Problems & Puzzles: Puzzles

 Puzzle 501. Three Consecutive primes a prime JM Bergot sent the following puzzle related to three consecutive primes: 7+(11+13)=31 31+(37+41)=109 109+(113+127)=349 349+(353+359)=1061 1061+(1063+1069)=3193 (Fail) Q. Can you find a larger (than 4) successful steps like these above?

My comment: This puzzle had been posed before (See Puzzle 421). That time JKA reached to one solution length 10

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Contributions came from Qu Shun Liang, Jan van Delden, Farideh Firoozbakht, Arkadiusz Wesolowski, Emmanuel Vantieghem, Seiji Tomita, Pat Costello, Fred Schneider & JC Rosa.

No one got a larger solution that the gotten by JKA in Puzzle 421.

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Qu Shun Liang wrote:

I found: 2543à7643à22961à68897à206699à620227à1860697(fail) length : 6

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Jan wrote:

Actually the given example has a length of 4, since the sum of the 5'th term is 3193 (instead of 3191) which is divisible by 31.

A small table [n,p]:

1 17
2 11
3 5
4 7
5 2543
6 249217
7 1783841
8 2494517

I tested p (at least) untill 10^8. Maximum sum of 3 primes to test for primeness [length n], would be around 3^(n+1)*p.
Which would take some time given large p and (larger) n and no special form for p.

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Farideh wrote:

Smallest 8 successful steps :

1. 2494517+(2494523+2494537)=7483577
2. 7483577+(7483601+7483603)=22450781
3. 22450781+(22450817+22450849)=67352447
4. 67352447+(67352449+67352507)=202057403
5. 202057403+(202057417+202057423)=606172243
6. 606172243+(606172253+606172283)=1818516779
7. 1818516779+(1818516851+1818516853)=5455550483
8. 5455550483+(5455550501+5455550539)=16366651523
9. 16366651523+(16366651537+16366651577)=49099954637 Fail

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Got a five some:

1. 2543 + (2549 + 2551) = 7643
2. 7643 + (7649 + 7669) = 22961
3. 22961 + (22963 + 22973) = 68897
4. 68897 + (68899 + 68903) = 206699
5. 206699 + (206749 + 206779) = 620227
6. 620227 + (620233 + 620237) = 1860697 (Fail)

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Emmanuel wrote:

I could find lots of longer similar chains. The longest I found was this one :

2494517 + 2494523 + 2494537 = 7483577 , prime.
7483577 + 7483601 + 7483603 = 22450781 , prime.
22450781 + 22450817 + 22450849 = 67352447 , prime.
67352447 + 67352449 + 67352507 = 202057403 , prime.
202057403 + 202057417 + 202057423 = 606172243 , prime.
606172243 + 606172253 + 606172283 = 1818516779 , prime.
1818516779 + 1818516851 + 1818516853 = 5455550483 , prime.
5455550483 + 5455550501 + 5455550539 = 16366651523 , prime.

A possible longer chain cannot be obtained if the smallest prime in question is smaller than the 10000000th prime

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Seiji wrote:

About Puzzle 501,I found two patterns(5,6).

2543 + 2549 + 2551 = 7643
7643 + 7649 + 7669 = 22961
22961 + 22963 + 22973 = 68897
68897 + 68899 + 68903 = 206699
206699 + 206749 + 206779 = 620227

...

249217 + 249229 + 249233 = 747679
747679 + 747713 + 747731 = 2243123
2243123 + 2243161 + 2243177 = 6729461
6729461 + 6729469 + 6729473 = 20188403
20188403 + 20188451 + 20188459 = 60565313
60565313 + 60565319 + 60565327 = 181695959

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Pat Costello wrote_

1783841 1783843 1783867 p1+p2+p3= 5351551 is prime
5351551 5351579 5351581 p1+p2+p3= 16054711 is prime
16054711 16054721 16054729 p1+p2+p3= 48164161 is prime
48164161 48164191 48164201 p1+p2+p3= 144492553 is prime
144492553 144492563 144492583 p1+p2+p3= 433477699 is prime
433477699 433477721 433477729 p1+p2+p3= 1300433149 is prime

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Fred Schneider wrote:

Here are the minimum terms for lengths 1 to 8. There are no longer sequences beginning below 3 billion.

Chain started from 17 of length: 1
1) 17 19 23

Chain started from 11 of length: 2
1) 11 13 17
2) 41 43 47

Chain started from 5 of length: 3
1) 5 7 11
2) 23 29 31
3) 83 89 97

Chain started from 7 of length: 4
1) 7 11 13
2) 31 37 41
3) 109 113 127
4) 349 353 359

Chain started from 2543 of length: 5
1) 2543 2549 2551
2) 7643 7649 7669
3) 22961 22963 22973
4) 68897 68899 68903
5) 206699 206749 206779

Chain started from 249217 of length: 6
1) 249217 249229 249233
2) 747679 747713 747731
3) 2243123 2243161 2243177
4) 6729461 6729469 6729473
5) 20188403 20188451 20188459
6) 60565313 60565319 60565327

Chain started from 1783841 of length: 7
1) 1783841 1783843 1783867
2) 5351551 5351579 5351581
3) 16054711 16054721 16054729
4) 48164161 48164191 48164201
5) 144492553 144492563 144492583
6) 433477699 433477721 433477729
7) 1300433149 1300433153 1300433171

Chain started from 2494517 of length: 8
1) 2494517 2494523 2494537
2) 7483577 7483601 7483603
3) 22450781 22450817 22450849
4) 67352447 67352449 67352507
5) 202057403 202057417 202057423
6) 606172243 606172253 606172283
7) 1818516779 1818516851 1818516853
8) 5455550483 5455550501 5455550539

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JC Rosa wrote:

I have found solutions for 5,6,7 and 8 successful steps.
The smallest solution for 8 steps is :
2494517+2494523+2494537=7483577
7483577+7483601+7483603=22450781
22450781+22450817+22450849=67352447
67352447+67352449+67352507=202057403
202057403+202057417+202057423=606172243
606172243+606172253+606172283=1818516779
1818516779+1818516851+1818516853=5455550483
5455550483+5455550501+5455550539=16366651523

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