Primes Formed By Concatenation

 Suppose we laid down the following criteria that prime numbers had to adhere to:

• formed from the \(n^{th}\) number and the  \(n^{th}\) prime
• no repeating digits
• sum of digits is a prime number

The criterion that there must be no repeating digits means that we must have a finite number of such primes because, as the primes get larger, digits must repeat. Concatenating the first number 1 and the first prime 2, we get 12 which is not prime. However, concatenating the second number 2 and the second prime 3, we get 23 which satisfies the criteria. The next possibility, 35, doesn't satisfy but 47 does. 

Writing a program that yields all the conforming primes up to ten million, yields the following select group where | represents the operation of concatenation: 

• 2 | 3 --> 23
• 4 | 7 --> 47
• 12 | 37 --> 1237
• 27 | 103 --> 27103
• 57 | 269 --> 57269
• 58 | 271 --> 58271
• 85 | 439 --> 85439
• 93 | 487 --> 93487
• 145 | 829 --> 145829
• 406 | 2791 --> 4062791
• 591 | 4327 --> 5914327
• 835 | 6421 --> 8356421

Interestingly, apart from 2, all the  \(n^{th}\) numbers are composite in the range up to 999. This investigation arose from the number associated with my diurnal age today, namely 27103. It turned out that this number is a member of OEIS A084667:

A084667

Primes which are a concatenation of \(n\) and prime(\(n\)).

For this sequence, we are only applying the first criterion. The initial members of the sequence are shown below with previous primes marked in bold:

23, 47, 613, 1237, 1759, 1861, 2383, 27103, 30113, 35149, 36151, 41179, 42181, 45197, 46199, 54251, 56263, 57269, 58271, 61283, 71353, 82421, 83431, 85439, 92479, 93487, 99523, 115631, 117643, 119653, 121661, 123677, 127709, 136769, 141811, 145829, 147853 

It can be noted that of the terms shown, the  \(n^{th}\) number in several cases is prime e.g. 1759, 2383 etc. I was interested in finding out how many primes survived once the second and third criteria were applied and 27103 survived as can be seen.