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= 2*3^2*7*11*13*17*23*103 and three more with 8 distinct prime factors

2148736590 = 2*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
(unless one would admit numbers to start with a zero on the left)

 
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 :  11909  (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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