Problems & Puzzles: Puzzles

Puzzle 110. Dropping-Digits Primes (by Patrick De Geest)

Here you are asked to find the least prime such that it remains prime after all the occurrences of each digit "d" are dropped at the same time, for each distinct digit of the original prime.

Example

Original prime: '1210778071' with 5 distinct digits (0, 1, 2, 7 & 8)

dropping all '0's leaves the prime 12177871
dropping all '1's leaves the prime 2077807
dropping all '2's leaves the prime 110778071
dropping all '7's leaves the prime 1210801
dropping all '8's leaves the prime 121077071

No leading zeros are allowed. This means that if a zero occurs in the original prime, at least two distinct digits must precede it.

Here are the least primes found by Patrick for k=2 to 7:

k, distinct digits

 Least original prime

2

23

3

 137

4

 6173

5

37019

6

5600239

7

476710937
8 8192454631 by Patrick De Geest (Nov/15/2000)
9

1645957688093 by Giovanni Resta (Oct. 2003)
10 78456580281239 by Giovanni Resta (Oct. 2003). See also Puzzle 235

Questions:

1. Can you complete the table
2. Do you devise a 'constructive' approach to get these primes (disregarding the objective of getting the least original prime for each k value)

Solution

 
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