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Old 2016-04-21, 16:23   #1
Trejack
 
Apr 2016

1A16 Posts
Post Prime Quadruplet Emirps

I've come up with the following question that is there a prime quadruplet emirp (of all 4 terms), n digits long?
There is a prime quadruplets frequency I forgot, but if anyone would be able to find one, say 32 digits, I could possibly have the same problem with twin prime, triples, and k-tuples, are all emirps.

Here is a small example for twins: 18911, 18913, 11981, 31981, ALL members are emirps too, and I thought this would be a challenging puzzle, yet hard. Thanks to all solvers.
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Old 2016-04-21, 17:18   #2
fivemack
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Feb 2006
Cambridge, England

23×3×5×53 Posts
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You have eight numbers (x, x+2, x+6, x+8, and their reverses) that need to be prime, log(n)^-8 is more than 10^-n for large enough n. So there are masses of examples, they'll be a bit tedious to find because of considering carries when doing the sieve for the reverses

no five-digit example
389561
1285511
36306071
126716201
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Old 2016-04-21, 18:04   #3
science_man_88
 
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Jul 2009
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Quote:
Originally Posted by Trejack View Post
I've come up with the following question that is there a prime quadruplet emirp (of all 4 terms), n digits long?
There is a prime quadruplets frequency I forgot, but if anyone would be able to find one, say 32 digits, I could possibly have the same problem with twin prime, triples, and k-tuples, are all emirps.

Here is a small example for twins: 18911, 18913, 11981, 31981, ALL members are emirps too, and I thought this would be a challenging puzzle, yet hard. Thanks to all solvers.
you could also ask it as are there any prime quadruplets of palindromic primes ? oh and @ fivemack sure but you can eliminate it down quite a bit just by knowing that you are looking for 4 consecutive primes that have to have an odd first digit:

Code:
5
7
727
733
739
743
751
919
1193
1201
1213
1217
1223
1229
1231
3371
3373
7177
9011
9769
9781
10039
10061
10067
11551
11699
11701
11777
11897
11903
11909
11923
11953
11959
12107
13147
13259
13693
14563
14891
14897
14923
15493
15497
15511
16561
18169
18719
18731
18743
18749
19219
19531
19661
31891
32467
34543
34549
34583
35117
35129
35141
35311
36097
36187
36251
38351
38903
39791
39799
39821
70241
70921
72227
72307
72313
72547
72551
74747
75721
77323
78607
78887
79379
79393
79397
79399
79531
90121
91183
92297
92479
92959
93581
94121
95111
95791
96263
96857
97397
98573
99397
102593
104059
105653
105667
106391
106397
108187
109849
109859
113189
118543
119773
119783
119797
121921
125627
125639
125641
125651
125821
126547
129587
129589
133709
139387
139393
139591
141157
143291
144407
144409
149561
149563
149579
149867
150217
155557
157061
157427
159697
162293
162343
162359
163781
163789
163811
166723
167381
167771
169067
169069
170689
170759
170761
171929
173249
175267
175463
176903
176921
176923
177743
178333
178349
178681
179173
181751
181757
181759
182011
186103
191627
193261
193283
193541
193877
193883
193993
194713
194839
195077
197383
197389
197419
198193
302123
302597
302767
303119
303571
303781
305867
307277
309599
311279
313763
314239
316073
316087
316501
317599
317663
323087
324031
324053
324893
324901
325163
325181
333923
333929
334759
334771
334777
334783
334787
336689
336703
336727
339671
339673
340429
341293
348053
350111
360169
361499
362431
362749
363719
366953
367687
375787
375799
375997
381103
381287
381859
381911
383083
386371
387503
389287
389297
389561
391537
392911
392923
392927
393361
393373
393377
393919
397597
397921
701383
701399
701593
701609
701837
703663
709381
712289
712301
714361
714377
714443
714577
714893
715361
715373
715397
717151
718813
720133
720869
722633
722639
724733
729877
736447
737897
738349
738953
741563
742499
743989
744019
744043
744071
744377
744389
747833
758987
760273
768841
773953
776801
777349
777353
777419
779003
779011
779021
780047
780049
786673
786691
786833
788449
788467
790271
790421
795761
798781
798799
799313
905213
910909
910939
910957
910981
918347
918353
919381
919393
919559
919679
920957
920963
920971
920999
922223
922237
922717
924643
928429
928453
928457
928463
940553
941449
941453
941461
941467
944821
944833
947641
947987
948581
948593
951089
954697
957241
960937
966431
971141
971143
971149
972029
972031
975089
975941
979273
979403
980717
980909
981391
981517
981523
982273
982703
983929
987029
991181
991927
993397
993527
994087
994093
995443
995447
995461
996539
996551
996563
999623
is my list of possible starting points below a million for example. doh now I realize I'm an idiot I can probably get my code working faster now.

Last fiddled with by science_man_88 on 2016-04-21 at 18:26
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Old 2016-04-22, 02:48   #4
LaurV
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8,753 Posts
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All the twins: (pari/gp, one minute exercise, there are many more if disparate primes, i.e. if "==2" test is omitted)
Code:
gp > p=2; while(p<10^5, np=nextprime(p+1); if(np-p==2&&isprime(rp=eval(concat(Vecrev(Str(p)))))&&isprime(rnp=eval(concat(Vecrev(Str(np))))),print(p","np","rp","rnp)); p=np)
3,5,3,5
5,7,5,7
11,13,11,31
71,73,17,37
149,151,941,151
179,181,971,181
311,313,113,313
1031,1033,1301,3301
1151,1153,1511,3511
1229,1231,9221,1321
3299,3301,9923,1033
3371,3373,1733,3733
3389,3391,9833,1933
3467,3469,7643,9643
3851,3853,1583,3583
7457,7459,7547,9547
7949,7951,9497,1597
9011,9013,1109,3109
9437,9439,7349,9349
10007,10009,70001,90001
10067,10069,76001,96001
10457,10459,75401,95401
10499,10501,99401,10501
10889,10891,98801,19801
11159,11161,95111,16111
11699,11701,99611,10711
11717,11719,71711,91711
11777,11779,77711,97711
11969,11971,96911,17911
12071,12073,17021,37021
12107,12109,70121,90121
13709,13711,90731,11731
13757,13759,75731,95731
13829,13831,92831,13831
13931,13933,13931,33931
14447,14449,74441,94441
14549,14551,94541,15541
14591,14593,19541,39541
15731,15733,13751,33751
16061,16063,16061,36061
16451,16453,15461,35461
17207,17209,70271,90271
17681,17683,18671,38671
17747,17749,74771,94771
17909,17911,90971,11971
18911,18913,11981,31981
19421,19423,12491,32491
19541,19543,14591,34591
30851,30853,15803,35803
31721,31723,12713,32713
32321,32323,12323,32323
32939,32941,93923,14923
33809,33811,90833,11833
34469,34471,96443,17443
34589,34591,98543,19543
34841,34843,14843,34843
34961,34963,16943,36943
35051,35053,15053,35053
35801,35803,10853,30853
36107,36109,70163,90163
37199,37201,99173,10273
37307,37309,70373,90373
37547,37549,74573,94573
37571,37573,17573,37573
38327,38329,72383,92383
38921,38923,12983,32983
39827,39829,72893,92893
70949,70951,94907,15907
70997,70999,79907,99907
71261,71263,16217,36217
71387,71389,78317,98317
72227,72229,72227,92227
72251,72253,15227,35227
72869,72871,96827,17827
74759,74761,95747,16747
75167,75169,76157,96157
75539,75541,93557,14557
76259,76261,95267,16267
78779,78781,97787,18787
78887,78889,78887,98887
79229,79231,92297,13297
79397,79399,79397,99397
79841,79843,14897,34897
92381,92383,18329,38329
92639,92641,93629,14629
93557,93559,75539,95539
94109,94111,90149,11149
94151,94153,15149,35149
94349,94351,94349,15349
94397,94399,79349,99349
94541,94543,14549,34549
94649,94651,94649,15649
95801,95803,10859,30859
96179,96181,97169,18169
97379,97381,97379,18379
97787,97789,78779,98779
98729,98731,92789,13789
time = 63 ms.
gp >

Last fiddled with by LaurV on 2016-04-22 at 02:49
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Old 2016-04-22, 02:55   #5
LaurV
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Jun 2011
Thailand

100010001100012 Posts
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On the same idea, consecutive triples, regardless of the distance:
Code:
gp > p=2; while(p<10^5, np=nextprime(p+1); nnp=nextprime(np+1);if(isprime(rp=eval(concat(Vecrev(Str(p)))))&&isprime(rnp=eval(concat(Vecrev(Str(np)))))&&isprime(rnnp=eval(concat(Vecrev(Str(nnp))))),print(p","np","nnp" - "rp","rnp","rnnp)); p=nnp)
2,3,5 - 2,3,5
5,7,11 - 5,7,11
11,13,17 - 11,31,71
149,151,157 - 941,151,751
179,181,191 - 971,181,191
727,733,739 - 727,337,937
739,743,751 - 937,347,157
751,757,761 - 157,757,167
919,929,937 - 919,929,739
1201,1213,1217 - 1021,3121,7121
1217,1223,1229 - 7121,3221,9221
1229,1231,1237 - 9221,1321,7321
1237,1249,1259 - 7321,9421,9521
1381,1399,1409 - 1831,9931,9041
1499,1511,1523 - 9941,1151,3251
1723,1733,1741 - 3271,3371,1471
3083,3089,3109 - 3803,9803,9013
3343,3347,3359 - 3433,7433,9533
3371,3373,3389 - 1733,3733,9833
3389,3391,3407 - 9833,1933,7043
3463,3467,3469 - 3643,7643,9643
7177,7187,7193 - 7717,7817,3917
7673,7681,7687 - 3767,1867,7867
9013,9029,9041 - 3109,9209,1409
9479,9491,9497 - 9749,1949,7949
9769,9781,9787 - 9679,1879,7879
9787,9791,9803 - 7879,1979,3089
10039,10061,10067 - 93001,16001,76001
10067,10069,10079 - 76001,96001,97001
10487,10499,10501 - 78401,99401,10501
11579,11587,11593 - 97511,78511,39511
11701,11717,11719 - 10711,71711,91711
11779,11783,11789 - 97711,38711,98711
11897,11903,11909 - 79811,30911,90911
11909,11923,11927 - 90911,32911,72911
11927,11933,11939 - 72911,33911,93911
11953,11959,11969 - 35911,95911,96911
11969,11971,11981 - 96911,17911,18911
12107,12109,12113 - 70121,90121,31121
12743,12757,12763 - 34721,75721,36721
13147,13151,13159 - 74131,15131,95131
13267,13291,13297 - 76231,19231,79231
13693,13697,13709 - 39631,79631,90731
13931,13933,13963 - 13931,33931,36931
14591,14593,14621 - 19541,39541,12641
14891,14897,14923 - 19841,79841,32941
14923,14929,14939 - 32941,92941,93941
15493,15497,15511 - 39451,79451,11551
15511,15527,15541 - 11551,72551,14551
15731,15733,15737 - 13751,33751,73751
16103,16111,16127 - 30161,11161,72161
16193,16217,16223 - 39161,71261,32261
16561,16567,16573 - 16561,76561,37561
17033,17041,17047 - 33071,14071,74071
17203,17207,17209 - 30271,70271,90271
17903,17909,17911 - 30971,90971,11971
18181,18191,18199 - 18181,19181,99181
18719,18731,18743 - 91781,13781,34781
18743,18749,18757 - 34781,94781,75781
18757,18773,18787 - 75781,37781,78781
19231,19237,19249 - 13291,73291,94291
19531,19541,19543 - 13591,14591,34591
19681,19687,19697 - 18691,78691,79691
30517,30529,30539 - 71503,92503,93503
30643,30649,30661 - 34603,94603,16603
31051,31063,31069 - 15013,36013,96013
31081,31091,31121 - 18013,19013,12113
31907,31957,31963 - 70913,75913,36913
32203,32213,32233 - 30223,31223,33223
32479,32491,32497 - 97423,19423,79423
32933,32939,32941 - 33923,93923,14923
33911,33923,33931 - 11933,32933,13933
34129,34141,34147 - 92143,14143,74143
34543,34549,34583 - 34543,94543,38543
34583,34589,34591 - 38543,98543,19543
35117,35129,35141 - 71153,92153,14153
35141,35149,35153 - 14153,94153,35153
35311,35317,35323 - 11353,71353,32353
36107,36109,36131 - 70163,90163,13163
36187,36191,36209 - 78163,19163,90263
36251,36263,36269 - 15263,36263,96263
37547,37549,37561 - 74573,94573,16573
37997,38011,38039 - 79973,11083,93083
38083,38113,38119 - 38083,31183,91183
38351,38371,38377 - 15383,17383,77383
38629,38639,38651 - 92683,93683,15683
38917,38921,38923 - 71983,12983,32983
39799,39821,39827 - 99793,12893,72893
39827,39829,39839 - 72893,92893,93893
39887,39901,39929 - 78893,10993,92993
70241,70249,70271 - 14207,94207,17207
70327,70351,70373 - 72307,15307,37307
70663,70667,70687 - 36607,76607,78607
70921,70937,70949 - 12907,73907,94907
71347,71353,71359 - 74317,35317,95317
71387,71389,71399 - 78317,98317,99317
71899,71909,71917 - 99817,90917,71917
72227,72229,72251 - 72227,92227,15227
72307,72313,72337 - 70327,31327,73327
72337,72341,72353 - 73327,14327,35327
72547,72551,72559 - 74527,15527,95527
72559,72577,72613 - 95527,77527,31627
74071,74077,74093 - 17047,77047,39047
74441,74449,74453 - 14447,94447,35447
74509,74521,74527 - 90547,12547,72547
74747,74759,74761 - 74747,95747,16747
75211,75217,75223 - 11257,71257,32257
75721,75731,75743 - 12757,13757,34757
76213,76231,76243 - 31267,13267,34267
76253,76259,76261 - 35267,95267,16267
76379,76387,76403 - 97367,78367,30467
76801,76819,76829 - 10867,91867,92867
77323,77339,77347 - 32377,93377,74377
77587,77591,77611 - 78577,19577,11677
78623,78643,78649 - 32687,34687,94687
78779,78781,78787 - 97787,18787,78787
78809,78823,78839 - 90887,32887,93887
78887,78889,78893 - 78887,98887,39887
79379,79393,79397 - 97397,39397,79397
79397,79399,79411 - 79397,99397,11497
79411,79423,79427 - 11497,32497,72497
79537,79549,79559 - 73597,94597,95597
79669,79687,79691 - 96697,78697,19697
79757,79769,79777 - 75797,96797,77797
90127,90149,90163 - 72109,94109,36109
90247,90263,90271 - 74209,36209,17209
90863,90887,90901 - 36809,78809,10909
91183,91193,91199 - 38119,39119,99119
92119,92143,92153 - 91129,34129,35129
92297,92311,92317 - 79229,11329,71329
92489,92503,92507 - 98429,30529,70529
92987,92993,93001 - 78929,39929,10039
93601,93607,93629 - 10639,70639,92639
94151,94153,94169 - 15149,35149,96149
94889,94903,94907 - 98849,30949,70949
95131,95143,95153 - 13159,34159,35159
95791,95801,95803 - 19759,10859,30859
96001,96013,96017 - 10069,31069,71069
96263,96269,96281 - 36269,96269,18269
96893,96907,96911 - 39869,70969,11969
97423,97429,97441 - 32479,92479,14479
98251,98257,98269 - 15289,75289,96289
98597,98621,98627 - 79589,12689,72689
98717,98729,98731 - 71789,92789,13789
99401,99409,99431 - 10499,90499,13499
time = 113 ms.
gp >
edit: in fact here the "p=nnp" at the end is wrong, because it can skip triples which are "shorter" than the printed. But I am too lazy to change now.

Last fiddled with by LaurV on 2016-04-22 at 02:57
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Old 2016-04-22, 03:43   #6
LaurV
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Jun 2011
Thailand

8,753 Posts
Default

When the distance is put in: (all to 1e9)
Code:
gp > p=2; d=8; while(p<10^9, np=nextprime(p+1); nnp=nextprime(np+1); nnnp=nextprime(nnp+1); if(nnnp-p<=d&&isprime(rp=
eval(concat(Vecrev(Str(p)))))&&isprime(rnp=eval(concat(Vecrev(Str(np)))))&&isprime(rnnp=eval(concat(Vecrev(Str(nnp)))))&&isprime
(rnnnp=eval(concat(Vecrev(Str(nnnp))))),print(p","np","nnp","nnnp" - "rp","rnp","rnnp","rnnnp)); p=np)
2,3,5,7 - 2,3,5,7
3,5,7,11 - 3,5,7,11
5,7,11,13 - 5,7,11,31
389561,389563,389567,389569 - 165983,365983,765983,965983
1285511,1285513,1285517,1285519 - 1155821,3155821,7155821,9155821
3200201,3200203,3200207,3200209 - 1020023,3020023,7020023,9020023
36306071,36306073,36306077,36306079 - 17060363,37060363,77060363,97060363
75681911,75681913,75681917,75681919 - 11918657,31918657,71918657,91918657
76605491,76605493,76605497,76605499 - 19450667,39450667,79450667,99450667
90561851,90561853,90561857,90561859 - 15816509,35816509,75816509,95816509
126716201,126716203,126716207,126716209 - 102617621,302617621,702617621,902617621
139984541,139984543,139984547,139984549 - 145489931,345489931,745489931,945489931
141272471,141272473,141272477,141272479 - 174272141,374272141,774272141,974272141
151851641,151851643,151851647,151851649 - 146158151,346158151,746158151,946158151
160436951,160436953,160436957,160436959 - 159634061,359634061,759634061,959634061
182746841,182746843,182746847,182746849 - 148647281,348647281,748647281,948647281
301397141,301397143,301397147,301397149 - 141793103,341793103,741793103,941793103
337425371,337425373,337425377,337425379 - 173524733,373524733,773524733,973524733
371610131,371610133,371610137,371610139 - 131016173,331016173,731016173,931016173
374964041,374964043,374964047,374964049 - 140469473,340469473,740469473,940469473
700788701,700788703,700788707,700788709 - 107887007,307887007,707887007,907887007
712457561,712457563,712457567,712457569 - 165754217,365754217,765754217,965754217
768415091,768415093,768415097,768415099 - 190514867,390514867,790514867,990514867
771810881,771810883,771810887,771810889 - 188018177,388018177,788018177,988018177
936019151,936019153,936019157,936019159 - 151910639,351910639,751910639,951910639
975697271,975697273,975697277,975697279 - 172796579,372796579,772796579,972796579
time = 6min, 40,889 ms.
gp >
edit: which is also not-optimal, because it computes the nextprimes 3 time more often than necessary, but well... doh... grr..

Last fiddled with by LaurV on 2016-04-22 at 03:46
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Old 2016-04-22, 06:10   #7
Trejack
 
Apr 2016

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Thanks LaurV,

I can use those results to record the (approximate) occurance of prime quadruplet (k-tuplet) emirps. As for verifying all four terms are emirps, I was unable to handle this using ntheory.
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Old 2016-04-22, 06:53   #8
paulunderwood
 
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To reverse a number use scalar reverse on the number.
Code:
perl -e 'print((scalar reverse 335)."\n")'
533

Last fiddled with by paulunderwood on 2016-04-22 at 06:54
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Old 2016-04-22, 17:40   #9
danaj
 
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Assuming I'm understanding the problem:

Code:
perl -Mntheory=:all -E 'for (sieve_prime_cluster(1,10**10,2,6,8)) { say if is_prime(reverse("".$_)) && is_prime(reverse("".$_+2)) && is_prime(reverse("".$_+6)) && is_prime(reverse("".$_+8)) }'
takes 0.5 seconds for 10^9, 5 seconds for 10^10, 60 seconds for 10^11. The cluster sieve gets all the quadruplets with that pattern (the value returned is the first), then checks primality of the reverses of each. It's a bit clumsy spelling out all the is_prime calls.

For 10^9 this seems to generate the same results as LaurV's barring 2 and 3 which don't match the pattern.

Some larger results:
10^20 + 5526684241
10^21 + 6826587001
10^22 + 23496012391
10^23 + 51139069771
10^28 + 34613950651
10^32 + 1181613772801

It's better for the large results to use a loop over the cluster sieve so it only spends a reasonable amount of time getting quadruplets before testing them. I'm envious of Python's yield for this (which lets one stream output instead of returning it in a big chunk). Something like:
Code:
perl -Mntheory=:all -E 'use bigint; my $s = 10**28; while (1) { say "-- $s"; for (sieve_prime_cluster($s,$s+1e10,2,6,8)) { say if is_prime(reverse("".$_)) && is_prime(reverse("".$_+2)) && is_prime(reverse("".$_+6)) && is_prime(reverse("".$_+8)) } $s += 1e10; }'
(my Windows machine is having issues with this, but it works fine on Linux -- not sure what's up there)

Even better is modifying the example threaded cluster sieve to restrict results with the reversal condition for whatever cluster is being used. That would be nice for larger clusters.

Last fiddled with by danaj on 2016-04-22 at 17:46
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Old 2016-04-22, 18:28   #10
R. Gerbicz
 
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"Robert Gerbicz"
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In the above there is a lot of no emirp (and broken codes), see for the definition: https://en.wikipedia.org/wiki/Emirp .
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Old 2016-04-22, 19:16   #11
danaj
 
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Quote:
Originally Posted by R. Gerbicz View Post
In the above there is a lot of no emirp (and broken codes), see for the definition: https://en.wikipedia.org/wiki/Emirp .
Which "the above" are you referring to?

I didn't check for palindromes but other than 5, it doesn't look like my code is outputing any (but it could). With C = {0,2,6,8} all the results p are such that p+c and reverse(p+c) are both prime for all c in C.

Last fiddled with by danaj on 2016-04-22 at 19:18
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