Problems & Puzzles: Puzzles

 Puzzle 397. 15984784979 Perhaps you have seen this curio about the prime 15984784979 Permute any two consecutive digits and you still have a prime number. [Blanchette] Question. Get non-trivial larger examples

Contributions came from Farideh Firoozbakht, J. K. Andersen & Giovanni Resta.

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Farideh wrote:

n = (16*10^419-1)/3 = 5.3(419)  is a solution.

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the nice number 15984784979 we have 11 distinct corresponding primes :
15894784979, 15948784979, 15984748979, 15984784799, 15984784979,
15984784997, 15984789479, 15984874979, 15987484979, 19584784979 & 51984784979.

Regarding to such nice solutions I defined the following two sequences
of smallest and largest n-digit such solutions.

a(n) for n= 2,3,..,10 :
13, 131, 1013, 15193, 102539, 2498213, 10568419, 270438293, 2020649713

b(n) for n= 2,3,..,10 :
97, 919, 9871, 95971, 918139, 9483437, 96931391, 640704091, 6473531413

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Andersen wrote:

The puzzle does not define "non-trivial". If consecutive digits must be distinct then 87563167093 is the next solution, and there are no more below 2*10^12.

If equal consecutive digits are allowed then 16023375131 is the next
solution.

Any repunit prime is by definition a solution. The largest known proven repunit prime is R(1031). It consists of 1031 1's. The record prp is R(109297), announced by Harvey Dubner on April 3 2007.

If at least the last third of the digits in a prime are 9's, then PrimeForm/GW can quickly prove primality. (8*10^2166+10^1345)/9-1 consists of 821 8's followed by 1345 9's. Permuting the adjacent 8 and 9 gives (8*10^2166+91*10^1344)/9-1. The two primes were found by PrimeForm/GW which proved them in seconds.

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Resta wrote:

I concentrate on numbers where all adjacents digits are distinct (if not, it is not difficult to find larger such primes, such as 11111111111111111111111119111111111 )

I searched prime numbers up to 22.000.000.000.000 and I found only one larger such prime: 87.563.167.093.

Of the same length, apart the cited 15.984.784.979, there is also 14.070.135.971.

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