ 2
600:− 2
171:at
83:'s
55:'s
39:or
2276::
2126:Pi
2105:10
1997:,
1948:,
1938:74
1936:,
1920:,
1910:71
1908:,
1886:,
1863:74
1861:,
1855:,
1804:44
1802:,
1759:,
1738:,
1728:39
1726:,
1669:,
1659:72
1657:,
1651:,
1570:^
1531:^
1392:−
1373:+
1369:−
1287:,
1167:+
1159:+
873:−
845:+
843:bx
841:+
839:ax
813:,
724:+
722:bx
720:+
718:ax
696:+
694:bx
692:+
690:ax
680:+
658:ac
640:,
632:+
630:bx
628:+
626:ax
589:.
564:+
562:bx
560:+
552:+
550:bx
548:+
546:ax
531:,
133::
90:−
2103:3
1987:e
1980:t
1973:v
1944::
1916::
1869::
1810::
1767::
1734::
1673::
1665::
1634:.
1438:x
1434:x
1407:x
1403:x
1398:x
1394:x
1390:x
1375:M
1371:k
1367:k
1363:n
1359:n
1355:n
1343:A
1336:p
1332:b
1328:a
1324:p
1309:)
1296:(
1281:p
1267:)
1264:p
1261:(
1248:a
1224:c
1220:p
1206:0
1203:=
1200:)
1197:p
1194:(
1181:b
1177:a
1173:p
1169:b
1165:a
1161:b
1157:a
1151:.
1137:)
1130:)
1113:(
1105:1
1095:1
1087:1
1082:(
1064:)
1056:1
1050:p
1046:p
1039:(
1032:p
1021:=
1018:A
1005:p
988:)
985:p
982:(
967:,
949:1
943:p
938:)
935:p
932:(
923:p
915:p
907:=
904:A
891:A
887:x
883:x
879:x
875:x
871:x
863:A
859:x
855:x
851:A
847:c
835:n
831:A
823:n
819:c
815:b
811:a
807:A
787:n
776:n
767:a
763:1
758:A
752:)
749:n
746:(
743:P
730:n
726:c
714:n
712:(
710:P
702:x
698:c
686:c
682:b
678:a
674:x
670:x
654:b
650:a
646:c
642:b
638:a
634:c
620:.
618:x
614:x
610:x
606:x
602:x
598:x
587:c
583:b
574:b
570:b
566:c
558:x
554:c
512:n
498:5
495:+
492:n
489:6
486:+
481:2
477:n
473:4
453:1
450:+
447:n
444:6
441:+
436:2
432:n
428:4
405:)
402:3
399:+
396:n
393:2
390:(
387:)
384:1
381:+
378:n
375:2
372:(
369:=
366:)
363:3
360:+
357:n
354:8
351:+
346:2
342:n
338:4
335:(
315:3
312:+
309:n
306:8
303:+
298:2
294:n
290:4
280:c
276:b
272:c
268:b
251:c
248:+
245:n
242:b
239:+
234:2
230:n
226:4
223:=
220:)
217:n
214:(
211:f
92:x
88:x
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
