Problems & Puzzles: Puzzles

Puzzle 1205 Similar set of primes

On Dec 22, 2024 Dmitry Kamenetsky wrote:

I have another interesting puzzle.

12323 and 36473 is a fascinating pair of primes. This is because produce a sequence of 14 terms, primes that follow them have exactly the same distance from them:

12323 12329 12343 12347 12373 12377 12379 12391 12401 12409 12413 12421 12433 12437

36473 36479 36493 36497 36523 36527 36529 36541 36551 36559 36563 36571 36583 36587

Q1. Can you find a pair of primes with a longer (>14) sequence of the same distances?

A few days latter he added:

I found 3 sets with 5 similar primes each giving a sequence of 9 terms:

(31, 51448351, 227344561, 476894491, 712136911)
31 37 41 43 47 53 59 61 67
51448351 51448357 51448361 51448363 51448367 51448373 51448379 51448381 51448387
227344561 227344567 227344571 227344573 227344577 227344583 227344589 227344591 227344597
476894491 476894497 476894501 476894503 476894507 476894513 476894519 476894521 476894527
712136911 712136917 712136921 712136923 712136927 712136933 712136939 712136941 712136947

(8219, 13378499, 26609897, 51770057, 653180039)
8219 8221 8231 8233 8237 8243 8263 8269 8273
13378499 13378501 13378511 13378513 13378517 13378523 13378543 13378549 13378553
28853399 28853401 28853411 28853413 28853417 28853423 28853443 28853449 28853453
82624109 82624111 82624121 82624123 82624127 82624133 82624153 82624159 82624163
653180039 653180041 653180051 653180053 653180057 653180063 653180083 653180089 653180093

(12197, 18254627, 26609897, 51770057, 294837077)
12197 12203 12211 12227 12239 12241 12251 12253 12263
18254627 18254633 18254641 18254657 18254669 18254671 18254681 18254683 18254693
26609897 26609903 26609911 26609927 26609939 26609941 26609951 26609953 26609963
51770057 51770063 51770071 51770087 51770099 51770101 51770111 51770113 51770123
294837077 294837083 294837091 294837107 294837119 294837121 294837131 294837133 294837143

Q2. Can you find another quintet of primes with a longer (>9) sequence of the same distances?

And one more result:

Finally a set of 4 primes that give a sequence of 10 terms!

(83, 1954343, 541937633, 273649613)
 

83 89 97 101 103 107 109 113 127 131
 

1954343 1954349 1954357 1954361 1954363 1954367 1954369 1954373 1954387 1954391


541937633 541937639 541937647 541937651 541937653 541937657 541937659 541937663 541937677 541937681
 

273649613 273649619 273649627 273649631 273649633 273649637 273649639 273649643 273649657 273649661

Q3. Can you find another quartet of primes with a longer (>10) sequence of the same distances?


From 18-24, Jan, 2025, contributions came from Vicente F. Izquierdo, Gennady Gusev, J. K. Andersen, Giorgos Kalogeropoulos, Oscar Volpatti

***

Vicente wrote:

Some results:
q1:
{13923331,184224511}
 
q2:
10 terms.
{6983,99930833,677992823,1223731793,1424144243}
   Sextet 10 terms.
   {2371609,32251339,380407939,496769029,542152249,1282612759}
 
q3:
11 terms.
{16825021,103534861,253290601,1056627001}

***

Gennady wrote:

An example from the puzzle condition that 12323 and 36473 produce a sequence of 14 terms, we can write as
12323 + d, 36473 + d are all Prime, where d= 0, 6, 20, 24, 50, 54, 56, 68, 78, 86, 90, 98, 110, 114.
 
Definition: A prime k-tuplet is a sequence of k consecutive prime numbers such that in some sense
 the difference between the first and the last is as small as possible.
 
This is a stricter requirement than in the puzzle and satisfies the puzzle condition.
 
The search results for prime k-tuplets are published on the website https://pzktupel.de.
Here is an example of 15 tuplets with predefined patterns from the page https://pzktupel.de/smarchive.php
 with link to the website https://oeis.org/A257304:
 
15-tuplets    (12 sets):
Numbers n such that n, n+2, n+6, n+8, n+12, n+18, n+20, n+26, n+30, n+32, n+36, n+42, n+48, n+50 and n+56 are all prime.
11, 44360646117391789301, 80846604473064395081, 85542881495337691541, 113615698477681825541, 116591588863353569081,
 140245881111654813611, 204185491710368653601, 227209370616659726411, 238931301405879137171, 441927344360107210721,
 457968514558418508761
 
And the largest (https://pzktupel.de/SMArchiv/smadditions.php) PRIME 21-TUPLETS  (4 sets)
d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84
29
39433867730216371575457664399 ( 29 digits, 8 Jan 2015, Raanan Chermoni & Jaroslaw Wroblewski )
138433730977092118055599751669 ( 30 digits, 8 Oct 2015, Raanan Chermoni & Jaroslaw Wroblewski )
248283957683772055928836513589 ( 30 digits, 1 Aug 2016, Raanan Chermoni & Jaroslaw Wroblewski.

***

J. K. Andersen wrote

Q1. Yahoo Groups has closed but in 2006 I posted a non-minimal case with 21 primes in the primenumbers group.
446863043340173267 + d and 534402442999154537 + d both give consecutive primes for d = 0, 2, 56, 62, 80, 110, 146, 150, 152, 170, 230, 234, 252, 264, 276, 290, 294, 296, 300, 302, 344.
This was tied in 2015 (see Q3) but may still be the record length.

Q2. 29 + d is a prime 20-tuplet for d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80.
https://pzktupel.de/KTHIST/kt020.php shows 30 other prime 20-tuplets with this pattern found by Raanan Chermoni & Jaroslaw Wroblewski.

Q3. 29 + d is a prime 21-tuplet for d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84.
https://pzktupel.de/Top10.php#largest21 shows three more 21-tuplets with this pattern starting at 39433867730216371575457664399, 138433730977092118055599751669 and 248283957683772055928836513589. Found by Raanan Chermoni & Jaroslaw Wroblewski in 2015 and 2016.

One hour later he added:

Q1. I tested the 20- and 21-tuplets by Chermoni & Wroblewski for extensions.

It gave one case with 22 primes by matching the two preceding primes n-42 and n-270 for two 20-tuplets starting at n = 466435879660522367654413675211 and n = 562324418721793120042174985351.

This means 466435879660522367654413674941 + d and 562324418721793120042174985081 + d is prime for d = 0, 228, 270, 272, 276, 278, 282, 290, 296, 300, 306, 308, 312, 318, 320, 326, 332, 336, 338, 342, 348, 350.

 

***

Giorgos wrote:

Q2. A set of 5 primes that give a sequence of 10 terms

 
(83, 1954343, 273649613, 541937633, 9447257903)

 
{83, 89, 97, 101, 103, 107, 109, 113, 127, 131},
{1954343, 1954349, 1954357, 1954361, 1954363, 1954367, 1954369, 1954373, 1954387, 1954391}, 
{273649613, 273649619, 273649627, 273649631, 273649633, 273649637, 273649639, 273649643, 273649657, 273649661}, 
{541937633, 541937639, 541937647, 541937651, 541937653, 541937657, 541937659, 541937663, 541937677, 541937681}, 
{9447257903, 9447257909, 9447257917, 9447257921, 9447257923, 9447257927, 9447257929, 9447257933, 9447257947, 9447257951}

 
Q3.  A set of 4 primes that give a sequence of 11 terms

 
(1954343, 20310930443, 141731496983, 180487617113)

 
{1954343, 1954349, 1954357, 1954361, 1954363, 1954367, 1954369, 1954373, 1954387, 1954391, 1954411}, 
{20310930443, 20310930449, 20310930457, 20310930461, 20310930463, 20310930467, 20310930469, 20310930473, 20310930487, 20310930491, 20310930511}, 
{141731496983, 141731496989, 141731496997, 141731497001, 141731497003, 141731497007, 141731497009, 141731497013, 141731497027, 141731497031, 141731497051}, 
{180487617113, 180487617119, 180487617127, 180487617131, 180487617133, 180487617137, 180487617139, 180487617143, 180487617157, 180487617161, 180487617181}

***

Oscar Volpatti wrote:

I chose an "exaustive search" approach.
Select target length L.
Given a sequence of L consecutive primes, define the distance pattern: 
[d_1,...,d_L]
where d_i is the distance between the first prime and the i-th prime of the sequence (so that d_1 is always 0).
Now choose some threshold pmax and let the starting prime vary between 3 and pmax;
"sort" the resulting distance patterns and detect repeated vectors;
store the largest set of starting primes with shared distance pattern;
in case of equal size, store the set (p1,..p_S) with smaller p_S.    

 
Solutions with length at least 9, found by choosing threshold pmax = 1.1*10^9.

 
Length 13 to 15, size 2.
(13923331,184224511)
[0,16,22,30,42,58,66,70,76,100,108,126,136,142,148]

 
Length 12, size 3.
(148942993,612514003,915909823)
[0,6,28,30,34,58,66,70,90,96,118,136]

 
Length 11, size 4.
(16825021,103534861,253290601,1056627001)
[0,6,16,22,28,36,42,48,58,70,78]

 
Length 10, size 5.
(25575353,27117803,191116253,221713403,241894253)
[0,6,14,20,30,38,44,60,74,84]

 
Length 9, size 12.
(10709,901427,1275539,22779767,25004129,78755249,89912087,94460477,115721909,217664789,263130479,601856789)
[0,2,14,20,24,30,44,62,72]

 
An obvious way to detect more and more solutions is simply to increase the threshold.  
One pair with length 16 found by choosing pmax = 2.2*10^9.
 
(1214369413,2159263363)
[0,6,10,24,46,78,88,96,118,124,126,144,186,196,214,228]

 
Another approach is to store distance patterns already repeating before current threshold, and to search for more repetitions among larger starting primes. 

 
Length 13, size 3
(67485413,181682573,140729531483)
[0,24,36,50,56,66,74,90,128,134,140,146,156]

 
Length 12, size 4.
(472387129,549566449,16818292069,20384208709)
[0,18,22,54,60,90,102,112,120,130,148,154]

 
Length 11,size 6.
(522190441,675130711,1583127031,5598730861,5854210981,13091225101)
[0,6,10,16,28,36,40,58,66,76,78]

***

 

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