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February 19th, 2011, 09:48 AM   #1
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Twin primes distributed around N mod 3 =0

Since there are quite a few prime number topics on this forum, here's a thought I developed a few years ago (but did not progress since then):

I started by postulating that all pairs of twin primes are equidistant around mod 3 = 0.

Examples:
* 29,31 and 41,43 around 36 (distance 5, 36 mod 3 = 0)
* 17,19 and 269, 271 around 144 (distance 125, 144 mod 3 = 0)

Didn't get much further with that, but with some good help we could potentially prove a weaker version of the postulate and that is: IF two pairs of twin primes (except the first pair 3,5) are equidistant around N (not necessarily a square), then N mod 3 MUST be 0.

Here's the prove:

Equidistant pairs of twin primes are of the shape (N-k-2, N-k) and (N+k, N+k+2)

Suppose that N mod 3 <> 0, then the remainder is either 1 or 2.

The case N mod 3 = 1 implies that:
k mod 3 must be <> 2 (since (N + k) mod 3 = 0 is not prime)
k mod 3 must be <> 1 (since (N - k) mod 3 = 0 is not prime)
so, k mod 3 must be 0, but then (N+k+2) mod 3 = 0 is not prime)
-> (N-k-2, N-k),(N+k, N+k+2) can't be all prime when N mod 3 = 1

The case N mod 3 = 2 implies that:
k mod 3 must be <> 1 (since (N + k) mod 3 = 0 and is not prime)
k mod 3 must be <> 2 (since (N - k) mod 3 = 0 and is not prime)
so, k mod 3 must be 0, but then (N-k-2) mod 3 = 0 and is not prime)
-> (N-k-2, N-k),(N+k, N+k+2) can't be all prime when N mod 3 = 2

This proves that (N-k-2, N-k),(N+k, N+k+2) can only be a set of equidistant twin primes in case N mod 3 = 0 (excluding the first twin pair 3,5)

Now we know that N mod 3 must be 0, how about k ?
k mod 3 <> 0 otherwise N+k mod 3 en N-k mod 3 would be 0
k mod 3 <> 1 otherwise N+k+2 and N-k-2 mod 3 would be 0
so, k mod 3 must be 2 which implies k = 3m + 2, m=0,1,2...
When N is even, k must be: k = 5, 11, 17 ...
When N is odd, k must be: k = 2, 8, 14 ...

This poses the inverse question whether all N mod 3 = 0 also have two equidistant pairs of twin primes? This has a surprising outcome. We learned (empirically tested up to N=1E6), that apparently only for N = 3, 6, 48, 198, 201, 258, 348, 393, 453, 558, 573, 633, 678, 1623 and 2103 no equidistant set
of twin primes exists. There is a related Sloane's series http://oeis.org/A072254. A further surprise is that k relative to N remains remarkably small (< 1%).

Obvious question is whether there is any N (mod 3 = 0) > 2103 that has no equidistant set of twin primes.

Any thoughts/ideas?
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February 19th, 2011, 12:23 PM   #2
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Re: Twin primes distributed around N mod 3 =0

...
al-mahed is offline  
February 19th, 2011, 12:33 PM   #3
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Re: Twin primes distributed around N mod 3 =0

it is clear that in any pair of twin primes p, q, such that p<q,

so call the distance between two pairs and then

clearly for a given integer , you want d/2+1 divisible by 3, so you want

just observe that

the question of is interesting, I'll test some values
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February 19th, 2011, 02:26 PM   #4
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Re: Twin primes distributed around N mod 3 =0

the simple algorithm and the list:

Code:
forstep(x=1,10000,2,if(isprime(x),if(isprime(x+2),forstep(y=1,x-2,2,if(isprime(y),if(isprime(y-2),if(ispower(y+(x-y)/2+1,2),print(y-2":"y"--"x":"x+2" square="y+(x-y)/2+1))))))))
Code:
3:5--11:13 square=9
11:13--17:19 square=16
17:19--29:31 square=25
5:7--41:43 square=25
17:19--107:109 square=64
59:61--137:139 square=100
101:103--137:139 square=121
17:19--179:181 square=100
59:61--179:181 square=121
5:7--191:193 square=100
41:43--197:199 square=121
137:139--197:199 square=169
191:193--197:199 square=196
11:13--227:229 square=121
107:109--227:229 square=169
149:151--239:241 square=196
239:241--269:271 square=256
3:5--281:283 square=144
107:109--281:283 square=196
227:229--281:283 square=256
197:199--311:313 square=256
41:43--347:349 square=196
227:229--347:349 square=289
197:199--521:523 square=361
5:7--569:571 square=289
149:151--569:571 square=361
227:229--569:571 square=400
197:199--599:601 square=400
101:103--617:619 square=361
179:181--617:619 square=400
347:349--617:619 square=484
3:5--641:643 square=324
59:61--659:661 square=361
137:139--659:661 square=400
137:139--827:829 square=484
227:229--827:829 square=529
419:421--827:829 square=625
521:523--827:829 square=676
107:109--857:859 square=484
197:199--857:859 square=529
821:823--857:859 square=841
227:229--1019:1021 square=625
659:661--1019:1021 square=841
5:7--1049:1051 square=529
197:199--1049:1051 square=625
617:619--1061:1063 square=841
857:859--1061:1063 square=961
827:829--1091:1093 square=961
197:199--1151:1153 square=676
17:19--1229:1231 square=625
71:73--1277:1279 square=676
641:643--1277:1279 square=961
1031:1033--1277:1279 square=1156
59:61--1289:1291 square=676
1019:1021--1289:1291 square=1156
617:619--1301:1303 square=961
29:31--1319:1321 square=676
599:601--1319:1321 square=961
137:139--1427:1429 square=784
617:619--1427:1429 square=1024
881:883--1427:1429 square=1156
1019:1021--1427:1429 square=1225
3:5--1451:1453 square=729
227:229--1451:1453 square=841
857:859--1451:1453 square=1156
197:199--1481:1483 square=841
827:829--1481:1483 square=1156
191:193--1487:1489 square=841
431:433--1487:1489 square=961
821:823--1487:1489 square=1156
71:73--1607:1609 square=841
311:313--1607:1609 square=961
1277:1279--1607:1609 square=1444
59:61--1619:1621 square=841
827:829--1619:1621 square=1225
11:13--1667:1669 square=841
641:643--1667:1669 square=1156
347:349--1697:1699 square=1024
197:199--1721:1723 square=961
521:523--1787:1789 square=1156
659:661--1787:1789 square=1225
1487:1489--1871:1873 square=1681
41:43--1877:1879 square=961
431:433--1877:1879 square=1156
569:571--1877:1879 square=1225
857:859--1877:1879 square=1369
1319:1321--1877:1879 square=1600
1481:1483--1877:1879 square=1681
1427:1429--1931:1933 square=1681
311:313--1997:1999 square=1156
1697:1699--1997:1999 square=1849
1871:1873--1997:1999 square=1936
17:19--2027:2029 square=1024
281:283--2027:2029 square=1156
419:421--2027:2029 square=1225
857:859--2027:2029 square=1444
1667:1669--2027:2029 square=1849
227:229--2081:2083 square=1156
1277:1279--2081:2083 square=1681
1787:1789--2081:2083 square=1936
1607:1609--2087:2089 square=1849
197:199--2111:2113 square=1156
179:181--2129:2131 square=1156
1229:1231--2129:2131 square=1681
2087:2089--2141:2143 square=2116
71:73--2237:2239 square=1156
41:43--2267:2269 square=1156
179:181--2267:2269 square=1225
617:619--2267:2269 square=1444
1091:1093--2267:2269 square=1681
1427:1429--2267:2269 square=1849
137:139--2309:2311 square=1225
1049:1051--2309:2311 square=1681
107:109--2339:2341 square=1225
857:859--2339:2341 square=1600
1019:1021--2339:2341 square=1681
1487:1489--2381:2383 square=1936
809:811--2549:2551 square=1681
1319:1321--2549:2551 square=1936
1277:1279--2591:2593 square=1936
227:229--2657:2659 square=1444
2141:2143--2657:2659 square=2401
2339:2341--2657:2659 square=2500
197:199--2687:2689 square=1444
2111:2113--2687:2689 square=2401
2309:2311--2687:2689 square=2500
2087:2089--2711:2713 square=2401
5:7--2729:2731 square=1369
2267:2269--2729:2731 square=2500
569:571--2789:2791 square=1681
1427:1429--2801:2803 square=2116
1997:1999--2801:2803 square=2401
227:229--2969:2971 square=1600
2027:2029--2969:2971 square=2500
197:199--2999:3001 square=1600
1229:1231--2999:3001 square=2116
1997:1999--2999:3001 square=2500
239:241--3119:3121 square=1681
1877:1879--3119:3121 square=2500
29:31--3167:3169 square=1600
191:193--3167:3169 square=1681
1061:1063--3167:3169 square=2116
2237:2239--3167:3169 square=2704
107:109--3251:3253 square=1681
617:619--3251:3253 square=1936
101:103--3257:3259 square=1681
2789:2791--3257:3259 square=3025
59:61--3299:3301 square=1681
569:571--3299:3301 square=1936
1697:1699--3299:3301 square=2500
2969:2971--3299:3301 square=3136
29:31--3329:3331 square=1681
1667:1669--3329:3331 square=2500
2687:2689--3359:3361 square=3025
857:859--3371:3373 square=2116
1427:1429--3371:3373 square=2401
1607:1609--3389:3391 square=2500
2657:2659--3389:3391 square=3025
227:229--3467:3469 square=1849
2801:2803--3467:3469 square=3136
3257:3259--3467:3469 square=3364
1877:1879--3527:3529 square=2704
2087:2089--3527:3529 square=2809
2729:2731--3539:3541 square=3136
137:139--3557:3559 square=1849
311:313--3557:3559 square=1936
857:859--3557:3559 square=2209
2711:2713--3557:3559 square=3136
3167:3169--3557:3559 square=3364
2687:2689--3581:3583 square=3136
197:199--3671:3673 square=1936
101:103--3767:3769 square=1936
461:463--3767:3769 square=2116
1031:1033--3767:3769 square=2401
1229:1231--3767:3769 square=2500
3671:3673--3767:3769 square=3721
17:19--3851:3853 square=1936
311:313--3917:3919 square=2116
881:883--3917:3919 square=2401
1487:1489--3917:3919 square=2704
1697:1699--3917:3919 square=2809
2129:2131--3917:3919 square=3025
3767:3769--3917:3919 square=3844
2339:2341--3929:3931 square=3136
227:229--4001:4003 square=2116
2267:2269--4001:4003 square=3136
2027:2029--4019:4021 square=3025
179:181--4049:4051 square=2116
1997:1999--4049:4051 square=3025
3389:3391--4049:4051 square=3721
137:139--4091:4093 square=2116
101:103--4127:4129 square=2116
1277:1279--4127:4129 square=2704
1487:1489--4127:4129 square=2809
2141:2143--4127:4129 square=3136
3557:3559--4127:4129 square=3844
71:73--4157:4159 square=2116
641:643--4157:4159 square=2401
2111:2113--4157:4159 square=3136
2801:2803--4157:4159 square=3481
3527:3529--4157:4159 square=3844
11:13--4217:4219 square=2116
197:199--4217:4219 square=2209
3467:3469--4217:4219 square=3844
569:571--4229:4231 square=2401
2729:2731--4229:4231 square=3481
4217:4219--4229:4231 square=4225
2027:2029--4241:4243 square=3136
1787:1789--4259:4261 square=3025
3929:3931--4259:4261 square=4096
1997:1999--4271:4273 square=3136
2687:2689--4271:4273 square=3481
3167:3169--4271:4273 square=3721
3917:3919--4271:4273 square=4096
461:463--4337:4339 square=2401
659:661--4337:4339 square=2500
1277:1279--4337:4339 square=2809
1931:1933--4337:4339 square=3136
3851:3853--4337:4339 square=4096
3767:3769--4421:4423 square=4096
1787:1789--4481:4483 square=3136
281:283--4517:4519 square=2401
3167:3169--4517:4519 square=3844
3671:3673--4517:4519 square=4096
3929:3931--4517:4519 square=4225
857:859--4547:4549 square=2704
1721:1723--4547:4549 square=3136
2087:2089--4637:4639 square=3364
2801:2803--4637:4639 square=3721
4337:4339--4637:4639 square=4489
149:151--4649:4651 square=2401
347:349--4649:4651 square=2500
1619:1621--4649:4651 square=3136
2309:2311--4649:4651 square=3481
2789:2791--4649:4651 square=3721
3539:3541--4649:4651 square=4096
2237:2239--4721:4723 square=3481
3467:3469--4721:4723 square=4096
11:13--4787:4789 square=2401
617:619--4787:4789 square=2704
827:829--4787:4789 square=2809
1481:1483--4787:4789 square=3136
197:199--4799:4801 square=2500
3389:3391--4799:4801 square=4096
2027:2029--4931:4933 square=3481
3257:3259--4931:4933 square=4096
29:31--4967:4969 square=2500
1301:1303--4967:4969 square=3136
1949:1951--5009:5011 square=3481
4787:4789--5009:5011 square=4900
3167:3169--5021:5023 square=4096
2339:2341--5099:5101 square=3721
3167:3169--5279:5281 square=4225
4517:4519--5279:5281 square=4900
4799:4801--5279:5281 square=5041
197:199--5417:5419 square=2809
2267:2269--5417:5419 square=3844
3557:3559--5417:5419 square=4489
827:829--5441:5443 square=3136
1997:1999--5441:5443 square=3721
4637:4639--5441:5443 square=5041
137:139--5477:5479 square=2809
569:571--5477:5479 square=3025
1481:1483--5477:5479 square=3481
2711:2713--5477:5479 square=4096
2969:2971--5477:5479 square=4225
3767:3769--5477:5479 square=4624
2687:2689--5501:5503 square=4096
1319:1321--5639:5641 square=3481
2549:2551--5639:5641 square=4096
4157:4159--5639:5641 square=4900
617:619--5651:5653 square=3136
1787:1789--5651:5653 square=3721
1301:1303--5657:5659 square=3481
2027:2029--5657:5659 square=3844
2789:2791--5657:5659 square=4225
4421:4423--5657:5659 square=5041
1697:1699--5741:5743 square=3721
4337:4339--5741:5743 square=5041
197:199--5849:5851 square=3025
419:421--5849:5851 square=3136
2339:2341--5849:5851 square=4096
4229:4231--5849:5851 square=5041
5099:5101--5849:5851 square=5476
179:181--5867:5869 square=3025
857:859--5867:5869 square=3364
1091:1093--5867:5869 square=3481
3929:3931--5867:5869 square=4900
4787:4789--5867:5869 square=5329
2309:2311--5879:5881 square=4096
3917:3919--5879:5881 square=4900
179:181--6089:6091 square=3136
137:139--6131:6133 square=3136
827:829--6131:6133 square=3481
5417:5419--6131:6133 square=5776
71:73--6197:6199 square=3136
1487:1489--6197:6199 square=3844
5657:5659--6197:6199 square=5929
3527:3529--6269:6271 square=4900
5279:5281--6269:6271 square=5776
659:661--6299:6301 square=3481
4649:4651--6299:6301 square=5476
599:601--6359:6361 square=3481
2087:2089--6359:6361 square=4225
1997:1999--6449:6451 square=4225
5099:5101--6449:6451 square=5776
3527:3529--6551:6553 square=5041
1619:1621--6569:6571 square=4096
1877:1879--6569:6571 square=4225
1787:1789--6659:6661 square=4225
269:271--6689:6691 square=3481
3389:3391--6689:6691 square=5041
4259:4261--6689:6691 square=5476
1487:1489--6701:6703 square=4096
197:199--6761:6763 square=3481
1427:1429--6761:6763 square=4096
4787:4789--6761:6763 square=5776
179:181--6779:6781 square=3481
659:661--6779:6781 square=3721
1667:1669--6779:6781 square=4225
3299:3301--6779:6781 square=5041
4157:4159--6791:6793 square=5476
857:859--6827:6829 square=3844
1619:1621--6827:6829 square=4225
2969:2971--6827:6829 square=4900
3251:3253--6827:6829 square=5041
4721:4723--6827:6829 square=5776
5651:5653--6827:6829 square=6241
569:571--6869:6871 square=3721
1319:1321--6869:6871 square=4096
11:13--6947:6949 square=3481
2027:2029--6947:6949 square=4489
4001:4003--6947:6949 square=5476
5849:5851--6947:6949 square=6400
6827:6829--6947:6949 square=6889
1229:1231--6959:6961 square=4096
1487:1489--6959:6961 square=4225
3119:3121--6959:6961 square=5041
5519:5521--6959:6961 square=6241
311:313--7127:7129 square=3721
1061:1063--7127:7129 square=4096
1319:1321--7127:7129 square=4225
3527:3529--7127:7129 square=5329
3821:3823--7127:7129 square=5476
4421:4423--7127:7129 square=5776
227:229--7211:7213 square=3721
4337:4339--7211:7213 square=5776
881:883--7307:7309 square=4096
1667:1669--7307:7309 square=4489
4241:4243--7307:7309 square=5776
4547:4549--7307:7309 square=5929
107:109--7331:7333 square=3721
857:859--7331:7333 square=4096
4217:4219--7331:7333 square=5776
2729:2731--7349:7351 square=5041
227:229--7457:7459 square=3844
1787:1789--7457:7459 square=4624
2339:2341--7457:7459 square=4900
4091:4093--7457:7459 square=5776
5021:5023--7457:7459 square=6241
7331:7333--7457:7459 square=7396
197:199--7487:7489 square=3844
1487:1489--7487:7489 square=4489
2309:2311--7487:7489 square=4900
2591:2593--7487:7489 square=5041
3167:3169--7487:7489 square=5329
3461:3463--7487:7489 square=5476
6959:6961--7487:7489 square=7225
137:139--7547:7549 square=3844
641:643--7547:7549 square=4096
1427:1429--7547:7549 square=4489
1697:1699--7547:7549 square=4624
4001:4003--7547:7549 square=5776
4931:4933--7547:7549 square=6241
2237:2239--7559:7561 square=4900
3389:3391--7559:7561 square=5476
599:601--7589:7591 square=4096
857:859--7589:7591 square=4225
3359:3361--7589:7591 square=5476
431:433--7757:7759 square=4096
1487:1489--7757:7759 square=4624
4721:4723--7757:7759 square=6241
6689:6691--7757:7759 square=7225
311:313--7877:7879 square=4096
569:571--7877:7879 square=4225
3671:3673--7877:7879 square=5776
6569:6571--7877:7879 square=7225
239:241--7949:7951 square=4096
2129:2131--7949:7951 square=5041
2999:3001--7949:7951 square=5476
179:181--8009:8011 square=4096
1787:1789--8009:8011 square=4900
3539:3541--8009:8011 square=5776
4787:4789--8009:8011 square=6400
6779:6781--8009:8011 square=7396
101:103--8087:8089 square=4096
3461:3463--8087:8089 square=5776
3767:3769--8087:8089 square=5929
6359:6361--8087:8089 square=7225
6701:6703--8087:8089 square=7396
227:229--8219:8221 square=4225
2729:2731--8219:8221 square=5476
3329:3331--8219:8221 square=5776
4259:4261--8219:8221 square=6241
6569:6571--8219:8221 square=7396
1787:1789--8291:8293 square=5041
2657:2659--8291:8293 square=5476
3257:3259--8291:8293 square=5776
7547:7549--8291:8293 square=7921
59:61--8387:8389 square=4225
857:859--8387:8389 square=4624
2267:2269--8387:8389 square=5329
3467:3469--8387:8389 square=5929
4091:4093--8387:8389 square=6241
17:19--8429:8431 square=4225
3119:3121--8429:8431 square=5776
4049:4051--8429:8431 square=6241
6359:6361--8429:8431 square=7396
4259:4261--8537:8539 square=6400
6947:6949--8537:8539 square=7744
8387:8389--8537:8539 square=8464
1481:1483--8597:8599 square=5041
3257:3259--8597:8599 square=5929
5849:5851--8597:8599 square=7225
347:349--8627:8629 square=4489
617:619--8627:8629 square=4624
1451:1453--8627:8629 square=5041
2027:2029--8627:8629 square=5329
3851:3853--8627:8629 square=6241
7211:7213--8627:8629 square=7921
2129:2131--8819:8821 square=5476
2729:2731--8819:8821 square=5776
137:139--8837:8839 square=4489
2111:2113--8837:8839 square=5476
2711:2713--8837:8839 square=5776
8087:8089--8837:8839 square=8464
2087:2089--8861:8863 square=5476
2687:2689--8861:8863 square=5776
5:7--8969:8971 square=4489
827:829--8969:8971 square=4900
5477:5479--8969:8971 square=7225
6869:6871--8969:8971 square=7921
7589:7591--8969:8971 square=8281
1949:1951--8999:9001 square=5476
2549:2551--8999:9001 square=5776
7559:7561--8999:9001 square=8281
3467:3469--9011:9013 square=6241
6827:6829--9011:9013 square=7921
7547:7549--9011:9013 square=8281
8627:8629--9041:9043 square=8836
5:7--9239:9241 square=4624
2309:2311--9239:9241 square=5776
3557:3559--9239:9241 square=6400
8429:8431--9239:9241 square=8836
1667:1669--9281:9283 square=5476
2267:2269--9281:9283 square=5776
8387:8389--9281:9283 square=8836
1607:1609--9341:9343 square=5476
659:661--9419:9421 square=5041
2129:2131--9419:9421 square=5776
8627:8629--9419:9421 square=9025
7127:7129--9431:9433 square=8281
641:643--9437:9439 square=5041
2111:2113--9437:9439 square=5776
3359:3361--9437:9439 square=6400
4337:4339--9437:9439 square=6889
5009:5011--9437:9439 square=7225
7487:7489--9437:9439 square=8464
8231:8233--9437:9439 square=8836
617:619--9461:9463 square=5041
1487:1489--9461:9463 square=5476
2087:2089--9461:9463 square=5776
1319:1321--9629:9631 square=5476
3167:3169--9629:9631 square=6400
1871:1873--9677:9679 square=5776
2801:2803--9677:9679 square=6241
3119:3121--9677:9679 square=6400
3767:3769--9677:9679 square=6724
1229:1231--9719:9721 square=5476
7949:7951--9719:9721 square=8836
29:31--9767:9769 square=4900
311:313--9767:9769 square=5041
2087:2089--9767:9769 square=5929
2711:2713--9767:9769 square=6241
5021:5023--9767:9769 square=7396
6791:6793--9767:9769 square=8281
9437:9439--9767:9769 square=9604
1091:1093--9857:9859 square=5476
1997:1999--9857:9859 square=5929
3917:3919--9857:9859 square=6889
4931:4933--9857:9859 square=7396
6701:6703--9857:9859 square=8281
149:151--9929:9931 square=5041
1019:1021--9929:9931 square=5476
1619:1621--9929:9931 square=5776
2549:2551--9929:9931 square=6241
4517:4519--9929:9931 square=7225
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February 19th, 2011, 10:27 PM   #5
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Re: Twin primes distributed around N mod 3 =0

Quote:
Originally Posted by Agno
I started by postulating that all pairs of twin primes are equidistant around mod 3 = 0.
What you're literally claiming is obviously false -- there are infinitely many primes above a number but only finitely many below. So what did you actually mean by this?

Quote:
Originally Posted by Agno
Didn't get much further with that, but with some good help we could potentially prove a weaker version of the postulate and that is: IF two pairs of twin primes (except the first pair 3,5) are equidistant around N (not necessarily a square), then N mod 3 MUST be 0.
Your terminology is nonstandard, but what you mean directly follows from the fact that twin primes are of the form 6n +/- 1. (Excluding 3,5 of course -- in fact I'll exclude it from now on.)

Quote:
Originally Posted by Agno
This poses the inverse question whether all N mod 3 = 0 also have two equidistant pairs of twin primes?
It's virtually certain that there are only finitely many exceptions. The twin primes are a fairly 'thick' sequence, being of (conjectural) density n/log^2 n; adding them to themselves should give almost all numbers not excluded by modular considerations.
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February 19th, 2011, 10:36 PM   #6
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Re: Twin primes distributed around N mod 3 =0

Quote:
Originally Posted by al-mahed
the simple algorithm and the list:

Code:
forstep(x=1,10000,2,if(isprime(x),if(isprime(x+2),forstep(y=1,x-2,2,if(isprime(y),if(isprime(y-2),if(ispower(y+(x-y)/2+1,2),print(y-2":"y"--"x":"x+2" square="y+(x-y)/2+1))))))))
Rather than looping through the odd numbers and testing if they're prime, loop over the primes directly:
Code:
forprime(x=3,100000,if(isprime(x+2),forprime(y=5,x-2,if(isprime(y-2)&&issquare(y+(x-y)/2+1),print(y-2":"y"--"x":"x+2" square="y+(x-y)/2+1)))))
This is much faster -- 6x faster in this case but it can get larger when the numbers are increased.

I also changed ispower(__, 2) to issquare(__) and if(__, if(__, __)) to if(__ && __, __) but those are mostly for readability; there's no particular performance difference.
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February 20th, 2011, 04:07 AM   #7
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Re: Twin primes distributed around N mod 3 =0

Quote:
Originally Posted by CRGreathouse
Quote:
Originally Posted by Agno
I started by postulating that all pairs of twin primes are equidistant around mod 3 = 0.
What you're literally claiming is obviously false -- there are infinitely many primes above a number but only finitely many below. So what did you actually mean by this?
What I meant was that all pairs of twin primes (except the first pair 3,5) are equidistant around , and that this always is a multiple of 3. So each twin pair always has an equidistant twin pair .

Examples:
* 5,7 en 11,13 (, distance 2)
* 29,31 en 41,43 (, distance 5)
* 59,61 en 101,103 (, distance 20)
* 17,19 en 269, 271 (, distance 125)

We tested a part of this conjecture ((i.e. that all squares mod 3 =0 always have at least 1 equidistant set of twin prime pairs around them, which does not imply that all twin prime pairs are 'touched') up to n = 1.000.000.000, and found no exceptions . So, indeed the density of twin primes seems such that they always 'circle' around a square mod 3 =0.

Few results:
Largest distance is (26646^2):
710009316: Twins (709837241,709837243) and (710181389,710181391) Distance: 172073 (ratio around 1/4126)

There are also still very small equidistances for very large squares:
640798596: Twins (640798589,640798591) and (640798601,640798603) Distance: 5
680114241: Twins (680114231,680114233) and (680114249,680114251) Distance: 8

Quote:
Originally Posted by CRGreathouse
Quote:
Originally Posted by Agno
Didn't get much further with that, but with some good help we could potentially prove a weaker version of the postulate and that is: IF two pairs of twin primes (except the first pair 3,5) are equidistant around N (not necessarily a square), then N mod 3 MUST be 0.
Your terminology is nonstandard, but what you mean directly follows from the fact that twin primes are of the form 6n +/- 1. (Excluding 3,5 of course -- in fact I'll exclude it from now on.)
Correct.

Quote:
Originally Posted by CRGreathouse
Quote:
Originally Posted by Agno
This poses the inverse question whether all N mod 3 = 0 also have two equidistant pairs of twin primes?
It's virtually certain that there are only finitely many exceptions. The twin primes are a fairly 'thick' sequence, being of (conjectural) density n/log^2 n; adding them to themselves should give almost all numbers not excluded by modular considerations.
If we could only prove that the density of twin primes is indeed such that no further exceptions can exist, since then the twin prime conjecture would also be proven...
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February 20th, 2011, 07:18 AM   #8
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Re: Twin primes distributed around N mod 3 =0

Charles! Thank you for the hint!

Agno, is trivial that n will be always divisible by 3 since in each pair of twins the smallest of them is of the form 3x+2 (if not, the bigger would be divisible by 3), as I posted, you don't need to check it. The ONLY exception will be the pair 3, 5.

That p is of the form 3x+2 is clear because, if not, if it were of the form 3x+1, then would be divisible by 3

so either p = 3 or p= 3x+2, since p=3 is not interesting here, we are daling with and

so call the distance between two pairs and ,

then , and clearly d is even, so for a given integer

the result you want to prove is that is divisible by 3 so you want

to prove it just observe that QED.
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February 20th, 2011, 07:45 AM   #9
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Re: Twin primes distributed around N mod 3 =0

Quote:
Originally Posted by al-mahed
Charles! Thank you for the hint!

Agno, is trivial that n will be always divisible by 3 since in each pair of twins the smallest of them is of the form 3x+2 (if not, the bigger would be divisible by 3), as I posted, you don't need to check it. The ONLY exception will be the pair 3, 5.

That p is of the form 3x+2 is clear because, if not, if it were of the form 3x+1, then would be divisible by 3

so either p = 3 or p= 3x+2, since p=3 is not interesting here, we are daling with and

so call the distance between two pairs and ,

then , and clearly d is even, so for a given integer

the result you want to prove is that is divisible by 3 so you want

to prove it just observe that QED.
al-mahed,

Got it. Many thanks for your clear explanation!
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