Problems & Puzzles: Puzzles

Puzzle 280.  3893257

p1, p2, p3, is a chain of prime numbers base 10 such that for i =>1, pi+1 = (pi )base 2

The larger chain I have found is one with 4 prime members starting in the prime p1 = 3893257 (the fourth member is a number composed by 230 digits).

Question: Find the earliest prime of a sequence of 5 prime numbers in chain.


Contributions came from Jim Fougeron, Faride Firoozbakht and Jon Wharf:

Fougeron and Firoozbakht found - independently - the earliest prime of a sequence of 5 prime numbers in chain as the requested: 9632552297.

Fougeron comments the following:

The first length 5 starts from p1=9632552297 p5 is 1203 digits long.

There were 657 chains of 4 prior to this chain of 5. I was expecting to have to check about 1120 before success, so I was glad this turned up early. A chain of 6 will most likely require about 4000 chains of 5 before finding one.

***

Jon Wharf did not find this solution but worked an interesting variation of this puzzle:

I did investigate the other bases below ten, naturally ignoring p1<b each time, and the largest chains I got in each case were:
base 3 transform: chain of 5 starting at 2119939
base 4 transform: chain of 5 starting at 5 (amazingly)
base 5 transform: chain of 5 starting at 8101
base 6 transform: chain of 5 starting at 2857
base 7 transform: chain of 5 starting at 1247599
base 8 transform: chain of 6 starting at 56263 (the best chain)
base 9 transform: chain of 5 starting at 266689
 
I looked at all numbers up to at least 1500000 for each of these bases. I may also try to code the opposite transforms, for example taking a base ten prime, regard the number as base 12 (say) and transform it back into base 10.

...

Results from the "other side", where I take a base b>10 and produce p(i+1) by converting p(i) from base b into base 10
base 11 transform: chain of 4 starting at 1451
base 12 transform: chain of 5 starting at 1297
base 13 transform: chain of 4 starting at 797
base 14 transform: chain of 7 starting at 19
base 15 transform: chain of 7 starting at 577
base 16 transform: chain of 6 starting at 61
base 17 transform: chain of 4 starting at 25169
base 18 transform: chain of 5 starting at 191
base 19 transform: chain of 4 starting at 1279
base 20 transform: chain of 6 starting at 13
base 21 transform: chain of 3 starting at 641 (worst result)
base 22 transform: chain of 5 starting at 17
 
The base 14 transform 7-chain is: 19, 23, 31, 43, 59, 79, 107
The base 15 transform 7-chain is: 577, 1237, 3877, 12037, 57427, 277687, 1898227

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