Problems & Puzzles: Puzzles

Puzzle 426. q = p +rad(p+1) Juan López González noticed that: Starting with the prime p1=3 and Using the recurrence relationship: pi+1 = pi + rad(pi +1) where rad(n) = Π(q), for every q|n Then every pi is prime for i=1 to 8, while p9 is composite. 3 + rad(4) = 5 is prime 5 + rad(6) = 11 is prime 11 + rad(12) = 17 is prime 17 + rad(18) = 23 is prime 23 + rad(24) = 29 is prime 29 + rad(30) = 59 is prime 59 + rad(60) = 89 is prime and 89 + rad(90) = 119 = 7 · 17 is not prime. He asks for other primes p1 with larger chain of primes. I have gotten the smallest prime for a chain of primes of length 9 & 10: Chain length p1 Author 8  3 JLG 9  8453231 CR 10  1339751783 CR Q1. Get chains of primes with length larger than 10.

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Contributions came from Farideh Firoozbakht & J. Wroblewski

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Farideh wrote:

I didn't find any such chains of length greater than 10.
I only found three other chains of length 10 starting with primes 1643700649,
1775196701 & 8252940359.
 

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Jarek Wroblewski wrote:

Here are larger L solutions, but for sure not the minimal ones for each L.

L=11

1 15288096001810787
2 15332798036903801
3 15645712282554899
4 17210283510810389
5 34420567021620779
6 51630850532431169
7 68841134043241559
8 86051417554051949
9 103261701064862339
10 120471984575672729
11 137682268086483119

L=12

1 327049196998870807
2 328788820387162673
3 334007690552038271
4 339226560716913869
5 678453121433827739
6 1017679682150741609
7 1356906242867655479
8 1696132803584569349
9 2035359364301483219
10 2374585925018397089
11 2713812485735310959
12 3053039046452224829

L=13

1 306638364685577566694909
2 308278142036837339778839
3 309917919388097112862769
4 321396360846915524450279
5 332874802305733936037789
6 665749604611467872075579
7 998624406917201808113369
8 1331499209222935744151159
9 1664374011528669680188949
10 1997248813834403616226739
11 2330123616140137552264529
12 2662998418445871488302319
13 2995873220751605424340109

L=14

1 560945376063531227
2 561112224777410921
3 570956298896312867
4 580800373015214813
5 590644447134116759
6 639864817728626489
7 1279729635457252979
8 1919594453185879469
9 2559459270914505959
10 3199324088643132449
11 3839188906371758939
12 4479053724100385429
13 5118918541829011919
14 5758783359557638409

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Farideh Firoozbakht wrote (Dec 28, 07)

The minimal chain with length 11 starting with prime
10643568319. So for L=11 we have the following table of primes.

1 10643568319
2 10691084249
3 10833632039
4 13542040049
5 16250448059
6 18958856069
7 21667264079
8 24375672089
9 27084080099
10 29792488109
11 32500896119

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