Problems & Puzzles: Puzzles

Puzzle 977. A special set of primes.

Lii Yosei published the following curio:

113 is the first member of the smallest three primes q, r, s > 100 such that 4*q, 3*r, 2*s are three consecutive natural numbers. These three primes, q=113, r=151, s=227, represent a quarter pound, one-third pound, and a half pound, respectively, in terms of grams, rounded to the nearest integer (a pound = 453.59237 grams).

What I saw in this curio was a set of three primes P={113, 151, 227} such that multiplied by a set of three consecutive integers n={4, 3, 2} produce a set of three consecutive integers N={452, 453, 454}

Q1. Find your largest trio of primes {q, r, s} alike the given above by Yosei.

Another issue suggested to my eyes by the Yosei's curio was to find a set of K of primes alike these three from Yosei, not any set of primes but the minimal set for each K>1

Here are my results for K=2, 3,..., 8

K=2
(p, n, N)
3 3 9
5 2 10
11 1 11
(Thanks Yosei!)*

K=3
(p, n, N)
5 4 20
7 3 21
11 2 22
23 1 23
(Thanks Yosei!)*

K=4
(p, n, N)
839  5 4195
1049 4 4196
1399 3 4197
2099 2 4198

K=5
(p, n, N)
1559 6 9354
1871 5 9355
2339 4 9356
3119 3 9357

K=6
(p, n, N)
1099799 7 7698593
1283099 6 7698594
1539719 5 7698595
1924649 4 7698596
2566199 3 7698597
3849299 2 7698598

K=7
(p, n, N)
1347569 8 10780552
1540079 7 10780553
1796759 6 10780554
2156111 5 10780555
2695139 4 10780556
3593519 3 10780557
5390279 2 10780558
10780559 1 10780559 (Thanks Yosei!)*

K=8
(p, n, N)
3330407359 9 29973666231
3746708279 8 29973666232
4281952319 7 29973666233
4995611039 6 29973666234
5994733247 5 29973666235
7493416559 4 29973666236
9991222079 3 29973666237
14986833119 2 29973666238

* rows in red added after an observation from Yosei.

 Q2. Find the minimal set of primes for K=9 to 15.

 

On Week Nov3-9, contributions came from J. K. Andersen. Seiji Tomita, Simon Cavegn,Gennady Gusev, Oscar Volpatti

***

Jens wrote:

If we add the natural condition in the red rows about also multiplying
by 1 then the minimal set is the primes given by Puzzle 206: (N-k)/k
primes. It's also https://oeis.org/A078502. You submitted my largest
solution with 14 primes to
https://primes.utm.edu/curios/page.php/7272877497848202240.html.

***

Seiji wrote:

About Puzzle 977.

Q1:
(p, n, N)
[300000000000000000000000000000000000000000000000000000000000000\
 00000000000000000000000000000001989779, 4, 1200000000000000\
 0000000000000000000000000000000000000000000000000000000000\
 0000000000000000000007959116]

[400000000000000000000000000000000000000000000000000000000000000\
 00000000000000000000000000000002653039, 3, 1200000000000000\
 0000000000000000000000000000000000000000000000000000000000\
 0000000000000000000007959117]

[600000000000000000000000000000000000000000000000000000000000000\
 00000000000000000000000000000003979559, 2, 1200000000000000\
 0000000000000000000000000000000000000000000000000000000000\
 0000000000000000000007959118]

Q2:

K=9
(p, n, N)
[13042093679,10, 130420936790]
[14491215199, 9, 130420936791]
[16302617099, 8, 130420936792]
[18631562399, 7, 130420936793]
[21736822799, 6, 130420936794]
[26084187359, 5, 130420936795]
[32605234199, 4, 130420936796]
[43473645599, 3, 130420936797]
[65210468399, 2, 130420936798]

***

Simon wrote:

Smaller K=3:
(p, n, N)
2, 4, 8
3, 3, 9
5, 2, 10
 
Q1:
Larger K=3:
(p, n, N)
86843633340121665350961995863162294284065313801620645150054015288670912479611610142
87795530787137035491030030175540895717054537879218321893290791590917159387238248577
55506609572692542885034568358935251185923152596213397466890182595820677527159009419
55102440773029952580798657013107009089759326659284201496546328504613517979230205054
95308804401109049066995697896717797614451007411725018261036854139380110837099259036
72860172221909518838815030652257429548119360707121973029391135640296891120488853397
348726078134988383932215660839534073694467979736997653, 4, 34737453336048666140384798
34526491771362612552064825806002160611546836499184464405715118212314854814196412012
07021635828682181515168732875731631663636686375489529943102202643829077017154013827
34357410047436926103848535898675607303832827101086360376782040976309211981032319462
80524280363590373066371368059861853140184540719169208202198123521760443619626798279
15868711904578040296469000730441474165575204433483970361469144068888763807535526012
26090297181924774428284878921175645425611875644819554135893949043125399535357288626
43358136294777871918947990612
11579151112016222046794932781754972571208708506882752686673868705156121663948214685
71706070771618271398804004023405452762273938383895776252438772212122287918298433143
67400881276359005718004609114524700158123087012828452995585357679442757003621201255
940136587697373270107731542684142678786345768879045601995395104672818023972306940073
270784058681453987559942638622903968192680098823000243480491388525068144494656787156
381356296254602511842004086967657273082581427616263070585484752039585482731847119646
4968104179984511909620881119378764925957306315996871, 3, 34737453336048666140384798345
264917713626125520648258060021606115468364991844644057151182123148548141964120120702
163582868218151516873287573163166363668637548952994310220264382907701715401382734357
410047436926103848535898675607303832827101086360376782040976309211981032319462805242
803635903730663713680598618531401845407191692082021981235217604436196267982791586871
190457804029646900073044147416557520443348397036146914406888876380753552601226090297
181924774428284878921175645425611875644819554135893949043125399535357288626433581362
94777871918947990613
17368726668024333070192399172632458856813062760324129030010803057734182495922322028
57559106157427407098206006035108179143410907575843664378658158318183431877447649715
51101321914538508577006913671787050237184630519242679493378036519164135505431801883
91020488154605990516159731402621401817951865331856840299309265700922703595846041010
99061760880221809813399139579343559522890201482345003652207370827876022167419851807
34572034444381903767763006130451485909623872141424394605878227128059378224097770679
4697452156269976767864431321679068147388935959473995307, 2, 3473745333604866614038479
8345264917713626125520648258060021606115468364991844644057
15118212314854814196412012070216358286821815151687328757316316636366863754895299431
02202643829077017154013827343574100474369261038485358986756073038328271010863603767
82040976309211981032319462805242803635903730663713680598618531401845407191692082021
98123521760443619626798279158687119045780402964690007304414741655752044334839703614
69144068888763807535526012260902971819247744282848789211756454256118756448195541358
9394904312539953535728862643358136294777871918947990614
 
Q2:
K=9
(p, n, N)
13042093679, 10, 130420936790
14491215199, 9, 130420936791
16302617099, 8, 130420936792
18631562399, 7, 130420936793
21736822799, 6, 130420936794
26084187359, 5, 130420936795
32605234199, 4, 130420936796
43473645599, 3, 130420936797
65210468399, 2, 130420936798
 
K=10
(p, n, N)
1438962008399, 11, 15828582092389
1582858209239, 10, 15828582092390
1758731343599, 9, 15828582092391
1978572761549, 8, 15828582092392
2261226013199, 7, 15828582092393
2638097015399, 6, 15828582092394
3165716418479, 5, 15828582092395
3957145523099, 4, 15828582092396
5276194030799, 3, 15828582092397
7914291046199, 2, 15828582092398
 
K=11
(p, n, N)
16841618053259, 12, 202099416639108
18372674239919, 11, 202099416639109
20209941663911, 10, 202099416639110
22455490737679, 9, 202099416639111
25262427079889, 8, 202099416639112
28871345234159, 7, 202099416639113
33683236106519, 6, 202099416639114
40419883327823, 5, 202099416639115
50524854159779, 4, 202099416639116
67366472213039, 3, 202099416639117
101049708319559, 2, 202099416639118
 
Found nothing for K=12: N up to 45494385475298747
Found nothing for K=13: N up to 48522381824236306
Found nothing for K=14: N up to 52378587226765785
Found nothing for K=15: N up to 7469605428405104

***

Gennady wrote:

First of all I'd like to note that some solutions can be improved:

for K=3:

2 4 8
3 3 9
5 2 10

for K=5:

1559 6 9354
1871 5 9355
2339 4 9356
3119 3 9357
4679 2 9358

Q1. I found several pairs of Sophie Germain primes  of kind k*2^n+1, 2*k*2^n+3 and chose that 4*k*2^n+5)/3 is prime.
My solution:

156925447*2^1000+1    4

672587868588209646405296259078950855772631084099366774886532228205\
    76376097880181547476764022131361261088550832962328684719649453\
    04448363430242618884062998965481477627264351175273810462479044\
    21727639351791954496199429165517805719250588460772294514797726\
    4831282343144622177847582176891322510117662770881823244292

(4*156925447*2^1000+5)/3   3

672587868588209646405296259078950855772631084099366774886532228205\
    76376097880181547476764022131361261088550832962328684719649453\
    04448363430242618884062998965481477627264351175273810462479044\
    21727639351791954496199429165517805719250588460772294514797726\
    4831282343144622177847582176891322510117662770881823244293

  2*156925447*2^1000+3   2

672587868588209646405296259078950855772631084099366774886532228205\
    76376097880181547476764022131361261088550832962328684719649453\
    04448363430242618884062998965481477627264351175273810462479044\
    21727639351791954496199429165517805719250588460772294514797726\
    4831282343144622177847582176891322510117662770881823244294

(310-digits numbers).
I used programs (newpgen, prp, pfgw, primo) to find numbers and to prove their primality.

Q2. I found solution only for K=9:

13042093679     10      130420936790
14491215199     9       130420936791
16302617099     8       130420936792
18631562399     7       130420936793
21736822799     6       130420936794
26084187359     5       130420936795
32605234199     4       130420936796
43473645599     3       130420936797
65210468399     2       130420936798

(Unfortunately, I didn't have time for more this week)

***

Oscar wrote:

I assumed that a suitable set of K primes is minimal if pmax is as small as possible.
In this sense, the published solutions for K= 2,5,7,8 are minimal, those for K = 3,4,6 can be improved:
 
K=2:   1 digit primes (same pmax, smaller pmin)
 
(p, n, N)
2  2  4
5  1  5
 
K=3,4:   1-2 digits primes
 
(p, n, N)
 2   4   8
 3   3   9
 5   2  10
11  1  11
 
K=6:   6 digits primes
(p, n, N)
346751  10  3467510
385279   9   3467511
433439   8   3467512
495359   7   3467513
577919   6   3467514
693503   5   3467515
 
For 9 <= K <= 14, I found the following minimal solutions: 
 
K=9:   10-11 digits primes
(p, n, N)
 6989954399   14  97859361586
 
 7527643199   13  97859361587
 8154946799   12  97859361588
 8896305599   11  97859361589
 9785936159   10  97859361590
10873262399   9   97859361591
12232420199   8   97859361592
13979908799   7   97859361593
16309893599   6   97859361594
 
K=10:   12 digits primes
(p, n, N)
138923104319  16  2222769669104
 
148184644607  15  2222769669105
158769262079  14  2222769669106
170982282239  13  2222769669107
185230805759  12  2222769669108
202069969919  11  2222769669109
222276966911  10  2222769669110
246974407679   9   2222769669111
277846208639   8   2222769669112
317538524159   7   2222769669113

 

K=11:   13 digits primes
(p, n, N)
3724065465119  16  59585047441904
3972336496127  15  59585047441905
4256074817279  14  59585047441906
4583465187839  13  59585047441907
4965420620159  12  59585047441908
5416822494719  11  59585047441909
5958504744191  10  59585047441910
6620560826879   9   59585047441911
7448130930239   8   59585047441912
8512149634559   7   59585047441913
9930841240319   6   59585047441914
 
K=12:   16-17 digits primes
(p, n, N)
 7334789590556999   14  102687054267797986
 7899004174445999   13  102687054267797987
 8557254522316499   12  102687054267797988
 9335186751617999   11  102687054267797989
10268705426779799  10  102687054267797990
 
11409672696421999   9   102687054267797991
12835881783474749   8   102687054267797992
14669579181113999   7   102687054267797993
17114509044632999   6   102687054267797994
20537410853559599   5   102687054267797995
25671763566949499   4   102687054267797996
34229018089265999   3   102687054267797997
 
K=13-14:   18 digits primes
(p, n, N)
152630661813293663  25  3815766545332341575
158990272722180899  24  3815766545332341576
165902893275319199  23  3815766545332341577
173443933878742799  22  3815766545332341578
181703168825349599  21  3815766545332341579
190788327266617079  20  3815766545332341580
200829818175386399  19  3815766545332341581
211987030296241199  18  3815766545332341582
224456855607784799  17  3815766545332341583
238485409083271349  16  3815766545332341584
254384436355489439  15  3815766545332341585
272554753238024399  14  3815766545332341586
293520503487103199  13  3815766545332341587
317980545444361799  12  3815766545332341588
 
For any other solution with K>=13, inequality pmax >= 6.3e17 holds.
 
For K = 15 I found no solutions.

***


Records   |  Conjectures  |  Problems  |  Puzzles