By Dirichlet’s theorem on primes in arithmetic progressions, the progression \(P m + 2\) contains infinitely many primes. Let \(q\) be such a prime factor of some \(a_m\). Since \(q\mid a_m\) and \(q mid P\), we have \(q mid (n_i+1)\) for all \(i\). Thus \(q\) is a new divisor not equal to any previous \(n_i\).
Therefore the process can continue indefinitely, producing an infinite sequence \((n_k)\). ∎
4. Conclusion
We have shown that for every integer \(k>1\) there exists a \(k\)-digit number with all digits different. The construction of the sequence \(\,N_k\,\) guarantees that such numbers exist for all values of \(k\).
Hence there are infinitely many integers whose decimal representation contains only distinct digits.
These are precisely the numbers \(1,2,\dots ,9,10,12,\dots ,98,102, 103,\dots \), and so on.
The set of such numbers is not a perfect arithmetic progression (there are gaps, e.g. between \(99\) and \(100\)). In base‑\(b\) there can be at most \(b^\,b\) distinct‑digit numbers. These integers are sometimes called pandigital* in the sense of using each digit only once, but not necessarily all digits.
The proof above shows that, for any finite alphabet, one can construct infinitely many words with no repeated symbols.
