Problems & Puzzles:
Puzzles
Puzzle 280. 3893257
p1, p2, p3, …
is a chain of prime
numbers base 10 such that for i =>1,
p_{i+1} = (p_{i} )_{base
2}
The larger
chain I have found is one with
4 prime members starting in the prime p_{1} = 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 7chain is: 19, 23, 31, 43, 59, 79, 107
The base 15 transform 7chain is: 577, 1237, 3877, 12037, 57427,
277687, 1898227
***
