Problems & Puzzles: Puzzles

 Puzzle 409. Consecutive primes triplets Anton Vrba poses the following puzzle: Find triplets of consecutive primes r1, r2 & r3 such that for a given value of d, q1=(r1+r2)/d and  q2=(r2+r3)/d are a pair of consecutive primes and  p=(q1+q2)/d is prime too. Examples have been found for d = 4, 6, 8, 12, 18 and 24. Solution for smallest p for a given d {p, q1, q2, r1, r2, r3} and the gaps {q2-q1, r2-r1, r3-r2} d =4 {6553, 13103, 13109, 26203, 26209, 26227} {6, 6, 18} d =6 {6689, 20063, 20071, 60169, 60209, 60217} {8, 40, 8} d =8 {85229, 340913, 340919, 1363631, 1363673, 1363679} {6, 42, 6} d =12 {59357, 356141, 356143, 2136839, 2136853, 2136863} {2, 14, 10} d =18 {86201, 775807, 775811, 6982247, 6982279, 6982319} {4, 32, 40} d =24 {9851, 118211, 118213, 1418513, 1418551, 1418561} {2, 38, 10} Questions Q1.  Can you find solutions for d=10, 14, 16, 20, 22, >24 or can you prove that no solutions exist for these d values?

Contributions came from Adam Stinchcombe, Bernardo Boncompagni. Ken Wilke, Gabor Levai, Farideh Firoozbakht & Fred Schneider.

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Of course, d has to be the gcd of r1+r2 and r2+r3, else (r1+r2)/d and (r2+r3)/d wouldn't be prime.

For d=60 we have {p, q1, q2, r1, r2, r3} = {79697,  2390909,  2390911,  71727251,  71727289,   71727371} with gaps {q2-q1, r2-r1, r3-r2} = {2,  38,  82}

for d=30 I find {p, q1, q2, r1, r2, r3} = {1553009, 23295133, 23295137, 349426949, 349427041, 349427069} and {q2-q1, r2-r1, r3-r2} = {4, 92, 28}

for d=36 I find {p, q1, q2, r1, r2, r3} = {1127741, 20299337, 20299339, 365388059, 365388073, 365388131} and {q2-q1, r2-r1, r3-r2} = {2, 14, 58}
for d=10 I find {p, q1, q2, r1, r2, r3} = {19094993, 95474959, 95474971, 477374759, 477374831, 477374879} and {q2-q1, r2-r1, r3-r2} = {12, 72, 48}

for d=14 I find {p, q1, q2, r1, r2, r3} = {10869109, 76083757, 76083769, 532586221, 532586377, 532586389} and {q2-q1, r2-r1, r3-r2} = {12, 156, 12}

for d=16 I find {p, q1, q2, r1, r2, r3} = {9022537, 72180293, 72180299, 577442311, 577442377, 577442407} and {q2-q1, r2-r1, r3-r2} = {6, 66, 30}

***

Bernardo wrote:

Following Anton's notation, here's the smallest p's for given d. All
other d's (22 is the smallest I couldn't find) can only yield a solution
for triplets whose smallest prime is larger than 16145632199.

d=4 {6553, 13103, 13109, 26203, 26209, 26227} {6, 6, 18}
d=6 {6689, 20063, 20071, 60169, 60209, 60217} {8, 40, 8}
d=8 {85229, 340913, 340919, 1363631, 1363673, 1363679} {6, 42, 6}
d=10 {22502429, 112512139, 112512151, 562560653, 562560737, 562560773}
{12, 84,36}
d=12 {59357, 356141, 356143, 2136839, 2136853, 2136863} {2, 14, 10}
d=14 {10869109, 76083757, 76083769, 532586221, 532586377, 532586389}
{12, 156, 12}
d=16 {9022537, 72180293, 72180299, 577442311, 577442377, 577442407} {6,
66, 30}
d=18 {86201, 775807, 775811, 6982247, 6982279, 6982319} {4, 32, 40}
d=20 {12896669, 128966687, 128966693, 1289666831, 1289666909,
1289666951} {6, 78, 42}
d=24 {9851, 118211, 118213, 1418513, 1418551, 1418561} {2, 38, 10}
d=28 {36649699, 513095783, 513095789, 7183340911, 7183341013,
7183341079} {6, 102, 66}
d=30 {1553009, 23295133, 23295137, 349426949, 349427041, 349427069} {4,
92, 28}
d=36 {1127741, 20299337, 20299339, 365388059, 365388073, 365388131} {2,
14, 58}
d=42 {36611411, 768839629, 768839633, 16145632199, 16145632219,
16145632367} {4, 20, 148}
d=48 {2239453, 53746871, 53746873, 1289924887, 1289924921, 1289924983}
{2, 34, 62}
d=60 {79697, 2390909, 2390911, 71727251, 71727289, 71727371} {2, 38, 82}
d=72 {10933753, 393615107, 393615109, 14170143797, 14170143907,
14170143941} {2, 110, 34}

***

Ken wrote:

Attached are the solutions I have found for the unsolved cases in puzzle 409. I constructed a sieve in UBASIC, a copy of the code is attached. In the table below are the solutions I have found for the cases d = 10, 14, 16 and 20 with the smallest solution listed first. I am still testing values for d =22 and have found no solutions below r1 = 3122554013. My code is straight forward and uses the NXTPRM function in UBASIC. Also the product of primes p <500 is used in conjunction with the gcd to weed out solutions divisible by small primes. As a result, I only need to test values generated of p1, q1 and r1 for primality. The table lists solutions in the same format used by the proposer; i.e. d, p1, q1, q2, r1, r2, r3, q2 - q1, r2 - r1, r3 - r2.

10 22502429 112512139 112512151 562560653 562560737 562560773 12 84 36
10 23225827 116129129 116129141 580645633 580645657 580645753 12 24 96
10 27330421 136652099 136652111 683260483 683260507 683260603 12 24 96
10 27492401 137461999 137462011 687309941 687310049 687310061
12 108 12

14 10869109 76083757 76083769 532586221 532586377 532586389
12 156 12

16 9022537 72180293 72180299 577442311 577442377 577442407 6 66 30
16 9249193 73993541 73993547 591948307 591948349 591948403 6 42 54
16 11454823 91638581 91638587 733108627 733108669 733108723 6 42 54
16 12175127 97401013 97401019 779208071 779208137 779208167 6 66 30

20 12896669 128966687 128966693 1289666831 1289666909 1289666951 6 78 42
20 17357897 173578967 173578973 1735789667 1735789673 1735789787 6 6 114
20 19099097 190990967 190990973 1909909649 1909909691 1909909769 6 42 78
20 19113841 191138407 191138413 1911384049 1911384091 1911384169 6 42 78

***

Gabor Levai wrote:

found the following smallest solutions (if r3 < 2^32):

d= 4 {6553, 13103, 13109, 26203, 26209, 26227} {6, 6, 18}
d= 6 {6689, 20063, 20071, 60169, 60209, 60217} {8, 40, 8}
d= 8 {85229, 340913, 340919, 1363631, 1363673, 1363679} {6, 42, 6}
d=10 {22502429, 112512139, 112512151, 562560653, 562560737, 562560773} {12, 84, 36}
d=12 {59357, 356141, 356143, 2136839, 2136853, 2136863} {2, 14, 10}
d=14 {10869109, 76083757, 76083769, 532586221, 532586377, 532586389} {12, 156, 12}
d=16 {9022537, 72180293, 72180299, 577442311, 577442377, 577442407} {6, 66, 30}
d=18 {86201, 775807, 775811, 6982247, 6982279, 6982319} {4, 32, 40}
d=20 {12896669, 128966687, 128966693, 1289666831, 1289666909, 1289666951} {6, 78, 42}
d=24 {9851, 118211, 118213, 1418513, 1418551, 1418561} {2, 38, 10}
d=30 {1553009, 23295133, 23295137, 349426949, 349427041, 349427069} {4, 92, 28}
d=36 {1127741, 20299337, 20299339, 365388059, 365388073, 365388131} {2, 14, 58}
d=48 {2239453, 53746871, 53746873, 1289924887, 1289924921, 1289924983} {2, 34, 62}
d=60 {79697, 2390909, 2390911, 71727251, 71727289, 71727371} {2, 38, 82}

The number of solutions (if r3 < 2^32):

d= 4 : 1186, d= 6 : 1927, d= 8 : 225, d=10 : 28, d=12 : 319, d=14 : 2, d=16 : 8, d=18 : 52
d=20 : 9, d=24 : 58, d=30 : 11, d=36 : 9, d=48 : 4, d=60 : 2

My observation: if d = k*12, then q1 and q2 are twins, but there are couterexamples:

d=12 {6842081, 41052481, 41052491, 246314863, 246314909, 246314983} {10, 46, 74}
d=12 {38448379, 230690269, 230690279, 1384141589, 1384141639, 1384141709} {10, 50, 70}
d=12 {39390233, 236341393, 236341403, 1418048339, 1418048377, 1418048459} {10, 38, 82}
d=12 {46262597, 277575577, 277575587, 1665453451, 1665453473, 1665453571} {10, 22, 98}
d=12 {52531153, 315186913, 315186923, 1891121473, 1891121483, 1891121593} {10, 10, 110}
d=12 {70955527, 425733157, 425733167, 2554398941, 2554398943, 2554399061} {10, 2, 118}
d=12 {79510703, 477064213, 477064223, 2862385249, 2862385307, 2862385369} {10, 58, 62}
d=12 {83240441, 499442641, 499442651, 2996655833, 2996655859, 2996655953} {10, 26, 94}
d=12 {86662187, 519973117, 519973127, 3119838653, 3119838751, 3119838773} {10, 98, 22}
d=12 {95241791, 571450741, 571450751, 3428704429, 3428704463, 3428704549} {10, 34, 86}

In (r1, r2, r3, q1, q2) are two twin primes:

d=24 {338609, 4063307, 4063309, 48759661, 48759707, 48759709} {2, 46, 2}
d=24 {693961, 8327531, 8327533, 99930371, 99930373, 99930419} {2, 2, 46}
d=36 {1461851, 26313317, 26313319, 473639671, 473639741, 473639743} {2, 70, 2}

***

Farideh wrote:

Smallest solution for d=30 is
{1553009,23295133,23295137,349426949,349427041,349427069} - {4,92,28}

and smallest solution for d=36 is
{1127741,20299337,20299339,365388059,365388073,365388131} - {2,14,58}.

***

Fred wrote:

Minimal solutions are:

d=10 {22502429, 112512139, 112512151, 562560653, 562560737, 562560773}
{12, 84, 36}
d=14 {10869109, 76083757, 76083769, 532586221, 532586377, 532586389}
{12, 156, 12}
d=16 {9022537, 72180293, 72180299, 577442311, 577442377, 577442407} {6, 66, 30}
d=20 {12896669, 128966687, 128966693, 1289666831, 1289666909,
1289666951} {6, 78, 42}
d=22 {628783537, 6916618901, 6916618913, 76082807809, 76082808013,
76082808073} {12, 204, 60}
d=28 {36649699, 513095783, 513095789, 7183340911, 7183341013,
7183341079} {6, 102, 66}
d=30 {1553009, 23295133, 23295137, 349426949, 349427041, 349427069} {4, 92, 28}
d=32 {190621351, 3049941613, 3049941619, 48799065787, 48799065829,
48799065979} {6, 42, 150}
d=36 {1127741, 20299337, 20299339, 365388059, 365388073, 365388131} {2, 14, 58}
d=40 {268423007, 5368460137, 5368460143, 107369202653, 107369202827,
107369202893} {6, 174, 66}
d=42 {36611411, 768839629, 768839633, 16145632199, 16145632219,
16145632367} {4, 20, 148}
d=44 {85593503, 1883057063, 1883057069, 41427255269,
41427255503,41427255533} {6, 234, 30}
d=48 {2239453, 53746871, 53746873, 1289924887, 1289924921, 1289924983} {2,34,62}
d=52 {2455450339, 63841708811, 63841708817, 1659884428981,
1659884429191, 1659884429293} {6, 210, 102}
d=54 {747226147, 20175105967, 20175105971, 544727861071, 544727861147,
544727861287} {4, 76, 140}
d=60 {79697, 2390909, 2390911, 71727251, 71727289, 71727371} {2, 38, 82}
d=66 {1727909423, 57021010957, 57021010961, 1881693361553,
1881693361609, 1881693361817} {4, 56, 208}
d=72 {10933753, 393615107, 393615109, 14170143797, 14170143907,
14170143941} {2, 110, 34}
d=78 {3748044589, 146173738969, 146173738973, 5700775819789,
5700775819793, 5700775820101} {4, 4, 308}
d=84 {45858721, 1926066281, 1926066283, 80894783731,
80894783873,80894783899} {2, 142, 26}
d=96 {67649011, 3247152527, 3247152529, 155863321259, 155863321333,
155863321451} {2, 74, 118}
d=108 {88564247, 4782469337, 4782469339, 258253344143, 258253344253,
258253344359} {2, 110, 106}

No solutions found for:
d=26,34,38,46,50,52,56,58,62,64,68,70,74,76,80,82,86,88,90,92,94,98,100,102,104,106

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