Which primes are alive

The concept of "alive" primes isn't standard mathematical terminology, but it might refer to several possible contexts. In recreational mathematics and programming, "living primes" or "survivor primes" sometimes describe primes that remain prime even when you repeatedly remove digits from either end—these "survive" the digit-removal process while maintaining primality. For example, 3797 is prime, and removing digits leaves 379, 37, and 7, all of which are also prime, making it "alive" through successive truncations.

Another interpretation involves "circular primes," which remain prime under digit rotation (like 197, 971, and 719 all being prime), or "emirp" primes, which spell different primes when reversed (like 13 and 31). In computer science and cryptography, people sometimes refer to primes "in use" or "active" primes that are currently being employed in encryption keys or cryptographic protocols as "alive" in a functional sense. More fancifully, some mathematics enthusiasts playfully discuss primes in terms of "behavior"—for instance, twin primes that appear close together, or prime gaps that seem to follow patterns—using anthropomorphic language that treats numbers as though they have agency. If you're asking about a specific context or definition of "alive" primes, clarifying which mathematical property or recreational number game you're referring to would help provide a more precise answer.