Problems & Puzzles: Puzzles

Puzzle 1200 14869 & 20357

Gaydos found (when?) that p=20357 is the smallest (five digits) prime number for which it and another smaller prime (q=14869) together are pandigital"-10 digits.

Q. Find all the pairs (p, q<p) such that the sum of each pair, p+q is the largest possible for each p.


From Dec 7 to 13, 2024, contributions came from Michael Branicky, JM Rebert, Adam Stinchcombe, Emmanuel Vantieghem

***

Michael wrote:

There are 142476 such pairs with (p, q<p) both prime, together comprising exactly 10 pandigital digits, with p+q maximized for each p.
More specifically, there are 1817, 8459, 20030, 39305, and 72865 where p has 5, 6, 7, 8, and 9 decimal digits, respectively.
They are all in the attached text file.

***

Rebert wrote:

Here are my 142476 results.

(Note by CR: Rebert got exactly the same results than Michael).

***

Adam wrote:

I get the largest sum of   183924   obtained for three different groups of primes:
{87523,96401}, {86423,97501} and  {86501,97423}   

CR: As far as I can see, you restricted your search to 5 digits prime p. Why?

I thought that was the puzzle, find two primes that divided the ten digits in half.  If not (any big prime p and any small prime such that p and q form exactly the 10 digits), I get a sum of 987654105 for the primes 987654103, 2

***

Emmanuel wrote:

If I correctly understand the question, there are  142475  valid couples.
The first  10  being :
(20357, 14869), (20359, 16487), (20369, 18457), (20389, 16547), (20479, 15683), (20483, 17659), (20543, 19867), (20549, 18637), (20563, 18947), (20593, 16487)
I send all of them in the annex.

***

 

Records   |  Conjectures  |  Problems  |  Puzzles