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:
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, 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) ... later she added this: p6= prime(1098392832) *** On December 5, 2004 Giovanni Resta added:
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