Problems & Puzzles: Puzzles

Puzzle 962. Claudio Meller 1519 entry. Claudio Meller in his entry 1519 ask for a larger solution than this one      59 x 2 + 59 + 2 = 179 Primo  59 x 3 + 59 + 3 =  239 Primo 59 x 5 + 59 + 5 =  359 Primo 59 x 7 + 59 + 7 =  479 Primo 59 x 11 + 59 + 11 = 719 Primo 59 x 13 + 59 + 13 = 839 Primo The fixed prime is 59 and the variable primes are 2, 3, ... 13 A larger solutions means here more than 6 rows. The variable primes need to be consecutive but not necessarily starting in 2. I found this one larger solution (9 rows)      3378469499 = 10709 * 315449 + 10709 + 315449 3378490919 = 10709 * 315451 + 10709 + 315451 3378598019 = 10709 * 315461 + 10709 + 315461 3378662279 = 10709 * 315467 + 10709 + 315467 3378812219 = 10709 * 315481 + 10709 + 315481 3378940739 = 10709 * 315493 + 10709 + 315493 3379197779 = 10709 * 315517 + 10709 + 315517 3379240619 = 10709 * 315521 + 10709 + 315521 3379304879 = 10709 * 315527 + 10709 + 315527 Q. Send your largest solution (more than 9 rows).

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Contributions came from Emmanuel Vantieghem, Shyam Sunder Gupta, Giovanni Resta, Simon Cavegn, Oscar Volpatti

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Emmanuel wrote on July 20, 2019

Here are ten lines :

 32271611*89 + 89 + 32271611 = 2904445079, prime
 32271619*89 + 89 + 32271619 = 2904445799, prime
 32271637*89 + 89 + 32271637 = 2904447419, prime
 32271641*89 + 89 + 32271641 = 2904447779, prime
 32271649*89 + 89 + 32271649 = 2904448499, prime
 32271677*89 + 89 + 32271677 = 2904451019, prime
 32271683*89 + 89 + 32271683 = 2904451559, prime
 32271703*89 + 89 + 32271703 = 2904453359, prime
 32271749*89 + 89 + 32271749 = 2904457499, prime
 32271779*89 + 89 + 32271779 = 2904460199, prime.

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Shyam wrote on July 25, 2019

A larger solutions means here more than 6 rows. The variable primes need to be consecutive but not necessarily starting in 2.

Q. Send your largest solution (more than 9 rows).

I found two solutions with 10 rows as given below:
A.
556148123 = 2 * 185382707 + 2 + 185382707
556148129 = 2 * 185382709 + 2 + 185382709
556148249 = 2 * 185382749 + 2 + 185382749
556148291 = 2 * 185382763 + 2 + 185382763
556148363 = 2 * 185382787 + 2 + 185382787
556148381 = 2 * 185382793 + 2 + 185382793
556148393 = 2 * 185382797 + 2 + 185382797
556148429 = 2 * 185382809 + 2 + 185382809
556148471 = 2 * 185382823 + 2 + 185382823
556148501 = 2 * 185382833 + 2 + 185382833
B.
2904445079 = 89 * 32271611 + 89 + 32271611
2904445799 = 89 * 32271619 + 89 + 32271619
2904447419 = 89 * 32271637 + 89 + 32271637
2904447779 = 89 * 32271641 + 89 + 32271641
2904448499 = 89 * 32271649 + 89 + 32271649
2904451019 = 89 * 32271677 + 89 + 32271677
2904451559 = 89 * 32271683 + 89 + 32271683
2904453359 = 89 * 32271703 + 89 + 32271703
2904457499 = 89 * 32271749 + 89 + 32271749
2904460199 = 89 * 32271779 + 89 + 32271779

I also found many solutions with 8 rows for variable primes starting from 2. The one solution is given below:

8387879 * 2 + 8387879 + 2 = 25163639
8387879 * 3 + 8387879 + 3 = 33551519
8387879 * 5 + 8387879 + 5 = 50327279
8387879 * 7 + 8387879 + 7 = 67103039
8387879 * 11 + 8387879 + 11 = 100654559
8387879 * 13 + 8387879 + 13 = 117430319
8387879 * 17 + 8387879 + 17 = 150981839
8387879 * 19 + 8387879 + 19 = 167757599

Fixed primes for some other solutions for 8 rows are
102392009, 402532679, 1646643179, 1826863769, 2013502259 and 2203060859.

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Giovanni wrote on July 26, 2019:

The best I can do is 12 rows:

  955889122515359759 = 2828074326968519 * 337 + 2828074326968519 + 337
  984169865785044959 = 2828074326968519 * 347 + 2828074326968519 + 347
  989826014438981999 = 2828074326968519 * 349 + 2828074326968519 + 349
1001138311746856079 = 2828074326968519 * 353 + 2828074326968519 + 353
1018106757708667199 = 2828074326968519 * 359 + 2828074326968519 + 359
1040731352324415359 = 2828074326968519 * 367 + 2828074326968519 + 367
1057699798286226479 = 2828074326968519 * 373 + 2828074326968519 + 373
1074668244248037599 = 2828074326968519 * 379 + 2828074326968519 + 379
1085980541555911679 = 2828074326968519 * 383 + 2828074326968519 + 383
1102948987517722799 = 2828074326968519 * 389 + 2828074326968519 + 389
1125573582133470959 = 2828074326968519 * 397 + 2828074326968519 + 397
1136885879441345039 = 2828074326968519 * 401 + 2828074326968519 + 401

I found another one of length 12 with fixed prime 961335281088989 and
variable primes {1231, 1237,..., 1307}.

Similarly I found one of length 11 with fixed prime 2337618622709 and
variable primes {197, 199,... , 257}

starting from 2 the best I can do is 10 rows:

  735939076949 = 245313025649 *  2 + 245313025649 +  2
  981252102599 = 245313025649 *  3 + 245313025649 +  3
1471878153899 = 245313025649 *  5 + 245313025649 +  5
1962504205199 = 245313025649 *  7 + 245313025649 +  7
2943756307799 = 245313025649 * 11 + 245313025649 + 11
3434382359099 = 245313025649 * 13 + 245313025649 + 13
4415634461699 = 245313025649 * 17 + 245313025649 + 17
4906260512999 = 245313025649 * 19 + 245313025649 + 19
5887512615599 = 245313025649 * 23 + 245313025649 + 23
7359390769499 = 245313025649 * 29 + 245313025649 + 29

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Simon wrote on July 26, 2019:

Searched fixed primes 2..13 up to consecutive primes 2E11.
Searched fixed primes 17..61 up to consecutive primes 7E10.
Searched fixed primes 67..109819 up to consecutive primes 4294967291.

Using C# and GMP4.2 on a hex core CPU (4930K, 22nm) at 3.6Ghz, 93 Watt, core temp 62C, room temp 31C.

Found about 100 solutions of 10 rows, 9 with 11 rows, and 2 with 12 rows.

29*13220209403+29+13220209403=396606282119
29*13220209421+29+13220209421=396606282659
29*13220209441+29+13220209441=396606283259
29*13220209447+29+13220209447=396606283439
29*13220209451+29+13220209451=396606283559
29*13220209549+29+13220209549=396606286499
29*13220209571+29+13220209571=396606287159
29*13220209619+29+13220209619=396606288599
29*13220209711+29+13220209711=396606291359
29*13220209741+29+13220209741=396606292259
29*13220209757+29+13220209757=396606292739
29*13220209759+29+13220209759=396606292799

 

25409*1533383321+25409+1533383321=38963270212019
25409*1533383359+25409+1533383359=38963271177599
25409*1533383377+25409+1533383377=38963271634979
25409*1533383387+25409+1533383387=38963271889079
25409*1533383399+25409+1533383399=38963272193999
25409*1533383419+25409+1533383419=38963272702199
25409*1533383431+25409+1533383431=38963273007119
25409*1533383437+25409+1533383437=38963273159579
25409*1533383447+25409+1533383447=38963273413679
25409*1533383471+25409+1533383471=38963274023519
25409*1533383513+25409+1533383513=38963275090739
25409*1533383549+25409+1533383549=38963276005499

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Oscar wrote on July 26, 2019

I found some solutions with 10 rows and with 11 rows
 

 2904445079 = 89 * 32271611 + 89 + 32271611
 2904445799 = 89 * 32271619 + 89 + 32271619
 2904447419 = 89 * 32271637 + 89 + 32271637
 2904447779 = 89 * 32271641 + 89 + 32271641
 2904448499 = 89 * 32271649 + 89 + 32271649
 2904451019 = 89 * 32271677 + 89 + 32271677
 2904451559 = 89 * 32271683 + 89 + 32271683
 2904453359 = 89 * 32271703 + 89 + 32271703
 2904457499 = 89 * 32271749 + 89 + 32271749
 2904460199 = 89 * 32271779 + 89 + 32271779
 

 0735939076949 = 245313025649 * 02 + 245313025649 + 02
 0981252102599 = 245313025649 * 03 + 245313025649 + 03
 1471878153899 = 245313025649 * 05 + 245313025649 + 05
 1962504205199 = 245313025649 * 07 + 245313025649 + 07
 2943756307799 = 245313025649 * 11 + 245313025649 + 11
 3434382359099 = 245313025649 * 13 + 245313025649 + 13
 4415634461699 = 245313025649 * 17 + 245313025649 + 17
 4906260512999 = 245313025649 * 19 + 245313025649 + 19
 5887512615599 = 245313025649 * 23 + 245313025649 + 23
 7359390769499 = 245313025649 * 29 + 245313025649 + 29
 

 8085756986999 = 13859 * 583387949 + 13859 + 583387949
 8085757679999 = 13859 * 583387999 + 13859 + 583387999
 8085757846319 = 13859 * 583388011 + 13859 + 583388011
 8085757984919 = 13859 * 583388021 + 13859 + 583388021
 8085758428439 = 13859 * 583388053 + 13859 + 583388053
 8085758899679 = 13859 * 583388087 + 13859 + 583388087
 8085759176879 = 13859 * 583388107 + 13859 + 583388107
 8085759509519 = 13859 * 583388131 + 13859 + 583388131
 8085759592679 = 13859 * 583388137 + 13859 + 583388137
 8085759814439 = 13859 * 583388153 + 13859 + 583388153
 8085759925319 = 13859 * 583388161 + 13859 + 583388161
 

 009752069256929 = 3250689752309 * 02 + 3250689752309 + 02
 013002759009239 = 3250689752309 * 03 + 3250689752309 + 03 
 019504138513859 = 3250689752309 * 05 + 3250689752309 + 05
 026005518018479 = 3250689752309 * 07 + 3250689752309 + 07
 039008277027719 = 3250689752309 * 11 + 3250689752309 + 11
 045509656532339 = 3250689752309 * 13 + 3250689752309 + 13
 058512415541579 = 3250689752309 * 17 + 3250689752309 + 17
 065013795046199 = 3250689752309 * 19 + 3250689752309 + 19
 078016554055439 = 3250689752309 * 23 + 3250689752309 + 23
 097520692569299 = 3250689752309 * 29 + 3250689752309 + 29
 104022072073919 = 3250689752309 * 31 + 3250689752309 + 31

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