Problems & Puzzles: Puzzles

Puzzle 281.  A follow up to Puzzle 24

Three years ago (13/7/2001) Jack Brennen found - for the Puzzle 24 of these pages, posed in 1998 - thee following prime: 50,006,393,431.

This prime is the earliest prime base 10 such that its eight representations of it in the bases form 2 to 9 are all of them prime numbers if taken as numbers base 10.

Later, on may of the last year (2003), Giovanni Resta found three more solutions: 727533146383, 2250332130313, 2651541199513.

Now I will modify a little bit that old puzzle.

Now I will ask for the earliest prime in base 10,  p10 such that the chain: p10, p9, p8, ..., p2 is composed only of primes base 10, and:

• p9 = (p10) base 9

• p8 = (p9) base 8

• ...

• p2 = (p3) base 2

In my own search up to 2^32, I have found that 829295767 is the earliest starting prime of a chain of seven primes as the asked.

Chain of length 7: p10, p9, ...p4.

829295767, 2123416831, 17644142377, 1163144510221, 2250201411320021,
4324414244442244220041, 322212311031220321131303031120102021

Question:

1. Find the earliest (non trivial) prime p10 for a chain as the asked above of length equal to a) 8 and b) 9.

Faride Firoozbakht wrote:

I couldn't find the solution. I only know the earliest (non trivial) prime p10 for a chain of length 8 is greater than 23684910373. It is interesting that the index of all the earliest (non trivial) prime p10 for chains of length 7 that I found is even.

p1= prime(42574190)
p2= prime(61526706)
p3= prime(86235406)
p4= prime(150123448)
p5= prime(247953846)

...

p6= prime(1098392832)
p7= prime(1125383168)
p8= prime(1256483046)
p9= prime(1424443227)

***

On December 5, 2004 Giovanni Resta added:

Two chains of length 8 starting from
537785307043
2660974895563
and there are no others chains of length 8 or more, for numbers below 3x10^12.

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