Prime Quadruplet Emirps

Trejack · 2016-04-21, 16:23

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.

fivemack · 2016-04-21, 17:18

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

science_man_88 · 2016-04-21, 18:04

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:

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.

LaurV · 2016-04-22, 02:48

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 >

LaurV · 2016-04-22, 02:55

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.

LaurV · 2016-04-22, 03:43

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..

Trejack · 2016-04-22, 06:10

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.

paulunderwood · 2016-04-22, 06:53

To reverse a number use scalar reverse on the number.

Code:

perl -e 'print((scalar reverse 335)."\n")'
533

danaj · 2016-04-22, 17:40

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.

R. Gerbicz · 2016-04-22, 18:28

In the above there is a lot of no emirp (and broken codes), see for the definition: https://en.wikipedia.org/wiki/Emirp .

danaj · 2016-04-22, 19:16

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.

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.

2016-04-22, 06:53	#8
paulunderwood Sep 2002 Database er0rr 111000001010₂ 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 2²·227 Posts	Assuming I'm understanding the problem: Code: perl -Mntheory=:all -E 'for (sieve_prime_cluster(1,1010,2,6,8)) { 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 = 1028; 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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2016-04-21, 17:18	#2
fivemack (loop (#_fork)) Feb 2006 Cambridge, England 2×3,191 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

2016-04-22, 06:10	#7
Trejack Apr 2016 2×13 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, 18:28	#10
R. Gerbicz "Robert Gerbicz" Oct 2005 Hungary 10110100101₂ Posts	In the above there is a lot of no emirp (and broken codes), see for the definition: https://en.wikipedia.org/wiki/Emirp .