To reduce the file sizes only the upper half plane roots are included in the following files. For each non-real root there is a corresponding complex conjugate root implied.
Root counts include upper and lower half planes 16000x16000 or 25000x25000 (Use Save Link As)
Named counts are for upper half plane only 20000x20000
Period | Roots Found (Named) | Missing | Data File |
---|---|---|---|
1 | 1 (1) | Complete | MRoots_1.txt |
2 | 1 (1) | Complete | MRoots_2.txt |
3 | 3 (1) | Complete | MRoots_3.txt |
4 | 6 (2) | Complete | MRoots_4.txt |
5 | 15 (2) | Complete | MRoots_5.txt |
6 | 27 (3) | Complete | MRoots_6.txt |
7 | 63 (3) | Complete | MRoots_7.txt |
8 | 120 (5) | Complete | MRoots_8.txt |
9 | 252 (5) | Complete | MRoots_9.txt |
10 | 495 (6) | Complete | MRoots_10.txt |
11 | 1,023 (5) | Complete | MRoots_11.txt |
12 | 2,010 (11) | Complete | MRoots_12.txt |
13 | 4,095 (6) | Complete | MRoots_13.txt |
14 | 8,127 (9) | Complete | MRoots_14.txt |
15 | 16,365 (12) | Complete | MRoots_15.txt (1.4MB) |
16 | 32,640 (14) | Complete | MRoots_16.txt (2.8MB) |
17 | 65,535 (8) | Complete | MRoots_17.txt (5.6MB) |
18 | 130,788 (19) | Complete | MRoots_18.txt (11.1MB) |
19 | 262,143 (9) | Complete | MRoots_19.txt (22.1MB) |
20 | 522,466 (22) | 1,304 | MRoots_20.txt (43.9MB) |
21 | 326,356 (18) | 722,153 | MRoots_21.txt (26.3MB) |
22 | 73,797 (15) | 2,022,330 | MRoots_22.txt (6.0MB) |
23 | 90,796 (11) | 4,103,507 | MRoots_23.txt (7.3MB) |
24 | 106,569 (39) | 8,279,871 | MRoots_24.txt (8.6MB) |
25 | 126,869 (18) | 16,650,331 | MRoots_25.txt (10.2MB) |
26 | 142,360 (18) | 33,407,975 | MRoots_26.txt (11.4MB) |
27 | 160,049 (25) | 66,948,559 | MRoots_27.txt (12.8MB) |
28 | 178,175 (33) | 134,031,355 | MRoots_28.txt (14.3MB) |
29 | 195,258 (14) | 268,240,197 | MRoots_29.txt (15.7MB) |
30 | 213,356 (52) | 536,640,649 | MRoots_30.txt (17.1MB) |
31 | 231,878 (15) | 1,073,509,945 | MRoots_31.txt (18.6MB) |
32 | 247,795 (41) | 2,147,203,085 | MRoots_32.txt (19.9MB) |
33 | 265,033 (30) | 4,294,701,236 | MRoots_33.txt (21.2MB) |
34 | 281,754 (24) | 8,589,587,301 | MRoots_34.txt (22.6MB) |
35 | 299,874 (36) | 17,179,569,231 | MRoots_35.txt (24.0MB) |
36 | 315,616 (79) | 34,359,289,664 | MRoots_36.txt (25.3MB) |
37 | 331,495 (18) | 68,719,145,240 | MRoots_37.txt (26.5MB) |
38 | 344,344 (27) | 137,438,346,983 | MRoots_38.txt (27.6MB) |
39 | 363,094 (36) | 274,877,539,751 | MRoots_39.txt (29.1MB) |
40 | 373,455 (78) | 549,754,916,025 | MRoots_40.txt (29.9MB) |
41 | 392,285 (20) | 1,099,511,235,490 | MRoots_41.txt (31.4MB) |
42 | 411,487 (78) | 2,199,021,787,334 | MRoots_42.txt (33.0MB) |
43 | 420,169 (21) | 4,398,046,090,934 | MRoots_43.txt (33.6MB) |
44 | 427,680 (55) | 8,796,090,497,370 | MRoots_44.txt (34.3MB) |
45 | 454,111 (76) | 17,592,185,573,669 | MRoots_45.txt (36.4MB) |
46 | 449,820 (33) | 35,184,367,444,707 | MRoots_46.txt (36.0MB) |
47 | 468,343 (23) | 70,368,743,709,320 | MRoots_47.txt (37.5MB) |
48 | 472,745 (135) | 140,737,479,461,335 | MRoots_48.txt (37.9MB) |
49 | 492,558 (39) | 281,474,976,218,034 | MRoots_49.txt (39.4MB) |
50 | 495,207 (70) | 562,949,936,148,393 | MRoots_50.txt (39.7MB) |
51 | 509,853 (48) | 1,125,899,906,267,232 | MRoots_51.txt (40.8MB) |
52 | 511,531 (66) | 2,251,799,779,619,279 | MRoots_52.txt (41.0MB) |
53 | 531,438 (26) | 4,503,599,626,839,057 | MRoots_53.txt (42.6MB) |
54 | 527,543 (115) | 9,007,199,186,973,769 | MRoots_54.txt (42.3MB) |
55 | 548,379 (60) | 18,014,398,508,932,566 | MRoots_55.txt (43.9MB) |
56 | 544,836 (117) | 36,028,796,884,201,284 | MRoots_56.txt (43.6MB) |
57 | 558,757 (54) | 72,057,594,037,107,032 | MRoots_57.txt (44.7MB) |
58 | 561,852 (42) | 144,115,187,806,858,563 | MRoots_58.txt (45.0MB) |
59 | 578,809 (29) | 288,230,376,151,132,934 | MRoots_59.txt (46.4MB) |
60 | 568,769 (228) | 576,460,751,765,458,021 | MRoots_60.txt (45.6MB) |
61 | 593,489 (30) | 1,152,921,504,606,253,486 | MRoots_61.txt (47.5MB) |
62 | 589,814 (45) | 2,305,843,008,139,362,313 | MRoots_62.txt (47.2MB) |
63 | 600,702 (114) | 4,611,686,018,425,738,374 | MRoots_63.txt (48.1MB) |
64 | 600,275 (122) | 9,223,372,034,706,691,885 | MRoots_64.txt (48.1MB) |
65 | 614,851 (72) | 18,446,744,073,708,932,654 | MRoots_65.txt (49.2MB) |
66 | 605,600 (130) | 36,893,488,143,121,434,181 | MRoots_66.txt (48.5MB) |
67 | 626,210 (33) | 73,786,976,294,837,580,253 | MRoots_67.txt (50.1MB) |
68 | 619,073 (88) | 147,573,952,581,085,859,257 | MRoots_68.txt (49.6MB) |
69 | 630,284 (66) | 295,147,905,179,348,001,265 | MRoots_69.txt (50.5MB) |
70 | 628,146 (156) | 590,295,810,341,525,145,759 | MRoots_70.txt (50.3MB) |
71 | 644,946 (35) | 1,180,591,620,717,410,658,477 | MRoots_71.txt (51.6MB) |
72 | 627,372 (311) | 2,361,183,241,400,453,854,548 | MRoots_72.txt (50.3MB) |
73 | 651,425 (36) | 4,722,366,482,869,644,562,270 | MRoots_73.txt (52.2MB) |
74 | 646,444 (54) | 9,444,732,965,670,570,304,211 | MRoots_74.txt (51.8MB) |
75 | 650,509 (140) | 18,889,465,931,478,563,410,691 | MRoots_75.txt (52.1MB) |
76 | 649,871 (99) | 37,778,931,862,819,722,106,219 | MRoots_76.txt (52.0MB) |
77 | 662,472 (90) | 75,557,863,725,914,322,755,577 | MRoots_77.txt (53.1MB) |
78 | 652,501 (156) | 151,115,727,451,553,734,728,464 | MRoots_78.txt (52.3MB) |
79 | 671,373 (39) | 302,231,454,903,657,293,005,170 | MRoots_79.txt (53.8MB) |
80 | 657,629 (270) | 604,462,909,806,764,830,848,931 | MRoots_80.txt (52.7MB) |
81 | 669,442 (125) | 1,208,925,819,614,629,106,927,870 | MRoots_81.txt (53.6MB) |
82 | 670,022 (60) | 2,417,851,639,228,158,837,114,553 | MRoots_82.txt (53.7MB) |
83 | 678,276 (41) | 4,835,703,278,458,516,698,146,427 | MRoots_83.txt (54.3MB) |
84 | 659,476 (342) | 9,671,406,556,914,834,239,522,834 | MRoots_84.txt (52.8MB) |
85 | 680,927 (96) | 19,342,813,113,834,066,794,552,338 | MRoots_85.txt (54.5MB) |
86 | 677,952 (63) | 38,685,626,227,663,735,543,408,575 | MRoots_86.txt (54.3MB) |
87 | 681,324 (84) | 77,371,252,455,336,266,912,078,481 | MRoots_87.txt (54.6MB) |
88 | 674,650 (195) | 154,742,504,910,663,738,268,693,550 | MRoots_88.txt (54.0MB) |
89 | 691,654 (44) | 309,485,009,821,345,068,724,089,401 | MRoots_89.txt (55.4MB) |
90 | 670,522 (404) | 618,970,019,642,672,544,725,861,858 | MRoots_90.txt(53.7MB) |
91 | 689,660 (108) | 1,237,940,039,285,380,274,898,430,405 | MRoots_91.txt(55.2MB) |
92 | 683,375 (121) | 2,475,880,078,570,725,365,425,476,235 | MRoots_92.txt(54.7MB) |
93 | 689,842 (90) | 4,951,760,157,141,521,098,522,065,227 | MRoots_93.txt(55.2MB) |
94 | 689,316 (69) | 9,903,520,314,282,971,830,448,126,811 | MRoots_94.txt(55.2MB) |
95 | 694,505 (108) | 19,807,040,628,566,084,398,385,030,920 | MRoots_95.txt(55.6MB) |
96 | 675,919 (459) | 39,614,081,257,132,028,057,135,493,041 | MRoots_96.txt (54.1MB) |
97 | 697,853 (48) | 79,228,162,514,264,337,593,543,252,482 | MRoots_97.txt (55.9MB) |
98 | 689,228 (153) | 158,456,325,028,528,393,712,110,492,660 | MRoots_98.txt (55.2MB) |
99 | 692,513 (190) | 316,912,650,057,057,350,369,880,141,283 | MRoots_99.txt (55.5MB) |
100 | 687,718 (294) | 633,825,300,114,114,137,798,396,969,882 | MRoots_100.txt (55.1MB) |
101 | 699,053 (50) | 1,267,650,600,228,229,401,496,702,506,322 | MRoots_101.txt (56.0MB) |
102 | 688,101 (208) | 2,535,301,200,456,457,677,084,909,010,944 | MRoots_102.txt (55.1MB) |
103 | 701,029 (51) | 5,070,602,400,912,917,605,986,812,120,474 | MRoots_103.txt (56.1MB) |
104 | 689,297 (234) | 10,141,204,801,825,832,960,173,811,268,343 | MRoots_104.txt (55.2MB) |
105 | 692,742 (312) | 20,282,409,603,651,670,423,930,069,659,213 | MRoots_105.txt (55.5MB) |
106 | 695,858 (78) | 40,564,819,207,303,336,344,294,874,505,677 | MRoots_106.txt (55.7MB) |
107 | 704,500 (53) | 81,129,638,414,606,681,695,789,004,439,563 | MRoots_107.txt (56.4MB) |
108 | 682,933 (531) | 162,259,276,829,213,354,384,344,395,256,907 | MRoots_108.txt (54.7MB) |
109 | 702,913 (54) | 324,518,553,658,426,726,783,156,019,873,342 | MRoots_109.txt (56.3MB) |
110 | 691,022 (260) | 649,037,107,316,853,435,551,913,528,882,883 | MRoots_110.txt (55.3MB) |
111 | 701,183 (108) | 1,298,074,214,633,706,907,132,555,362,127,102 | MRoots_111.txt (56.2MB) |
112 | 686,822 (405) | 2,596,148,429,267,413,778,236,451,144,926,618 | MRoots_112.txt (52.0MB) |
113 | 702,079 (56) | 5,192,296,858,534,827,628,530,496,328,518,016 | MRoots_113.txt (56.2MB) |
114 | 690,436 (234) | 10,384,593,717,069,655,185,003,261,181,130,465 | MRoots_114.txt (55.3MB) |
115 | 695,753 (132) | 20,769,187,434,139,310,514,121,985,311,990,312 | MRoots_115.txt (55.7MB) |
116 | 695,040 (154) | 41,538,374,868,278,620,884,128,782,557,209,850 | MRoots_116.txt (55.7MB) |
117 | 694,550 (228) | 83,076,749,736,557,242,056,487,666,388,919,790 | MRoots_117.txt (55.6MB) |
118 | 695,182 (87) | 166,153,499,473,114,483,824,745,506,382,636,145 | MRoots_118.txt (55.7MB) |
119 | 697,568 (144) | 332,306,998,946,228,968,225,951,765,069,322,977 | MRoots_119.txt (55.9MB) |
120 | 676,430 (932) | 664,613,997,892,457,935,875,442,228,072,396,330 | MRoots_120.txt (54.2MB) |
121 | 698,779 (105) | 1,329,227,995,784,915,872,903,807,060,279,644,773 | MRoots_121.txt (56.0MB) |
122 | 695,398 (90) | 2,658,455,991,569,831,744,654,692,615,953,146,777 | MRoots_122.txt (55.7MB) |
123 | 694,234 (120) | 5,316,911,983,139,663,491,615,227,141,609,056,291 | MRoots_123.txt (55.6MB) |
124 | 691,092 (165) | 10,633,823,966,279,326,980,924,613,473,028,371,558 | MRoots_124.txt (55.3MB) |
125 | 693,207 (162) | 21,267,647,932,558,653,966,460,912,964,468,042,793 | MRoots_125.txt (55.5MB) |
126 | 679,916 (606) | 42,535,295,865,117,307,928,310,137,711,520,620,848 | MRoots_126.txt (54.5MB) |
127 | 694,834 (63) | 85,070,591,730,234,615,865,843,651,857,941,358,029 | MRoots_127.txt (55.6MB) |
128 | 686,519 (365) | 170,141,183,460,469,231,722,463,931,679,028,643,401 | MRoots_128.txt (55.0MB) |
>128 | 885 (444) | - | MRoots_More.txt |
Searching is performed using a quadtree subdivision of the grid which increases search resolution in areas of higher root density. Within a search area first a map of the function magnitude is made at moderate resolution, then points of local minimum are refined using 640 bit fixed point math. Root searching requires twice the precision of the desired result due to multiplying very small numbers. As roots are added the quadtree is subdivided further and the search continues. Only the upper half plane is searched due to symmetry.
Quadtree depth was purposely limited on higher periods to reduce run time since the number of roots becomes unmanageable.
Higher periods have only been searched in selected areas and lower quadtree depth.
Quadtree