I just love primes!
Number Theory
Level
2
What is the smallest prime number that contains all the prime digits ?
The answer is 2357.
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This is the 3rd Smarandache-Wellin Prime, as in 2 , 2 3 , 2 3 5 7 , which is a prime number formed by concactenating successive primes. The 4th one is
2 3 5 7 1 1 1 3 1 7 1 9 2 3 2 9 3 1 3 7 4 1 4 3 4 7 5 3 5 9 6 1 6 7 7 1 7 3 7 9 8 3 8 9 9 7 1 0 1 1 0 3 1 0 7 1 0 9
1 1 3 1 2 7 1 3 1 1 3 7 1 3 9 1 4 9 1 5 1 1 5 7 1 6 3 1 6 7 1 7 3 1 7 9 1 8 1 1 9 1 1 9 3 1 9 7 1 9 9 2 1 1 2 2 3 2
2 7 2 2 9 2 3 3 2 3 9 2 4 1 2 5 1 2 5 7 2 6 3 2 6 9 2 7 1 2 7 7 2 8 1 2 8 3 2 9 3 3 0 7 3 1 1 3 1 3 3 1 7 3 3 1 3 3 7
3 4 7 3 4 9 3 5 3 3 5 9 3 6 7 3 7 3 3 7 9 3 8 3 3 8 9 3 9 7 4 0 1 4 0 9 4 1 9 4 2 1 4 3 1 4 3 3 4 3 9 4 4 3 4 4 9
4 5 7 4 6 1 4 6 3 4 6 7 4 7 9 4 8 7 4 9 1 4 9 9 5 0 3 5 0 9 5 2 1 5 2 3 5 4 1 5 4 7 5 5 7 5 6 3 5 6 9 5 7 1 5 7 7
5 8 7 5 9 3 5 9 9 6 0 1 6 0 7 6 1 3 6 1 7 6 1 9 6 3 1 6 4 1 6 4 3 6 4 7 6 5 3 6 5 9 6 6 1 6 7 3 6 7 7 6 8 3 6 9 1
7 0 1 7 0 9 7 1 9
This happens because the larger a random number is, the less likely that it's a prime. So, a lot of primes have to keep being added until there is a "hit".
Meanwhile, of the 2 4 possible permutations of 2 , 3 , 5 , 7 , exactly 8 are primes
2 3 5 7 , 2 7 5 3 , 3 2 5 7 , 3 5 2 7 , 5 2 3 7 , 5 2 7 3 , 7 2 5 3 , 7 5 2 3