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 2016-04-21, 16:23 #1 Trejack   Apr 2016 2×13 Posts 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.
 2016-04-21, 17:18 #2 fivemack (loop (#_fork))     Feb 2006 Cambridge, England 22×3×232 Posts 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
2016-04-21, 18:04   #3
science_man_88

"Forget I exist"
Jul 2009
Dumbassville

26×131 Posts

Quote:
 Originally Posted by Trejack 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
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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
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176903
176921
176923
177743
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181751
181757
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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

 2016-04-22, 02:48 #4 LaurV Romulan Interpreter     Jun 2011 Thailand 2·3·1,423 Posts 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
 2016-04-22, 02:55 #5 LaurV Romulan Interpreter     Jun 2011 Thailand 215A16 Posts 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
 2016-04-22, 03:43 #6 LaurV Romulan Interpreter     Jun 2011 Thailand 215A16 Posts 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
 2016-04-22, 06:10 #7 Trejack   Apr 2016 328 Posts 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.
 2016-04-22, 06:53 #8 paulunderwood     Sep 2002 Database er0rr 2·3·5·107 Posts 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
 2016-04-22, 17:40 #9 danaj   "Dana Jacobsen" Feb 2011 Bangkok, TH 17·53 Posts 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
 2016-04-22, 18:28 #10 R. Gerbicz     "Robert Gerbicz" Oct 2005 Hungary 7×191 Posts In the above there is a lot of no emirp (and broken codes), see for the definition: https://en.wikipedia.org/wiki/Emirp .
2016-04-22, 19:16   #11
danaj

"Dana Jacobsen"
Feb 2011
Bangkok, TH

17×53 Posts

Quote:
 Originally Posted by R. Gerbicz 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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