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German

Do primes arise from wrong binomic formulae?

When at school, we had to learn formulae called "binomische Formeln" in German; I do not know the English term for them. Search with Google led me to the binomial theorem which is very near, but more general (also for higher powers, not only for quadratic expressions), and which includes the binomial coefficients while the formulae we had to learn had only an explicit 2.

Perhaps you remember the three formulae

          (a + b)² = a² + 2ab + b²
          (a - b)² = a² - 2ab + b²
          (a + b) (a - b) = a² - b²
The first two are very similar; you only have to remember that the minus sign is only with the middle term, precisely not with the b². The third formula was called the "mixed binomic forumla", and it has the minus with the quadratic term b². Its right side is simpler than the left, opposite to the first two formulae.

The difficulty with these formulae was not to use them left to right, but to use them backwards, right to left. As an example, we should simplify a fraction like

          a² - b²                       
          -------  , giving  1/3 (a + b)
          3a - 3b

There seems no connection to prime numbers - or is there a hidden connection? One can test and insert some numbers for a and b. We limit the discussion to positive integers a and b, having a > b, in the following.

Searching for primes, it is not an good idea to start with an expression a² - b²; the mixed formula gives exactly a factorisation, and therefore, the result of a² - b² is not a prime, except for the special case a - b = 1. With this special case, one can reach all odd numbers, and therefore all primes but 2. This is noted on the margin only, here.

Numbers from the first two formulae are not suitable either; they are squares, and a square is not a prime.

It happened now and then that we computed errorneously (a + b)² = a² + b²; I call it error 1. The tentation was to do it like the multiplication rule (the distributive law) which states that (a + b) * 2 = a * 2 + b * 2.

Error 2 is to omit the coefficient 2 of the middle term, i.e. to 'compute' (a + b)² = a² + ab + b². And error 3 is to set the minus sign also with the b², i.e. to 'compute' (a - b)² = a² - ab - b².

Intresting (at least for me) is that these three error terms

          a² + b²
          a² + ab + b²
          a² - ab - b²
yield rather many prime numbers, and that one can see some schema or some type of order with it. This is shown in the following table. For some primes, it can happen that we get them by two different terms. In this case, they are assigned to the first "error term" in the above order. For instance, one gets 13 both as 3² + 2² = 9 + 4 (error 1 term), and also as 3² + 3 * 1 + 1² (error 2 term).

Primes resulting from one of the "error terms" are colored in the following table. The table is very economic, maybe it can be called spartanic. Numbers divisible by 2, 3 or 5 are completely omitted. For a range of 30 numbers. there are only 8 possibilities to find a prime. I started the table with these 8 colums. This gives a table which is almost bewildering by its colorfulness. Then I realized that there is only one color for a column when I double the range of each line to 60 numbers, containing 16 places for primes. That is how you see the table now.

Beyond that: if one prime nuber of a column is colored, so are all other primes in the same column, and they get the same color. Eight columns out of the 16 contain primes of form a²+b², four primes of form a²+ab+b², and two such of form a²-ab-b². The last two two columns contain primes which do not fit to any of the three "error formulae".

One can even find the primes of form a² + b² in the OEIS (Online Encyclopedia of Integer Sequences), under A002144. Primes of form a²+ab+b² are under A068229, and those of form a² -ab - b² under A141302. And you find also the primes not fitting to any "error formula", under A107169.


 
Legend
Prime can be written as a²+b² Prime can be written as a²+ab+b² Prime can be written as a²-ab-b² Not a prime, a product a*b
Prime without simple binomic expression

+ 1+ 7+ 11+ 13+ 17+ 19+ 23+ 29+31+37+ 41+ 43+ 47+ 49+ 53+ 59
0
1
2_14_13_24_13_2
23
5_25_16_15_46_1
47
7 * 77_29_2
606_57_29_18_37 * 117_3
83
8_57 * 139_410_19_2
107
10_38_77 * 17
12011 * 117_612_17 * 1911_410_311 * 1310_79_511_67 * 2311_3
167
13 * 1313_217_5
18010_911 * 1715_212_714_113_27 * 2911 * 1914_17 * 3113 * 1711_6
227
15_213_816_1
24015_413 * 1917_211 * 2316_17 * 37
263
13_1010_914_916_513_67 * 4117 * 1717_213 * 23
3007 * 4317_121_513_1214_1111 * 2917 * 197 * 4711_1016_911 * 317 * 49
347
18_517_824_7
36019 * 1913_97 * 5318_713 * 2915_7
383
17_1017 * 2319_620_113 * 3111 * 3720_37 * 5921_1
42015_147 * 6124_517_1219 * 2318_5
443
20_711 * 4121_419_1021_1
467
7 * 6711 * 4323_2
48013 * 3721_227_717 * 297 * 7118_7
503
22_57 * 7311 * 4720_1117_917 * 3123 * 2313 * 4111 * 49
54021_1014_1319 * 297 * 7919_1413 * 43
563
20_1321_524_17 * 8311 * 53
587
19 * 3123_825_1
60024_523_313 * 4718_1719_1622_57 * 8917 * 3715_1413 * 4925_418_11
647
11 * 5922_1333_10
66025_623 * 2911 * 6123_1226_17 * 97
683
13 * 5319_1117 * 4126_519 * 377 * 10122_1523 * 3135_11
7207 * 10318_1317 * 4327_211 * 6723_7
743
7 * 10721_1026_920_197 * 10913 * 5925_1222_1719 * 41
78011 * 7127_27 * 11313 * 6126_1117 * 4711 * 7328_525_619 * 4325_1419_14
827
27_1017 * 4932_5
84029 * 2911 * 7723 * 3723_1829_423_10
863
11 * 7913 * 6729_625_1621_13
887
7 * 12719 * 4729 * 31
90017 * 5326_740_1311 * 837 * 13118_1713 * 7123_2019 * 4924_1929_1023 * 41
947
13 * 7328_137 * 137
96031 * 3127_739_117 * 13931_411 * 89
983
23 * 4326_931_613 * 7717 * 5919 * 5328_1523_2233_2
+ 1+ 7+ 11+ 13+ 17+ 19+ 23+ 29+31+37+ 41+ 43+ 47+ 49+ 53+ 59
102030_1113 * 7939_1032_317 * 6122_157 * 14932_529_67 * 15131_1031_311 * 9730_1329 * 3713 * 83
108023 * 4721_1736_533_229_167 * 157
1103
25_2211 * 10126_2119 * 5933_123 * 4927_2011 * 10317 * 67
11407 * 16331 * 3735_233_813 * 8919 * 61
1163
7 * 16725_1411 * 10734_513 * 91
1187
29 * 4132_1311 * 109
120025_2417 * 717 * 17327_2231_1623 * 53
1223
35_229_1034_917 * 7311 * 11329 * 4332_157 * 17936_1
126013 * 977 * 18131 * 4119 * 6734_1133_5
1283
35_826_1536_126_2527_14
1307
17 * 7713 * 10143_10
132036_523_1911 * 12131 * 437 * 19113 * 10317 * 7919 * 717 * 19323 * 5931_2029 * 47
1367
37 * 3737_27 * 197
138034_1519 * 7313 * 1077 * 19911 * 12725_1823 * 6128_2517 * 8313 * 10929 * 4931_11
1427
30_2337_839_2
144011 * 13137_241_538_331 * 4730_1319 * 7713 * 11335_67 * 21135_1638_1
1487
33_2038_743_7
150019 * 7911 * 13748_1317 * 8937 * 4131 * 49
1523
11 * 13926_1929 * 5323 * 6734_917 * 9135_1832_2340_1
15607 * 22338_347_1113 * 12119 * 8337_5
1583
7 * 22737 * 4334_2140_17 * 229
1607
40_338_1353_17
162039_1037_67 * 23323 * 7131_2611 * 14931 * 5317 * 9713 * 12736_1911 * 15126_21
1667
38_157 * 23923 * 73
168041 * 417 * 24119 * 8937_1841_430_1713 * 13135_2229 * 5917 * 10140_1141_111 * 15719 * 9138_1737 * 47
174030_2933_1417 * 10332_277 * 25138_741 * 4329 * 6123 * 7739_1613 * 13737_9
1787
42_511 * 1637 * 257
180035_2413 * 13949_1037 * 4923 * 7917 * 107
1823
31 * 5930_1911 * 1677 * 26319 * 97
1847
43 * 4317 * 10913 * 143
186031_3038_953_1433_2841_1427_237 * 26940_1731 * 617 * 27135_2611 * 173
1907
23 * 8343_819 * 101
192017 * 11341 * 4745_242_1313 * 1497 * 27729 * 6743_1026_2519 * 10337 * 5313 * 1517 * 28111 * 17938_2345_1
19807 * 28343_311 * 18143_1234_2942_5
2003
41 * 4939_1044_943 * 4717 * 119
2027
45_219 * 10748_5
+ 1+ 7+ 11+ 13+ 17+ 19+ 23+ 29+31+37+ 41+ 43+ 47+ 49+ 53+ 59
204013 * 15723 * 897 * 29342_1717 * 12129 * 71
2063
38_2519 * 10931 * 6741_2037_14
2087
45_823 * 9160_19
210011 * 19143 * 4947_233_3229 * 7313 * 16311 * 19340_2341_936_2946_543_619 * 1137 * 30737_2817 * 127
216044_1511 * 19713 * 16741 * 537 * 31135_1837 * 5911 * 1997 * 31313 * 16931 * 7131_23
2207
47 * 4747_27 * 317
222045_1417 * 13123 * 9729 * 7746_1138_15
2243
13 * 17335_1937 * 6119 * 11931 * 73
2267
37_3047_843 * 53
228045_1637_1729 * 7942_2344_1919 * 12147 * 4947_1034_217 * 33111 * 21123 * 10113 * 17917 * 13743_2257_13
234046_1542_1149_113 * 18141_267 * 33717 * 13923 * 10346_544_2135_3441_1331 * 7742_2537_3256_11
240049 * 4929 * 8363_1919 * 12749_441 * 59
2423
7 * 34717 * 14349_640_297 * 349
2447
31 * 7911 * 22359_14
246023 * 10734_237 * 35348_1346_1937 * 6713 * 19119 * 13147 * 5311 * 22741 * 6149_223 * 10913 * 1937 * 35911 * 229
252036_3519 * 13367_2217 * 14943 * 5933_25
2543
50_750_146_2113 * 19711 * 23317 * 1517 * 36731 * 8357_10
258029 * 8913 * 19955_748_1749 * 5323 * 11319 * 13747_207 * 37351_450_1143 * 6137 * 7111 * 23943_2829 * 91
264019 * 13943_1411 * 2417 * 37949_1650_3
2663
17 * 15749_539_347 * 38338_21
2687
40_3347_2253_2
270037 * 7351_259_1152_319 * 14343_157 * 38952_549_623 * 11946_2513 * 21141 * 6743_3052_731 * 89
276011 * 25142_1717 * 16347 * 5944_297 * 39723 * 12150_1731_3051_1449_2034_277 * 40153 * 5329 * 9760_11
282031 * 9111 * 25719 * 14948_2341_3417 * 167
2843
37 * 7745_1451_1650_197 * 40947 * 6119 * 15117 * 16971_23
288043 * 6733_2949 * 5911 * 26344_3113 * 223
2903
53_1041 * 7154_123 * 12737 * 79
2927
29 * 1017 * 41957_5
294017 * 1737 * 42113 * 22753_1246_2911 * 269
2963
40_3754_113 * 22911 * 27119 * 15729 * 10349 * 6141 * 7372_23
+ 1+ 7+ 11+ 13+ 17+ 19+ 23+ 29+31+37+ 41+ 43+ 47+ 49+ 53+ 59
300051_2031 * 9761_1023 * 1317 * 43138_25
3023
13 * 2337 * 43354_1155_417 * 17911 * 27745_3243 * 7123 * 133
306055_641_2237 * 837 * 43917 * 18145_17
3083
55_811 * 28119 * 1637 * 44329 * 10713 * 23947_3011 * 28365_14
312040_3953 * 5931 * 10113 * 24156_143 * 737 * 44947 * 6723 * 13741 * 7729 * 10953_6
3167
55_1219 * 16717 * 187
318045_3446_1757_131 * 10323 * 1397 * 457
3203
53_2019 * 16956_955_1411 * 2937 * 46150_2753 * 6141 * 79
32407 * 46317 * 19176_2557_256_1142_2313 * 2517 * 46735_3129 * 11317 * 19349 * 6719 * 17323 * 14337 * 8969_17
330049_3057_143 * 7757_831 * 10750_13
3323
52_2541_2547 * 7113 * 25754_7
3347
17 * 1977 * 47959_2
336056_1537 * 9175_2358_311 * 30731 * 10917 * 19958_555_643 * 7919 * 17941 * 83
3407
7 * 48758_713 * 263
342011 * 31123 * 14947 * 7352_277 * 49119 * 18111 * 31343_4029 * 11944_3950_3147_19
3467
45_3823 * 15149 * 71
348059 * 5911 * 31775_227 * 49913 * 26945_2231 * 11329 * 12151_1459_67 * 50313 * 271
3527
48_3558_1360_1
354054_2558_353 * 6719 * 18749_3457_57 * 50943 * 8335_3449 * 7359_1038_3117 * 21137 * 9753_2859 * 61
360013 * 27749_1823 * 15743_4244_4147 * 77
3623
19 * 19154_1146_3911 * 33143_267 * 52141 * 8913 * 28161_1
36607 * 52319 * 19372_1748_3759_1413 * 28329 * 12731 * 11939_3149_3655_2623 * 16111 * 33753_3047 * 7967_10
372061 * 6147_2241 * 9157_2237 * 10143_2719 * 19723 * 16331 * 12117 * 22156_2553 * 71
3767
60_1349 * 7781_26
378019 * 1997 * 54117 * 22352_3346_4129 * 131
3803
13 * 29337 * 10311 * 34761_1042_2943 * 897 * 54753_3211 * 349
384023 * 16761_271_1462_329 * 13317 * 227
3863
53 * 7349 * 7954_3159_2011 * 35323 * 16960_1717 * 2297 * 557
390047 * 8362_181_2543 * 9161_1457_10
3923
52_3551_1931 * 1277 * 56359_7
3947
11 * 35959 * 6737 * 107
396017 * 23358_919 * 20929 * 13741 * 9723 * 1737 * 56958_2513 * 3077 * 57149_4039_34
4007
19 * 21162_1379_22
+ 1+ 7+ 11+ 13+ 17+ 19+ 23+ 29+31+37+ 41+ 43+ 47+ 49+ 53+ 59
402050_3942_3129 * 13937 * 10911 * 3677 * 57713 * 31155_3261_559_2431 * 13117 * 23949 * 8313 * 31352_3772_13
408053 * 7761 * 6765_258_2717 * 24163_211 * 3737 * 58739_3523 * 17913 * 31731 * 133
4127
60_2362_1771_11
414041 * 10129 * 1437 * 59348_4359_2657_1323 * 18111 * 37943 * 9764_937 * 11347 * 8953 * 7959 * 717 * 59919 * 221
420051_407 * 60173_1311 * 38364_1138_3741 * 10365_241_3419 * 22365_457_1431 * 1377 * 60753_3868_5
426065_617 * 25172_1157_3247 * 9111 * 389
4283
65_87 * 61361_2423 * 18713 * 33159 * 7331 * 13919 * 2277 * 617
432029 * 14962_761 * 717 * 61949_4455_1843 * 10150_4319 * 22966_149 * 8953_2111 * 39717 * 25762_2329 * 151
438013 * 33741 * 10769_523 * 19161_2653 * 8337 * 11953_4011 * 4017 * 63165_1466_119 * 23343 * 10331 * 14323 * 193
444060_2939_3883_2361 * 7364_1949 * 91
4463
41 * 10917 * 26337 * 12165_1662_97 * 64167 * 6767_211 * 409
45007 * 64357_1713 * 34748_4749_4647_30
4523
7 * 64723 * 19713 * 34919 * 23959 * 77
4547
65_1829 * 15747 * 97
456060_3141_377 * 65317 * 26923 * 19919 * 241
4583
13 * 35359_1554_4143 * 10751_2617 * 27111 * 4197 * 65931 * 149
462061_307 * 66111 * 42141 * 11359_3458_17
4643
68_565_656_3959 * 7949_2913 * 35929 * 16168_775_11
468031 * 15143 * 10969_119 * 24761 * 7737 * 127
4703
17 * 2777 * 67353 * 8964_2546_3329 * 16352_4558_377 * 677
474011 * 43147 * 10180_1749 * 9767 * 7153_2511 * 43319 * 25113 * 36717 * 2817 * 68363_11
4787
55_4268_1372_5
480065_2423 * 20917 * 28367_1856_4161 * 7953 * 9111 * 43969_17 * 69147 * 10329 * 16737 * 13113 * 37323 * 21143 * 113
486069_1031 * 15785_2211 * 44361_3441 * 11919 * 25767_2067 * 7359 * 8329 * 16969_27 * 70170_317 * 28988_25
492037 * 13313 * 37989_2662_3364_2911 * 449
4943
49 * 10146_3569_1441 * 1217 * 709
4967
60_3767_2213 * 383
498017 * 29357_2231 * 16163_3219 * 26362_15
5003
65_2850_3129 * 17370_1158_2111 * 45747 * 1077 * 71987_23
+ 1+ 7+ 11+ 13+ 17+ 19+ 23+ 29+31+37+ 41+ 43+ 47+ 49+ 53+ 59
504071 * 7149 * 10375_731 * 16313 * 38945_3761 * 8337 * 13711 * 46171_659_4023 * 221
5087
7 * 72711 * 46384_19
510051_5049_3319 * 26953_4843 * 11970_347 * 10923 * 2237 * 73311 * 46753 * 9737 * 139
5147
19 * 27168_2367 * 77
516013 * 39742_4187_227 * 73931 * 16745_3871 * 7370_1729 * 17966_297 * 74343 * 12141 * 12772_513 * 40117 * 307
522023 * 22758_2381_1472_771_1431 * 16949 * 10729 * 18159 * 897 * 75170_1919 * 27723 * 22911 * 47967_2896_31
528060_4117 * 31137 * 14367 * 7971_167 * 757
5303
53_5047 * 11313 * 40917 * 31354_297 * 76173 * 7373_219 * 281
534049 * 10967_1179_1053 * 10111 * 48723 * 23331 * 17359 * 9141 * 13119 * 28365_347 * 769
5387
17 * 31773_880_11
540011 * 49157_267 * 77363_3859_4443_4229 * 18761 * 8946_3969_2671_2061_2113 * 41960_4341 * 13353 * 103
546043 * 12771 * 7775_213 * 42174_173_2
5483
11 * 49919 * 28923 * 23974_571_6
5507
7 * 78737 * 14977_5
552065_3663_1983_1411 * 50349 * 11329 * 19123 * 24131 * 17961 * 9174_967 * 8347_3919 * 29363_4058_477 * 797
558066_3537 * 15196_2947 * 11929 * 19311 * 50913 * 43171 * 7931 * 18141 * 13773 * 7767_1417 * 33113 * 43343 * 13179_7
564075_453_3381_1073_1861_4458_277 * 80965_3853 * 1077 * 81123 * 24746_4147 * 12175_862_4341 * 139
570074_1513 * 43988_1929 * 19771_2643 * 13359 * 9717 * 33711 * 52156_5170_2957_297 * 82157_5011 * 52313 * 443
57607 * 82373 * 7929 * 19923 * 25153 * 10975_2
5783
7 * 82770_1131 * 18776_57 * 829
5807
37 * 15773_2223 * 253
582075_1469_1349 * 11919 * 30713 * 44963_22
5843
68_3571_1076_970_3141 * 143
5867
62_457 * 83995_26
588075_1629 * 20343 * 13771 * 8376_1117 * 347
5903
19 * 31123 * 25761 * 9731 * 19169_14
5927
77 * 7717 * 349100_31
594013 * 45719 * 31311 * 54157_5237 * 16159 * 10167 * 8947 * 1277 * 85343 * 13959_5031 * 193
5987
53 * 11313 * 4617 * 857
+ 1+ 7+ 11+ 13+ 17+ 19+ 23+ 29+31+37+ 41+ 43+ 47+ 49+ 53+ 59
600017 * 35377_184_117 * 85911 * 54713 * 46319 * 31777_1037 * 16366_417 * 86374_7
6047
23 * 26362_4773 * 83
606029 * 20973_913 * 46777_1259 * 10347_4377 * 7967_4049_4167 * 9174_2517 * 35931 * 19741 * 14973_2829 * 211
612064_4511 * 55781_578_719 * 3237 * 877
6143
43 * 14359_3047 * 13161 * 10178_17 * 88131 * 19958_5337 * 167
61807 * 88323 * 26941 * 15111 * 56371_3475_7
6203
7 * 88746_4576_2161_5049 * 12713 * 47973_3023 * 27117 * 367
624079 * 7962_2747 * 13337 * 16979_411 * 569
6263
70_3759_3179_611 * 57161 * 103
6287
19 * 33131 * 203105_34
630075_2653 * 11995_2359 * 10774_2971 * 89
6323
77_2013 * 48771_3617 * 37357_3411 * 5777 * 90773_3297_25
636069_4058_3323 * 27778_177 * 91170_1713 * 49158_5577 * 8359_5437 * 17319 * 33743 * 14929 * 22153 * 12149 * 131
642070_3977_659 * 1097 * 91941 * 15747 * 13717 * 37980_761_3011 * 58771 * 9123 * 28129 * 22363_5068_4331 * 209
648080_913 * 49987_1143 * 15173 * 8967 * 977 * 92923 * 28317 * 38349 * 13380_1111 * 59361 * 10765_4847 * 13913 * 503
654031 * 21173_1499_2672_3779 * 837 * 937
6563
80_1375_1181_470_4129 * 2277 * 94111 * 59919 * 34787_10
660041 * 16174_1311 * 60117 * 38913 * 50958_3537 * 1797 * 94719 * 34961_5429 * 22973 * 9123 * 28961 * 10962_53108_35
666081_1059 * 1137 * 95363_5211 * 60778_741 * 16380_1755_3937 * 18174_3577_919 * 35378_2549 * 13783_2
672047 * 14331 * 21753 * 12782_376_3123 * 29311 * 61317 * 39743 * 15729 * 23380_1969_2267 * 1017 * 96713 * 52193_17
678075_3411 * 61797_2267_487 * 97113 * 523
6803
11 * 61949 * 13917 * 40119 * 35961_33
6827
77_3068_477 * 977
684080_2141 * 16731 * 22177 * 8961_5619 * 361
6863
62_5575_1423 * 2997 * 98363_3171 * 9783 * 8361 * 113101_26
690067 * 10367_2687_731 * 22379_2637 * 18743 * 16141 * 16929 * 2397 * 99111 * 63153 * 131
6947
82_1517 * 409104_29
696081_2073_1884_119 * 36771_447 * 997
6983
29 * 24181_574_3976_3547 * 14977 * 9143 * 16382_17109_34
+ 1+ 7+ 11+ 13+ 17+ 19+ 23+ 29+31+37+ 41+ 43+ 47+ 49+ 53+ 59
702059 * 11967_2779 * 8913 * 54131 * 22770_23
7043
53 * 13311 * 64184_123 * 3077 * 100937 * 19175_3811 * 64388_7
708073 * 9719 * 3737 * 101341 * 17347 * 15131 * 229
7103
70_4713 * 54711 * 64764_5517 * 419
7127
80_277 * 101959 * 121
714037 * 1937 * 102193_1423 * 31117 * 42182_529 * 24767 * 10771 * 10184_1143 * 16711 * 653
7187
79 * 9167_5223 * 313
720019 * 37951_47111_3583_187 * 103153_4531 * 23385_27 * 103381_2613 * 55774_19
7247
11 * 65982_2361 * 119
726053 * 13743 * 16911 * 6617 * 103919 * 38329 * 251
7283
37 * 19723 * 31776_3949 * 14967 * 109
7307
78_3571 * 10313 * 563
732061_6017 * 43192_1163_5829 * 25341 * 1797 * 104982_2550_497 * 105117 * 43337 * 19953 * 13985_1273 * 10147 * 157
738061 * 12183 * 8919 * 38972_4713 * 56949 * 15111 * 67331 * 23975_1984_1941 * 18113 * 5717 * 106123 * 32368_5343 * 173
74407 * 106311 * 67789_529 * 25776_4165_3317 * 43977 * 9731 * 24186_985_167 * 1069
7487
80_3359 * 12795_14
750013 * 57762_3737 * 20311 * 68386_1173 * 103
7523
77_4017 * 44379_3671_5019 * 397
7547
85_1883 * 9199_19
756075_4447 * 16167 * 11387_264_5953 * 143
7583
65_5865_3471 * 10711 * 69169_29
7607
7 * 108723 * 33119 * 401
762086_1529 * 26313 * 58717 * 4497 * 109177_18
7643
68_557 * 109331 * 24747 * 16379 * 9741 * 18787_1083_287 * 1097
768084_2571_27108_2949 * 15743 * 17978_17
7703
13 * 59311 * 70181_347 * 110359_42
7727
59 * 13137 * 20971 * 109
774075_4661 * 12723 * 33788_386_1975_227 * 110917 * 45719 * 40977 * 10131 * 25143 * 18113 * 59983_3088_711 * 709
780029 * 26937 * 21173 * 10713 * 60164_617 * 1117
7823
73_5041 * 19117 * 46179_4031 * 25359 * 13347 * 16767_5829 * 271
78607 * 112359_4317 * 46368_5774_4975_23
7883
49 * 16113 * 60753 * 14985_267 * 1129
7907
11 * 71941 * 193111_31
792089 * 8978_1977 * 10378_4389_417 * 46747 * 16982_3574_2573 * 10919 * 41954_4931 * 25713 * 61367 * 11979 * 101
798023 * 34749 * 16361 * 13172_5311 * 72719 * 42153 * 15185_2889_184_3113 * 61771 * 11323 * 34937 * 21729 * 27796_11
+ 1+ 7+ 11+ 13+ 17+ 19+ 23+ 29+31+37+ 41+ 43+ 47+ 49+ 53+ 59
804043 * 18713 * 61983 * 9787_227 * 115167_3511 * 73365_627 * 115341 * 19780_4159 * 137
8087
67_6082_3789 * 91
810090_167 * 121112_3161 * 13389_1423 * 353
8123
11 * 73947 * 17379 * 1037 * 116317 * 479
8147
29 * 28131 * 26341 * 199
816081_4066_37116_3511 * 74337 * 22182_1549 * 16719 * 43190_17 * 117159 * 13913 * 63129 * 28372_5543 * 19199_14
822090_1119 * 433105_2277_4886_2977 * 107
8243
73 * 11337 * 22323 * 35911 * 75169_347 * 118190_1388_2317 * 487
828091 * 9173_29103_1978_4791_443 * 19323 * 3617 * 118759_4691_653 * 15741 * 20311 * 75775_5213 * 64131 * 269
834019 * 43917 * 4917 * 119387_2861 * 13713 * 643
8363
88_2511 * 76176_5129 * 28983 * 101
8387
90_1777 * 10937 * 227
840031 * 2717 * 120113 * 64747 * 17919 * 44373_30
8423
77_5055_5159 * 14323 * 36757_49
8447
71 * 11979 * 10711 * 769
846090_1991_243 * 19737 * 22949 * 17361 * 13917 * 49913 * 6537 * 121329 * 29374_5511 * 77347 * 18167 * 12792_77 * 1217
852085_3663_4319 * 44953 * 16191_1687_10
8543
83 * 10317 * 50343 * 1997 * 122391_313 * 65941 * 20982_4323 * 373
858066_6531 * 27771 * 12113 * 66189_2657_507 * 122980_4779 * 1097 * 123137 * 23367_39
8627
90_2389 * 9753 * 163
864071_6087_1141 * 21117 * 50911 * 7877 * 1237
8663
85_3829 * 29981_4691_2019 * 45773 * 11992_1573_58115_31
870077 * 11383_1831 * 28193_823 * 37965_4261 * 14343 * 20374_3184_4179_507 * 1249
8747
13 * 67392_1719 * 461
876075_5611 * 79749 * 17931 * 28367 * 13190_7
8783
47 * 18759 * 14919 * 46313 * 67789_9
8807
23 * 3837 * 1259125_41
882089_3091 * 9795_273 * 12194_193_237 * 23968_6553 * 16717 * 52194_591_6
8867
49 * 18119 * 46713 * 683
888083 * 10782_2117 * 52378_5341 * 21711 * 80929 * 30759 * 15167 * 13337 * 24111 * 81158_5179 * 11373_6082_477 * 1277
894090_2923 * 389105_177 * 127953 * 16931 * 289
8963
88_3579_2647 * 1917 * 128313 * 69143 * 20989 * 10123 * 391109_22
+ 1+ 7+ 11+ 13+ 17+ 19+ 23+ 29+31+37+ 41+ 43+ 47+ 49+ 53+ 59
900080_5174_33123_3887_3871 * 12729 * 3117 * 128995_211 * 8217 * 129195_466_4383 * 10993_2011 * 82399_7
906041 * 22167_4247 * 19343 * 21129 * 3137 * 129731 * 29361 * 14959_5111 * 82719 * 47979_277 * 130178_5513 * 70111 * 829
91207 * 130394_323 * 39793_2271_6437 * 24741 * 2237 * 130770_3979_5485_4477 * 11989 * 10353 * 17373_6267 * 137
918091_3078_2991 * 10129 * 31717 * 54183_22
9203
80_5361 * 15113 * 70995_1423 * 401
9227
11 * 8397 * 1319111_23
924096_57 * 132129 * 31919 * 48776_5947 * 19759 * 15731 * 29973 * 12794_2195_1662_4937 * 2517 * 132777_5817 * 547
930071 * 13141 * 22797_167 * 13977 * 12195_3
9323
19 * 49143 * 21796_1185_4681_2613 * 71995_1847 * 19949 * 191
936037 * 25329 * 323121_3491 * 10379_5683 * 11311 * 85341 * 22986_1971_6679 * 11991_1123 * 40997 * 9797_2124_37
942086_4511 * 857125_3893_2891_3490_1371 * 13311 * 85913 * 72749 * 19394_2578_31
9467
17 * 55797_8103_10
948019 * 49953 * 179105_1311 * 86376_6159 * 16143 * 22137 * 25779_3031 * 30789_4089 * 1077 * 136113 * 73382_53108_17
954047 * 20373_38104_1141 * 23319 * 50379 * 12173 * 1317 * 136717 * 56361 * 15767 * 14337 * 259
9587
43 * 22353 * 18129 * 331
960095_2413 * 7397 * 137398_359 * 16363_50
9623
98_595_623 * 41931 * 31186_2111 * 87780_5749 * 19713 * 743
966070_697 * 138119 * 50917 * 56994_2993_1023 * 42192_3511 * 88181_5689 * 10931 * 31317 * 57173 * 13311 * 883120_31
972075_6471 * 13737 * 26397_1891 * 10795_7
9743
82_5549 * 19911 * 88743 * 22713 * 751
9767
88_4529 * 33777 * 127
978090_4171_42123_347 * 139997 * 10141 * 239
9803
17 * 57765_4999_461 * 16147 * 20931 * 31798_1592_37103_7
984013 * 75743 * 229111_1959 * 16789_4483_277 * 140971 * 13991_1583 * 11941 * 24174_39
9887
31 * 31913 * 76119 * 521
990099_1098_353 * 18723 * 43147 * 21191 * 109
9923
85_5261_5419 * 52371_7061 * 16349 * 20390_4337 * 26923 * 433
99607 * 142387_2259 * 16982_5711 * 90717 * 58767 * 1497 * 142797 * 10313 * 76973 * 1377 * 1429
10007
100_331 * 32343 * 233
+ 1+ 7+ 11+ 13+ 17+ 19+ 23+ 29+31+37+ 41+ 43+ 47+ 49+ 53+ 59
1002011 * 91137 * 2717 * 143379 * 12789_4678_3583 * 12113 * 77323 * 43789 * 11394_3529 * 347
10067
87_507 * 1439120_29
1008017 * 59377 * 131108_1393_3823 * 43987_23
10103
11 * 91985_2667 * 15129 * 34953 * 19141 * 2477 * 144798_23117_25
1014085_5473 * 139107_1171 * 1437 * 145173_42
10163
100_137 * 145396_3195_3417 * 59961 * 16723 * 44397_2847 * 217
10200101 * 10159 * 173115_227 * 145917 * 60111 * 929
10223
53 * 19313 * 78729 * 35377 * 13391_18
10247
37 * 27783_58116_23
1026031 * 33159_58104_587_5243 * 23919 * 54191 * 113100_1741 * 2517 * 1471101_10101_111 * 93761 * 16992_4317 * 607
1032095_3623 * 449132_4198_2779_6449 * 211
10343
79 * 13111 * 94194_3913 * 79743 * 2417 * 148180_6341 * 25397 * 107
103807 * 148347 * 221119_2619 * 54737 * 28170_47101 * 1037 * 148729 * 35911 * 94717 * 6137 * 1489
10427
102_597_3273 * 143
1044053 * 19731 * 3377 * 1493102_7101_1663_55
10463
29 * 36137 * 28399_2647 * 22311 * 953
10487
17 * 6177 * 1499129_37
1050090_4979 * 13323 * 45773_7213 * 80967 * 15717 * 619100_2385_2941 * 25783 * 12713 * 81153 * 19977 * 13761 * 173113_17
1056059 * 17998_931 * 34197 * 1097 * 151171 * 14919 * 55785_5889 * 11979_66101_2023 * 461
10607
103 * 103103_241 * 259
1062043 * 24762_57117_2249 * 21711 * 96773_4529 * 36723 * 46365_5481_647 * 152366_53
10667
47 * 22713 * 82159 * 181
1068011 * 97179_38109_1037 * 28919 * 56313 * 82377 * 139103_1069_507 * 153171 * 15194_1717 * 631100_2783_62131_38
1074023 * 46711 * 97713 * 827103_1231 * 34753 * 20347 * 22989 * 12189_2513 * 82991_5041 * 26367 * 16195_4243 * 251136_43
108007 * 1543101 * 10719 * 56911 * 98329 * 37331 * 34979 * 13791 * 11974_4589_5437 * 2937 * 1549
10847
19 * 57197_38108_7
1086094_4583_347 * 155383 * 13173 * 14943 * 253
10883
80_6799_1017 * 64111 * 99191_2313 * 83990_537 * 155961 * 179
1092067 * 16349 * 22317 * 64329 * 377104_11102_531 * 35382_6547 * 23399_3497 * 11319 * 57711 * 9977 * 156798_37111_11
1098079 * 13963_5829 * 37988_577 * 157117 * 647
11003
101 * 10991 * 12123 * 479103 * 10773 * 151
11027
41 * 26959 * 18783 * 133
+ 1+ 7+ 11+ 13+ 17+ 19+ 23+ 29+31+37+ 41+ 43+ 47+ 49+ 53+ 59
1104061 * 18198_1343 * 2577 * 157989_5695_1837 * 29985_6270_5153 * 2097 * 158381_38
11087
13 * 853103_2211 * 1009
1110017 * 65329 * 38341 * 27177_7286_6185_3349 * 22731 * 359105_143 * 25913 * 85711 * 101371 * 15793_5019 * 587125_29
1116080_6913 * 859132_3797_42104_197 * 159753 * 21167 * 16731 * 36191_5423 * 48717 * 6597 * 160111 * 101982_6713 * 863
1122049 * 229103 * 10911 * 102147 * 23917 * 661105_2
11243
7 * 160794_21104_21106_57 * 160919 * 59359 * 19192_53141_46
1128029 * 38978_437 * 161323 * 49179 * 14398_1589 * 12743 * 263101_10106_985_6467 * 16947 * 24195_487 * 161929 * 391
1134011 * 10317 * 1621131_3593_5241 * 27737 * 30711 * 1033100_3783 * 13731 * 36719 * 59999_1459 * 1937 * 1627103_28133_37
1140013 * 87761 * 187117_17101 * 11349 * 23319 * 601
11423
11 * 103971 * 16194_5117 * 67382_39
11447
107 * 10713 * 8817 * 1637
1146073 * 15791_27137_4177 * 14923 * 49913 * 883
11483
92_5589_30101_3653 * 21774_4937 * 31117 * 67729 * 397143_47
1152041 * 28194_2313 * 88719 * 60783 * 13911 * 104997 * 119107_1090_2991 * 12711 * 105131 * 37343 * 26923 * 50371 * 163132_35
1158037 * 31377_4667 * 173107_12106_197 * 165741 * 28347 * 24717 * 68396_4986_6559 * 19777 * 15129 * 401103_32103 * 113
116407 * 166319 * 61361 * 19143 * 271104_2989 * 131107 * 1097 * 166711 * 1061106_21100_417 * 166931 * 377108_511 * 1063125_26
11700105_2623 * 50949 * 23953 * 22179_7463_6219 * 61737 * 31766_5997 * 12159 * 19993_2617 * 69131 * 37973 * 16111 * 1069
1176019 * 61941 * 28779 * 14961 * 193104_31107_3
11783
83_7013 * 90747 * 251101_4037 * 319
11807
49 * 24198_4753 * 223
1182090_6198_19128_29108_1389 * 13387_3513 * 91141 * 2897 * 169371 * 16729 * 409103_11
11867
83 * 14331 * 3837 * 1697
11880109 * 10997_2147 * 2537 * 1699109_473 * 163
11903
97_5043 * 27717 * 70191 * 13167_59
11927
79 * 151107_22123_22
1194095_5413 * 91937 * 323108_1711 * 108778_477 * 170988_6571_5559 * 203109_1023 * 521
11987
19 * 63167 * 17971 * 169
Prepared by Matthias Engelhardt
Mail an Matthias Engelhardt
 
last change: 2014-02-01
Address of page: http://nqueens.de/otherTopics/WrongBinomic.en.html