Problems & Puzzles: Puzzles

Puzzle 926. pandigital and prime numbers

In the table below I have organized some of the facts that I have just collected or that I have computed about pandigitals and prime numbers.

Pandigitals are one one the two classes: a) composed of the ten decimal digits, 0 to 9, or b) composed of the decimal digits 1 to 9. The first are named just "pandigitals" and the later are named "zero-less pandigitals".

Q1. I invite you to work over the three unknown facts corresponding to the last three rows of the table.

Q2. Do you know some other interesting fact about pandigitals and prime numbers?

 Concept Zero-less pandigitals Pandigitals Smallest 123456789 1023456789 Largest 987654321 9876543210 Quantity 9! = 362880 10! = 3628800 Smallest prime factor 2 2 The only prime that divides all the members of the set 3 3 Largest prime factor 109739359 1097393447 Smallest prime that is not factor of any member of the set 44449 111119 The largest prime that only divides just one member of the set 109739359 1097393447 Members of the set with the largest quantity of primes divisors 589234176 =2^16*3^5*37, (22) 6398410752=2^21*3^3*113, (25) Members of the set with the largest quantity of distinct primes divisors 725638914= *3^2*7*11*13*17*23*103 and three more with 8 distinct prime factors 2148736590 = *3^2*5*7*11*13*17*23*61 and 16 more with 9 distinct prime factors Sum from 2 to P giving a pandigital P=92857, 110921 & 112997 (394521678, 547128369 & 572469138, respectively) P=155853, … 441461 (1063254978, …, 7803615924) (13 primes and pandigital values) The smallest prime that only divides just one member of the set of pandigitals Unknown Unknown The quantity of the primes that that only divides just one member of the set of pandigitals Unknown (Please send the whole list of primes and the corresponding pandigital) Unknown  (Please send the whole list of primes and the corresponding pandigital) The quantity of the distinct prime factors involved in all the set of pandigitals Unknown Unknown

Contributions came from Emmanuel Vantieghem

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Emmanuel wrote:

First of all, the number of pandigitals is not 10!  but 9*9! = 3265920

Further :

The set of zero-less pandigitals :
* Number of distinct primes involved in the set : 144319
* Number of primes that divide just one number of the set : 109076
smallest such element :  44909  (dividing 142675893)

The set of pandigitals :
* Number of distinct primes involved in the set : 1102173
*Number of primes that divide just one number of the set : 834218
smallest such element :  293339 (dividing 1795234680)

As you can see, the number of primes that divide just only pandigital is quite big, so I send you the results in annex 1annex 2.
There you will find the prime couples  {p,m}  where  p  is the prime in question and  m  the associated pandigital

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Pierandrea Formusa wrote on jan 1, 2019:

 Concept Zero-less pandigitals Pandigitals Smallest (zero-less) pandigital that is between two twin primes 123457968 1023567948 Largest (zero-less) pandigital that is between two twin primes 987614532 9876530142 Quantity of (zero-less) pandigitals that are between two twin primes 2595 26763

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