Friday, 30 May 2014

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. 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)
  3. 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.
  4. 1,357,202 to 1,357,332 – the first of two sequences of 131 composite numbers in the second million
  5. the 16,719th century – the first with no primes whatsoever – and surrounding prime-free numbers.

31398 to 31468

Number“Small” Factors“Large” Factors
31,399171847
31,403311013
31,40977641
31,411101311
31,41789353
31,421132417
31,42376767
31,427112857
31,42953593
31,433174343
31,439149211
31,441231367
31,447134159
31,45174493
31,45371443
31,45783379
31,459163193
31,46373431
Square root of 31,468 is 177.3922207989967

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

58789 to 58888

Number“Small” Factors“Large” Factors
58,793737227
58,799134523
58,801127463
58,807731271
58,811232557
58,813103571
58,817115347
58,819131449
58,82359997
58,82989661
58,831
is a prime
58,837173461
58,841292029
58,8431919163
58,84783709
58,849771201
58,853229257
58,85971829
58,861115351
58,867371591
58,871173463
58,873113521
58,877713647
58,87997607
58,8831153101
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
155,897722271
155,8993147107
155,9031114173
155,9091367179
155,911722273
155,917236779
155,921
is a prime
155,923413803
155,927241647
155,929211739
155,9331929283
155,939722277
155,941179173
155,9471114177
155,951277563
155,953722279
155,957831879
155,959263593
155,963236781
155,96911111289
155,971198209
155,977612557
155,981722283
155,9831511033
155,9871313923
155,989389401
155,993473319
155,999257607
156,001732137
Square root of 156,001 is 394.9696190848101

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

2684th century

Number “Small” Factors “Large” Factors
268,3011124391
268,303738329
268,3071320639
268,309713779
268,3131571709
268,3192511069
268,3211791499
268,3271511777
268,331738333
268,3331320641
268,3371929487
268,339536183
268,343
is a prime
268,3491491801
268,3511272113
268,3571012657
268,361377253
268,363437979
268,3671131787
268,3691671607
268,373775477
268,3791715787
268,381349769
268,3877231667
268,391594549
268,393311863
268,3972391123
268,399972767
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
370,267479773
370,2711141821
370,2734379109
370,2771723947
370,27971313313
370,283379977
370,2893491061
370,2911919489
370,2973531049
370,30129113113
370,3033671009
370,307752901
370,309675527
370,313477879
370,319547677
370,321752903
370,3271073461
370,3311361467
370,3333710009
370,33711131257
370,3391991861
370,343596277
370,3497191277
370,3511792069
370,3571331919
370,361383967
370,3637157337
370,36719101193
370,3692316103
Square root of 370,373 is 608.5827799075488

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

3,710th and early 3,711st centuries

Number “Small” Factors “Large” Factors
370,901421881
370,90313103277
370,9071672221
370,9097114817
370,913735081
370,919
is a prime
370,9212316127
370,927419047
370,9311133721
370,933596287
370,9377192789
370,9392912791
370,9433471069
370,949
is a prime
370,9517197269
370,9571721821
370,961438627
370,963409907
370,96723127127
370,9691073467
370,9731013673
370,9797767113
370,9811328537
370,9873491063
370,99117139157
370,993752999
370,99711291163
370,9993737271
371,0033531051
371,009419049
371,011577643
371,017563659
371,021753003
371,0233111193
Square root of 371,024 is 609.1173942681329

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

13,573rd century

Number “Small” Factors “Large” Factors
1,357,2072359009
1,357,2113143781
1,357,21311139491
1,357,2172575281
1,357,2194767431
1,357,2237414729
1,357,22917292753
1,357,23110313177
1,357,2377193891
1,357,2411499109
1,357,24311312011
1,357,2473074421
1,357,24912710687
1,357,2532359011
1,357,2591379907
1,357,2613493889
1,357,2675712377
1,357,2713736683
1,357,2733143783
1,357,2775325609
1,357,27971117627
1,357,2832295927
1,357,2897318593
1,357,29113131797
1,357,2971779841
1,357,30111163757
1,357,3031971437
1,357,3077712731
1,357,3097271867
1,357,3134728879
1,357,3191817499
1,357,3217971999
1,357,3274493023
1,357,3311779843
Square root of 1,357,332 is 1165.04592184171

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

16,719th century and surrounds

Number “Small” Factors “Large” Factors
1,671,78713128599
1,671,791111919421
1,671,79310316231
1,671,79717432287
1,671,79931199271
1,671,8037238829
1,671,8095992791
1,671,81113712203
1,671,8177241991
1,671,8212957649
1,671,8231918753
1,671,8276127407
1,671,8291987991
1,671,83312891297
1,671,83913128603
1,671,84112231367
1,671,8472372689
1,671,8516724953
1,671,85310116553
1,671,857111141337
1,671,8597238837
1,671,8633594657
1,671,8698320143
1,671,8714873433
1,671,8777921163
1,671,8813315051
1,671,8834359659
1,671,8877238841
1,671,8895213209
1,671,89323157463
1,671,8991798347
1,671,90171121713
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 16,719th century, the smallest devoid of primes, only 33 primes of 207 are needed to do factor 24 numbers. This result from the 16,719th century and 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. Hence, perhaps the observed gaps are more impressive than the actual use of small prime factors would suggest.

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