Problems & Puzzles: Puzzles

Puzzle 1190  Friedman Prime numbers 995347 is the largest k=6 digits prime Friedman number [which can be written in some non-trivial way using its own digits, together with the symbols + – × / ^ ( ) and concatenation. In this example, 995347=4*( 7+(9+3)^5 )-9 ] Q. Find the largest prime Friedman numbers for k=7, 8, 9 & 10.

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From 21-28 Sept. 2024, contributions came from Emmanuel Vantieghem

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Emmanuel wrote:

Here is what I could find about Puzzle 1190 :
 

   k         prime                      Formula

  7      9834497          ( 8 ( 9 - 9 + 7 ))^4 - 3 + 4
  8      99897353       ( 8 ( ( 9 + 9 - 7 ) 5 + 3 ) )^3 + 9
  9      992436551     ( 9 ( 3 + 4) ( 2 + 5 - 6 ) )^5 + 9 - 1
 10     9993948257   ( 2 ( 4^5 + 9) + 9*9 + 7 )^3 - 7 ( 9 - 8)

But I think there may be bigger solutions.

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Later, on Set 30, 2024, Adam Stinchcombe, wrote:

From  https://erich-friedman.github.io/mathmagic/0800.html   I find a list of 7 digit Friedman numbers which include the prime 9999991= (9+1)^(9-(9+9)/9) - 9. This is the largest 7 digit prime, so it must be the largest 7 digit Friedman prime.

 

Sticking with that theme, first prime under 10^8 is 99999989 which is  (9+9/9)^8-(9+(9+9)/9)
 

The first prime under 10^9 is the 9 digit 999999937 which is equal to       (3+7)^9-9*(9-9/9-9/9)
 

Not the first prime under 10^10 but close is 9999999001 which is
10^(9+9/9)-999+0*9

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