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October 30th, 2014, 02:51 PM   #1
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Primes

The numbers 11;13;17 and 19 are all prime and all between 10 and 20. Does it ever happen for any n>10 that there are four primes p such that n < p < n+ 10?

I suppose not, because primes are getting increasingly rare, however, I do not know how to prove this on. Could you help me out?
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October 30th, 2014, 05:30 PM   #2
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101, 103, 107 and 109 are all prime. I think it's an open question whether there are infinitely many such.
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October 30th, 2014, 05:50 PM   #3
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Quote:
Originally Posted by matheist View Post
The numbers 11;13;17 and 19 are all prime and all between 10 and 20. Does it ever happen for any n>10 that there are four primes p such that n < p < n+ 10?

I suppose not, because primes are getting increasingly rare, however, I do not know how to prove this on. Could you help me out?
I wrote a little program and found these. I only went up to 800,000. If there are infinitely many of these then there are infinitely many twin primes, an unsolved problem. But it should be easier to prove that there are only finitely many of these quartets. Or maybe not.

What would Gauss and Euler have done if they'd had computers? Probably gone to work as software engineers and never done any math.

3 5 7 11
5 7 11 13
11 13 17 19
101 103 107 109
191 193 197 199
821 823 827 829
1481 1483 1487 1489
1871 1873 1877 1879
2081 2083 2087 2089
3251 3253 3257 3259
3461 3463 3467 3469
5651 5653 5657 5659
9431 9433 9437 9439
13001 13003 13007 13009
15641 15643 15647 15649
15731 15733 15737 15739
16061 16063 16067 16069
18041 18043 18047 18049
18911 18913 18917 18919
19421 19423 19427 19429
21011 21013 21017 21019
22271 22273 22277 22279
25301 25303 25307 25309
31721 31723 31727 31729
34841 34843 34847 34849
43781 43783 43787 43789
51341 51343 51347 51349
55331 55333 55337 55339
62981 62983 62987 62989
67211 67213 67217 67219
69491 69493 69497 69499
72221 72223 72227 72229
77261 77263 77267 77269
79691 79693 79697 79699
81041 81043 81047 81049
82721 82723 82727 82729
88811 88813 88817 88819
97841 97843 97847 97849
99131 99133 99137 99139
101111 101113 101117 101119
109841 109843 109847 109849
116531 116533 116537 116539
119291 119293 119297 119299
122201 122203 122207 122209
135461 135463 135467 135469
144161 144163 144167 144169
157271 157273 157277 157279
165701 165703 165707 165709
166841 166843 166847 166849
171161 171163 171167 171169
187631 187633 187637 187639
194861 194863 194867 194869
195731 195733 195737 195739
201491 201493 201497 201499
201821 201823 201827 201829
217361 217363 217367 217369
225341 225343 225347 225349
240041 240043 240047 240049
243701 243703 243707 243709
247601 247603 247607 247609
247991 247993 247997 247999
257861 257863 257867 257869
260411 260413 260417 260419
266681 266683 266687 266689
268811 268813 268817 268819
276041 276043 276047 276049
284741 284743 284747 284749
285281 285283 285287 285289
294311 294313 294317 294319
295871 295873 295877 295879
299471 299473 299477 299479
300491 300493 300497 300499
301991 301993 301997 301999
326141 326143 326147 326149
334421 334423 334427 334429
340931 340933 340937 340939
346391 346393 346397 346399
347981 347983 347987 347989
354251 354253 354257 354259
358901 358903 358907 358909
361211 361213 361217 361219
375251 375253 375257 375259
388691 388693 388697 388699
389561 389563 389567 389569
392261 392263 392267 392269
394811 394813 394817 394819
397541 397543 397547 397549
397751 397753 397757 397759
402131 402133 402137 402139
402761 402763 402767 402769
412031 412033 412037 412039
419051 419053 419057 419059
420851 420853 420857 420859
427241 427243 427247 427249
442571 442573 442577 442579
444341 444343 444347 444349
452531 452533 452537 452539
463451 463453 463457 463459
465161 465163 465167 465169
467471 467473 467477 467479
470081 470083 470087 470089
477011 477013 477017 477019
490571 490573 490577 490579
495611 495613 495617 495619
500231 500233 500237 500239
510611 510613 510617 510619
518801 518803 518807 518809
536441 536443 536447 536449
536771 536773 536777 536779
539501 539503 539507 539509
549161 549163 549167 549169
559211 559213 559217 559219
563411 563413 563417 563419
570041 570043 570047 570049
572651 572653 572657 572659
585911 585913 585917 585919
594821 594823 594827 594829
597671 597673 597677 597679
607301 607303 607307 607309
622241 622243 622247 622249
626621 626623 626627 626629
632081 632083 632087 632089
632321 632323 632327 632329
633461 633463 633467 633469
633791 633793 633797 633799
654161 654163 654167 654169
657491 657493 657497 657499
661091 661093 661097 661099
663581 663583 663587 663589
664661 664663 664667 664669
666431 666433 666437 666439
680291 680293 680297 680299
681251 681253 681257 681259
691721 691723 691727 691729
705161 705163 705167 705169
715151 715153 715157 715159
734471 734473 734477 734479
736361 736363 736367 736369
739391 739393 739397 739399
768191 768193 768197 768199
773021 773023 773027 773029
795791 795793 795797 795799
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Last edited by Maschke; October 30th, 2014 at 05:57 PM.
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October 30th, 2014, 05:52 PM   #4
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Quote:
Originally Posted by Petek View Post
101, 103, 107 and 109 are all prime. I think it's an open question whether there are infinitely many such.
Yes. But 'everyone' knows that it's true. Bateman, Horn, & Stemmler have a conjecture which not only implies that there are infinitely many but gives their density.

One of my college professors was very interested in this particular case (four primes in a decade) but I don't know if he ever published anything on it.
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