Introduction

The problem of finding the smallest value of n such that 1234567...n is prime is a fascinating exploration in the realm of number theory. While such a prime has yet to be discovered, this article delves into the details, offering insights into Smarandache primes and the challenges associated with this problem.

Understanding Smarandache Primes

Smarandache primes are a special type of prime numbers that arise from sequences of ascending digits in various numerical bases. They are primes formed by concatenating ascending digits starting from 1 up to a certain number. While this concept is intriguing, the existence of such primes remains an open question, particularly for large sequences.

Exploring Ascending Digit Sequences

To understand the problem better, we need to explore ascending digit sequences in various bases. Let's start with the simplest cases:

Base 2: The only ascending digit sequence is "1". This is trivially not a prime number as it consists of a single digit.

Base 3: The sequence "12" in base 3 is equal to 5 in base 10. This is the smallest solution among the few bases we've explored.

Base 10: There is no solution in base 10, as the number 1234567 to the nth digit does not yield a prime number for any n less than 1,000,000.

Exploring Larger Bases

As we move to larger bases, the problem becomes more complex. Here are some examples:

Base 11: The sequence "12" is 13 in base 10. This is a valid solution.

Base 12: The sequence "12345" is 24677 in base 10, which is quite large.

Base 14: The sequence "123" is 227 in base 10.

Base 15: The sequence "12" is 17 in base 10.

For base 16 (hexadecimal), the sequence "123456789ABCD" is 320255973501901, and starting "123" yields the least n which is 13 in hexadecimal.

Computational Challenges

Computing such solutions can be computationally intensive. For instance, attempting to check for pseudoprimes using PariGP, the script runs for a long time as there are no pseudoprimes for the first 4544 numbers using concatenation.

Conclusion

The problem of finding the smallest value of n such that 1234567...n is prime remains open, especially for larger sequences. Smarandache primes are a fascinating area of study, offering both challenge and intrigue in the world of number theory.

For further exploration, readers are encouraged to delve into computational number theory and the intricacies of ascending digit sequences in various bases.