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 A surprising prime gap (Posted on 2016-07-27)
N is a positive integer. If you list the positive integers in rows of N such that the first row contains 1 to N, the second row contains N+1 to 2*N, and so on, then the first row with no prime is immediately followed by another row with no prime. What is N?

 See The Solution Submitted by Math Man Rating: 5.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
 re(3): My solution | Comment 6 of 8 |
(In reply to re(2): My solution by Ady TZIDON)

I don't know about "many", but certainly some, especially as you need a gap twice as big, and positioned appropriately, as the gap sought for a double row.

Just see the double-row gaps I previously listed:

gap                  ranges
17 15692 15708    15709 15725
18 16147 16164    16165 16182
21 19615 19635    19636 19656
22 25477 25498    25499 25520
25 31401 31425    31426 31450
27 31402 31428    31429 31455
31 31404 31434    31435 31465
39 155923 155961    155962 156000
41 155924 155964    155965 156005
42 360655 360696    360697 360738
47 370267 370313    370314 370360
48 370273 370320    370321 370368

that were disqualified because there were gaps that long before, that were only single-row, not double-row.

Just to take at random, the gap of 22 from the above list.  Single rows of gap 22 far exceed double such rows:

22 7261 7282
22 7371 7392
22 7767 7788
22 7965 7986
22 8471 8492
22 8977 8998
22 10011 10032
22 10803 10824
22 10913 10934
22 11749 11770
22 12013 12034
22 12167 12188
22 13069 13090
22 13971 13992
22 14257 14278
22 14565 14586
22 15687 15708
22 16149 16170
22 16765 16786
22 18019 18040
22 18811 18832
22 18877 18898
22 18921 18942
22 19097 19118
22 19185 19206
22 19339 19360
22 19625 19646
22 20571 20592
22 20813 20834
22 21033 21054
22 22199 22220
22 22309 22330
22 23475 23496
22 23695 23716
22 24289 24310
22 25191 25212
22 25279 25300
22 25477 25498
22 25477 25498    25499 25520
22 25499 25520
22 26269 26290
22 26599 26620
22 27149 27170
22 27215 27236
22 27303 27324
22 27369 27390
22 27589 27610
22 27853 27874
22 28249 28270
22 28359 28380
22 29503 29524
22 29767 29788
22 30405 30426
22 30603 30624
22 30735 30756
22 31197 31218
22 31285 31306
22 31417 31438
22 31417 31438    31439 31460
22 31439 31460
22 31923 31944
22 32275 32296
22 32539 32560
22 32847 32868
22 33265 33286
22 33969 33990
22 34079 34100
22 34079 34100    34101 34122
22 34079 34122
22 34101 34122
22 34189 34210
22 34783 34804
22 35179 35200
22 35465 35486
22 35619 35640
22 35619 35640    35641 35662
22 35619 35662
22 35641 35662
22 35685 35706
22 35685 35706    35707 35728
22 35707 35728
22 35773 35794
22 36411 36432
22 36499 36520
22 37665 37686
22 37753 37774
22 37929 37950
22 38017 38038
22 38127 38148
22 38479 38500
22 39051 39072
22 39271 39292
22 39469 39490
22 40041 40062
22 40305 40326
22 40393 40414
22 40437 40458
22 40657 40678
22 41691 41712
22 41779 41800
22 42109 42130
22 42615 42636
22 43341 43362
22 43341 43362    43363 43384
22 43363 43384
22 43803 43824
22 43803 43824    43825 43846
22 43803 43846
22 43825 43846
22 44133 44154
22 44309 44330
22 44419 44440
22 44595 44616
22 44661 44682
22 45211 45232
22 45453 45474
22 45783 45804
22 45915 45936
22 46355 46376
22 46773 46794
22 46971 46992
22 47983 48004
22 48687 48708
22 48687 48708    48709 48730
22 48687 48730
22 48709 48730
22 49501 49522
22 49567 49588
22 49897 49918
22 49963 49984
22 50183 50204
22 50469 50490
22 50557 50578
22 50601 50622
22 50799 50820
22 51085 51106
22 51723 51744
22 52339 52360
22 52405 52426
22 53021 53042
22 53241 53262
22 53329 53350
22 53659 53680
22 54803 54824
22 55023 55044
22 55177 55198
22 55265 55286
22 55551 55572
22 56013 56034
22 56057 56078
22 56211 56232
22 56277 56298
22 57003 57024
22 57531 57552
22 57949 57970
22 58279 58300
22 58455 58476
22 58807 58828
22 58851 58872
22 59291 59312
22 59841 59862
22 60171 60192
22 60545 60566
22 60545 60566    60567 60588
22 60567 60588
22 60831 60852
22 60963 60984
22 61183 61204
22 61887 61908
22 62239 62260
22 62437 62458
22 62943 62964
22 63251 63272
22 63867 63888
22 63955 63976
22 64241 64262
22 64351 64372
22 64461 64482
22 64527 64548
22 64637 64658
22 64725 64746
22 65451 65472
22 65737 65758
22 66001 66022
22 66111 66132
22 66199 66220
22 66243 66264
22 66309 66330
22 66661 66682
22 68179 68200
22 68289 68310
22 68839 68860
22 68971 68992
22 69037 69058
22 69081 69102
22 69169 69190
22 69279 69300
22 69345 69366
22 69565 69586
22 69631 69652
22 69785 69806
22 69961 69982
22 70401 70422
22 70731 70752
22 70797 70818
22 71611 71632
22 72117 72138
22 72183 72204
22 72315 72336
22 72579 72600
22 73151 73172
22 73195 73216
22 73393 73414
22 73789 73810
22 74625 74646
22 74977 74998
22 75043 75064
22 75087 75108
22 75439 75460
22 75945 75966
22 76055 76076
22 76165 76186
22 77111 77132
22 77815 77836
22 77903 77924
22 78739 78760
22 79003 79024
22 79509 79530
22 79707 79728
22 79707 79728    79729 79750
22 79707 79750
22 79729 79750
22 80081 80102
22 80873 80894
22 81247 81268
22 81379 81400
22 81467 81488
22 81577 81598
22 81775 81796
22 82083 82104
22 82083 82104    82105 82126
22 82083 82126
22 82105 82126
22 82435 82456
22 82677 82698
22 82853 82874
22 82941 82962
22 83139 83160
22 83315 83336
22 83359 83380
22 83513 83534
22 83667 83688
22 84019 84040
22 84591 84612
22 84833 84854
22 84877 84898
22 84921 84942
22 85163 85184
22 85273 85294
22 85339 85360
22 85383 85404
22 85757 85778
22 85867 85888
22 85955 85976
22 86043 86064
22 86087 86108
22 86637 86658
22 86791 86812
22 86879 86900
22 86879 86900    86901 86922
22 86879 86922
22 86901 86922
22 87451 87472
22 87561 87582
22 88045 88066
22 88133 88154
22 88265 88286
22 88353 88374
22 88441 88462
22 88551 88572
22 88617 88638
22 88683 88704
22 88749 88770
22 89167 89188
22 89695 89716
22 89695 89716    89717 89738
22 89695 89738
22 89717 89738
22 90443 90464
22 90553 90574
22 90751 90772
22 90949 90970
22 91037 91058
22 91257 91278
22 91345 91366
22 91543 91564
22 92511 92532
22 92599 92620
22 93017 93038
22 93347 93368
22 93435 93456
22 93655 93676
22 93765 93786
22 94491 94512
22 94623 94644
22 94909 94930
22 95041 95062
22 95657 95678
22 95657 95678    95679 95700
22 95679 95700
22 95833 95854
22 96295 96316
22 96559 96580
22 96603 96624
22 96867 96888
22 97043 97064
22 97329 97350
22 97615 97636
22 97747 97768
22 97791 97812
22 98275 98296
22 99199 99220
22 99441 99462
22 100079 100100
22 100563 100584
22 100871 100892
22 100959 100980
22 101223 101244
22 101223 101244    101245 101266
22 101223 101266
22 101245 101266
22 102411 102432
22 102565 102586
22 102719 102740
22 103247 103268
22 103489 103510
22 103621 103642
22 103731 103752
22 103929 103950
22 104061 104082
22 104259 104280
22 104435 104456
22 105073 105094
22 105293 105314
22 105469 105490
22 105579 105600
22 105623 105644
22 105777 105798
22 106041 106062
22 106041 106062    106063 106084
22 106063 106084
22 106459 106480
22 106503 106524
22 106547 106568
22 106547 106568    106569 106590
22 106547 106590
22 106569 106590
22 107141 107162
22 107383 107404
22 107383 107404    107405 107426
22 107383 107426
22 107405 107426
22 107515 107536
22 107515 107536    107537 107558
22 107515 107558
22 107537 107558
22 107801 107822
22 108307 108328
22 108593 108614
22 108681 108702
22 108835 108856
22 109231 109252
22 109693 109714
22 109759 109780
22 110089 110110
22 110133 110154
22 110199 110220
22 110375 110396
22 110375 110396    110397 110418
22 110375 110418
22 110397 110418
22 110683 110704
22 110991 111012
22 111057 111078
22 111277 111298
22 111387 111408
22 111541 111562
22 111673 111694
22 112509 112530
22 112707 112728
22 113235 113256
22 113235 113256    113257 113278
22 113235 113278
22 113257 113278
22 113301 113322
22 113389 113410
22 113851 113872
22 114115 114136
22 114379 114400
22 114423 114444
22 114511 114532
22 115369 115390
22 115435 115456
22 115699 115720
22 115941 115962
22 116051 116072
22 116205 116226
22 116557 116578
22 116601 116622
22 116755 116776
22 117283 117304
22 117459 117480
22 117591 117612
22 117855 117876
22 118009 118030
22 118097 118118
22 118317 118338
22 118933 118954
22 118977 118998
22 118977 118998    118999 119020
22 118999 119020
22 119197 119218
22 119329 119350
22 119395 119416
22 119461 119482
22 120253 120274
22 120451 120472
22 120517 120538
22 120781 120802
22 121089 121110
22 121199 121220
22 121639 121660
22 121793 121814
22 121793 121814    121815 121836
22 121815 121836
22 121969 121990
22 122365 122386
22 122409 122430
22 122629 122650
22 122893 122914
22 124037 124058
22 124389 124410
22 124609 124630
22 124873 124894
22 125071 125092
22 125159 125180
22 125555 125576
22 125841 125862
22 125973 125994
22 126281 126302
22 126369 126390
22 126655 126676
22 126787 126808
22 126875 126896
22 127007 127028
22 127425 127446
22 127557 127578
22 127887 127908
22 128063 128084
22 128525 128546
22 128569 128590
22 128635 128656
22 128723 128744
22 128789 128810
22 129141 129162
22 129317 129338
22 129559 129580
22 129647 129668
22 129977 129998
22 130593 130614
22 130659 130680
22 130703 130724
22 130747 130768
22 130879 130900
22 130879 130900    130901 130922
22 130879 130922
22 130901 130922
22 131077 131098
22 131385 131406
22 131451 131472
22 131649 131670
22 131803 131824
22 131869 131890
22 131979 132000
22 132023 132044
22 132177 132198
22 132441 132462
22 132551 132572
22 132771 132792
22 132771 132792    132793 132814
22 132771 132814
22 132793 132814
22 132991 133012
22 133123 133144
22 133453 133474
22 133607 133628
22 133739 133760
22 133893 133914
22 134003 134024
22 134377 134398
22 134531 134552
22 134531 134552    134553 134574
22 134531 134574
22 134553 134574
22 134641 134662
22 135675 135696
22 135807 135828
22 135939 135960
22 135983 136004
22 136071 136092
22 136577 136598
22 136665 136686
22 136819 136840
22 136907 136928
22 137039 137060
22 137281 137302
22 137545 137566
22 137677 137698
22 137875 137896
22 138117 138138
22 138689 138710
22 138997 139018
22 139041 139062
22 139217 139238
22 139635 139656
22 140031 140052
22 140075 140096
22 140493 140514
22 140559 140580
22 140691 140712
22 140955 140976
22 140999 141020
22 141329 141350
22 141373 141394
22 141417 141438
22 141879 141900
22 142275 142296
22 142473 142494
22 142627 142648
22 142913 142934
22 143001 143022
22 143001 143022    143023 143044
22 143023 143044
22 143067 143088
22 143199 143220
22 143749 143770
22 143837 143858
22 143925 143946
22 144079 144100
22 144673 144694
22 144849 144870
22 145223 145244
22 145311 145332
22 145553 145574
22 145729 145750
22 145839 145860
22 145905 145926
22 146455 146476
22 146895 146916
22 147049 147070
22 147115 147136
22 147423 147444
22 147885 147906
22 147951 147972
22 148039 148060
22 148831 148852
22 148963 148984
22 149645 149666
22 149645 149666    149667 149688
22 149667 149688
22 150019 150040
22 150261 150282
22 150349 150370
22 150745 150766
22 151097 151118
22 151735 151756
22 151911 151932
22 151977 151998
22 152153 152174
22 152329 152350
22 152571 152592
22 153011 153032
22 153011 153032    153033 153054
22 153011 153054
22 153033 153054
22 153165 153186
22 153209 153230
22 153385 153406
22 153781 153802
22 153847 153868
22 153957 153978
22 154463 154484
22 154903 154924
22 154947 154968
22 155343 155364
22 155475 155496
22 155629 155650
22 155937 155958
22 155937 155958    155959 155980
22 155937 156002
22 155959 155980
22 155959 155980    155981 156002
22 155959 156002
22 155981 156002
22 156179 156200
22 156377 156398
22 156443 156464
22 156553 156574
22 156751 156772
22 156861 156882
22 157367 157388
22 157587 157608
22 157697 157718
22 157961 157982
22 158049 158070
22 158467 158488
22 158665 158686
22 158709 158730
22 158819 158840
22 158885 158906
22 159127 159148
22 159237 159258
22 159237 159258    159259 159280
22 159259 159280
22 159633 159654
22 159875 159896
22 159941 159962
22 160051 160072
22 160095 160116
22 160271 160292
22 160513 160534
22 160557 160578
22 160909 160930

or for 48:

48 69265 69312
48 82081 82128
48 124369 124416
48 126865 126912
48 132769 132816
48 155953 156000
48 162145 162192
48 173377 173424
48 175153 175200
48 178705 178752
48 189073 189120
48 190417 190464
48 195601 195648
48 203473 203520
48 212305 212352
48 218305 218352
48 221329 221376
48 223153 223200
48 225169 225216
48 255265 255312
48 256225 256272
48 265633 265680
48 277105 277152
48 285889 285936
48 290257 290304
48 294577 294624
48 298945 298992
48 305665 305712
48 308161 308208
48 323281 323328
48 329809 329856
48 338065 338112
48 354481 354528
48 360097 360144
48 360673 360720
48 361585 361632
48 367969 368016
48 370273 370320
48 370273 370320    370321 370368
48 370321 370368
48 395329 395376
48 396769 396816
48 401425 401472
48 402145 402192
48 404881 404928
48 419089 419136
48 427633 427680
48 440593 440640
48 448465 448512
48 458257 458304
48 463105 463152
48 473041 473088
48 473665 473712
48 492145 492192
48 492913 492960
48 510625 510672
48 514993 515040
48 520465 520512
48 529873 529920
48 530449 530496
48 531409 531456
48 531745 531792
48 540913 540960
48 544033 544080
48 561841 561888
48 563665 563712
48 576817 576864
48 593713 593760
48 595633 595680
48 604081 604128
48 618721 618768
48 621169 621216
48 624865 624912
48 626065 626112
48 633505 633552
48 637009 637056
48 637969 638016
48 642289 642336
48 642625 642672
48 645265 645312
48 646465 646512
48 650113 650160
48 653377 653424
48 668353 668400
48 672385 672432
48 673681 673728
48 675649 675696
48 682081 682128
48 682849 682896
48 693889 693936
48 694129 694176
48 694657 694704
48 698113 698160
48 698593 698640
48 714289 714336
48 714385 714432
48 714625 714672
48 724657 724704
48 732385 732432
48 736513 736560
48 749473 749520
48 750865 750912
48 755905 755952
48 762673 762720
48 768673 768720
48 769489 769536
48 770689 770736
48 779425 779472
48 789889 789936
48 791713 791760
48 794257 794304
48 795361 795408
48 801889 801936
48 803761 803808
48 815761 815808
48 818737 818784
48 823057 823104
48 828289 828336
48 834385 834432
48 835153 835200
48 838273 838320
48 840385 840432
48 840769 840816
48 841489 841536
48 843937 843984
48 844657 844704
48 851905 851952
48 857761 857808
48 860161 860208
48 863425 863472
48 873793 873840
48 875041 875088
48 879745 879792
48 880609 880656
48 887761 887808
48 894817 894864
48 896209 896256
48 899329 899376
48 912289 912336
48 912673 912720
48 914065 914112
48 927889 927936
48 928705 928752
48 932689 932736
48 935905 935952
48 946609 946656
48 948193 948240
48 949057 949104
48 951505 951552
48 954769 954816
48 956641 956688
48 961345 961392
48 962689 962736
48 963505 963552
48 966049 966096
48 975745 975792
48 979585 979632
48 982993 983040
48 984769 984816
48 987745 987792
48 997825 997872

 Posted by Charlie on 2016-07-31 18:45:10

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