The 16,719th century and other prime-free sequences

Although I have not read about primes in the past few months, I still  enjoy the topic whenever I get around to it, and in the past day I — feeling tired of languages and football at the moment – had a read about Oscar Hoppe, the mathematician who during the World War I years proved the number 1111111111111111111 (R₁₉) to be prime.

Whilst I do not understand how Oscar Hoppe did it except by showing that R₁₉ could not be expressed as the difference of nonconsecutive squares, I thought I should try and do something I had wanted to do many years ago. That is to see how many of the possible prime factors actually divide numbers in some major prime gaps or sequences with very few primes, in order to see what size prime gap would theoretically be possible if all “small” primes (defined as those under the square root) divided a number within a sequence of a numbers of an arbitrary size.

The sequences examined are:

1. 31,398 to 31,468 — a highly persistent prime gap, the largest with five-digit numbers and the most persistent above 1,361
2. 58,790 to 58,888 — almost 100 numbers with just one prime, at a size one-third that of the first 100 consecutive numbers with only one prime
3. 155,900 to 155,999 and 268,300 to 268,399 — the first two centuries with one prime only (the first with two primes is actually from 302,000 to 302,099)
4. 370,262 to 370,372 and 370,900 to 371,020 — the first being the smallest prime gap of over 100 and the second the second century with only two primes.
5. 1,357,202 to 1,357,332 — the first of two sequences of 131 composite numbers in the second million
6. the 16,719^th century — the first with no primes whatsoever — and surrounding prime-free numbers.

31,398 to 31,468

Number	“Small” Factors			“Large” Factors
31,399	17			1847
31,403	31			1013
31,409	7	7		641
31,411	101			311
31,417	89			353
31,421	13			2417
31,423	7	67	67
31,427	11			2857
31,429	53			593
31,433	17	43	43
31,439	149			211
31,441	23			1367
31,447	13	41	59
31,451	7			4493
31,453	71			443
31,457	83			379
31,459	163			193
31,463	73			431

Square root of 31,468 is 177.3922207989967

Total number of prime factors under square root required: 18 of 38 (47.36 percent).

58,790 to 58,888

Number	“Small” Factors			“Large” Factors
58,793	7	37	227
58,799	13			4523
58,801	127			463
58,807	7	31		271
58,811	23			2557
58,813	103			571
58,817	11			5347
58,819	131			449
58,823	59			997
58,829	89			661
58,831	is a prime
58,837	17			3461
58,841	29			2029
58,843	19	19	163
58,847	83			709
58,849	7	7		1201
58,853	229			257
58,859	71			829
58,861	11			5351
58,867	37			1591
58,871	17			3463
58,873	113			521
58,877	7	13		647
58,879	97			607
58,883	11	53	101

Square root of 58,888 is 242.6684981615867

Total number of prime factors under square root required: 23 of 46 (50.00 percent).

155,894 to 156,001

Number	“Small” Factors			“Large” Factors
155,897	7			22271
155,899	31	47	107
155,903	11			14173
155,909	13	67	179
155,911	7			22273
155,917	23			6779
155,921	is a prime
155,923	41			3803
155,927	241			647
155,929	211			739
155,933	19		29	283
155,939	7			22277
155,941	17			9173
155,947	11			14177
155,951	277			563
155,953	7			22279
155,957	83			1879
155,959	263			593
155,963	23			6781
155,969	11		11	1289
155,971	19			8209
155,977	61			2557
155,981	7			22283
155,983	151			1033
155,987	13		13	923
155,989	389			401
155,993	47			3319
155,999	257			607
156,001	73			2137

Square root of 156,001 is 394.9696190848101

Total number of prime factors under square root required: 23 of 74 (31.08 percent).

2684^th century

Number	“Small” Factors			“Large” Factors
268,301	11			24391
268,303	7			38329
268,307	13			20639
268,309	71			3779
268,313	157			1709
268,319	251			1069
268,321	179			1499
268,327	151			1777
268,331	7			38333
268,333	13			20641
268,337	19	29	487
268,339	53	61	83
268,343	is a prime
268,349	149			1801
268,351	127			2113
268,357	101			2657
268,361	37			7253
268,363	43	79	79
268,367	11		31	787
268,369	167			1607
268,373	7		7	5477
268,379	17			15787
268,381	349			769
268,387	7		23	1667
268,391	59			4549
268,393	311			863
268,397	239			1123
268,399	97			2767

Square root of 268,402 is 518.0752841045402

Total number of prime factors under square root required: 29 of 95 (30.52 percent).

First century gap

Number	“Small” Factors				“Large” Factors
370,267	479				773
370,271	11		41		821
370,273	43		79	109
370,277	17		23		947
370,279	7	13	13	313
370,283	379				977
370,289	349				1061
370,291	19				19489
370,297	353				1049
370,301	29		113	113
370,303	367				1009
370,307	7				52901
370,309	67				5527
370,313	47				7879
370,319	547				677
370,321	7				52903
370,327	107				3461
370,331	13		61	467
370,333	37				10009
370,337	11		131	257
370,339	199				1861
370,343	59				6277
370,349	7		191	277
370,351	179				2069
370,357	13		31		919
370,361	383				967
370,363	7		157	337
370,367	19		101	193
370,369	23				16103

Square root of 370,373 is 608.5827799075488

Total number of prime factors under square root required: 38 of 108 (35.18 percent).

3,710^th and early 3,711^st centuries

Number	“Small” Factors				“Large” Factors
370,901	421				881
370,903	13		103	277
370,907	167				2221
370,909	7		11		4817
370,913	73				5081
370,919	is a prime
370,921	23				16127
370,927	41				9047
370,931	11				33721
370,933	59				6287
370,937	7		19		2789
370,939	29				12791
370,943	347				1069
370,949	is a prime
370,951	7		197	269
370,957	17				21821
370,961	43				8627
370,963	409				907
370,967	23		127	127
370,969	107				3467
370,973	101				3673
370,979	7	7	67	113
370,981	13				28537
370,987	349				1063
370,991	17		139	157
370,993	7				52999
370,997	11		29		1163
370,999	37	37	271
371,003	353				1051
371,009	41				9049
371,011	577				643
371,017	563				659
371,021	7				53003
371,023	311				1193

Square root of 371,024 is 609.1173942681329

Total number of prime factors under square root required: 33 of 108 (30.56 percent).

13,573^rd century

Number	“Small” Factors			“Large” Factors
1,357,207	23			59009
1,357,211	31			43781
1,357,213	11		13	9491
1,357,217	257			5281
1,357,219	47	67	431
1,357,223	7		41	4729
1,357,229	17		29	2753
1,357,231	103			13177
1,357,237	7			193891
1,357,241	149			9109
1,357,243	113			12011
1,357,247	307			4421
1,357,249	127			10687
1,357,253	23			59011
1,357,259	137			9907
1,357,261	349			3889
1,357,267	571			2377
1,357,271	37			36683
1,357,273	31			43783
1,357,277	53			25609
1,357,279	7		11	17627
1,357,283	229			5927
1,357,289	73			18593
1,357,291	13	131	797
1,357,297	17			79841
1,357,301	11	163	757
1,357,303	19			71437
1,357,307	7		71	2731
1,357,309	727			1867
1,357,313	47			28879
1,357,319	181			7499
1,357,321	7		97	1999
1,357,327	449			3023
1,357,331	17			79843

Square root of 1,357,332 is 1165.04592184171

Total number of prime factors under square root required: 33 of 189 (17.46 percent).

16,719^th century and surrounds

Number	“Small” Factors				“Large” Factors
1,671,787	13				128599
1,671,791	11	19	19	421
1,671,793	103				16231
1,671,797	17		43		2287
1,671,799	31		199	271
1,671,803	7				238829
1,671,809	599				2791
1,671,811	137				12203
1,671,817	7		241	991
1,671,821	29				57649
1,671,823	191				8753
1,671,827	61				27407
1,671,829	19				87991
1,671,833	1289				1297
1,671,839	13				128603
1,671,841	1223				1367
1,671,847	23				72689
1,671,851	67				24953
1,671,853	101				16553
1,671,857	11	11	41	337
1,671,859	7				238837
1,671,863	359				4657
1,671,869	83				20143
1,671,871	487				3433
1,671,877	79				21163
1,671,881	331				5051
1,671,883	43		59	659
1,671,887	7				238841
1,671,889	521				3209
1,671,893	23		157	463
1,671,899	17				98347
1,671,901	7		11		21713

Square root of 1,671,907 is 1293.022428266424

Total number of prime factors under square root required: 36 of 207 (17.39 percent).

It is clearly noticeable how, as our numbers become bigger and bigger, the proportion of primes under the square root required to factor long sequences of composite numbers becomes smaller and smaller.

In the sequence from 1,328 to 1,360, every prime less than the square root is used to factor a number which cannot be proven composite by elementary divisibility tests for 2, 3 and 5. In the 16,719^th century, the smallest devoid of primes, only 33 primes of 207 are needed to factor 26 numbers. This result from the 16,719^th century, alongside similar cases with smaller numbers, suggests that if every possible factor had a multiple of a prime larger than the square root, then gaps of over one hundred could occur easily with five-digit numbers.

The “accuracy” required to match small and large prime factors for the largest possible sequence of composite numbers — as happens in the fourteenth century — however, must be quite impossible to replicate even with six-digit numbers, owing to the fact that necessary factors are often much larger than those used from 1,328 to 1,360, so that a “matching” prime is less likely because the density of primes falls logarithmically with number size. Hence, perhaps the observed gaps are more impressive than the actual use of small prime factors would suggest.