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This question is bourn out of my recent answer to a question asked here

can a Car Registration Number, a combination of prime, be prime?

Now this is true for certain prime numbers , e.g., $3, 7, 109$ and $673$ that if you concatenate any two of these numbers in any order , the resulting number will be a prime . As in this case, Concatenating $7$ at the end of $673$ results in $6737$ which is a prime. This lead me to make a conjecture, that concatenation of the pair of twin prime, taken in order (with larger prime being concatenated after the smaller of the twin prime numbers ),would never yield a Prime. Is this conjecture true?

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The concatenation of the primes in a twin prime pair where the first prime is larger than $3$ is divisible by $3$, so your conjecture is true. This is because the first must be $3k+2$ and the second $3k+4$. Concatenating involves multiplying by a power of $10$, which doesn't change the congruence class mod $3$, and adding. Thus the congruence class of the concatenation is the same as that of the sum, which is $0$, i.e. it is divisible by $3$.

Matt Samuel
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