617 = 1!^2 + 2!^2 + 3!^2 + 4!^2. [Meller]
617 is the only three-digit prime such that, if 2 is subtracted from its 4th power, the resulting number is prime and if 2 is subtracted from the 4th power of this number, the result is still prime. I.e., 617, 617^4-2, and (617^4-2)^4-2 are all primes. [Opao]
The smallest prime that has an expression in more than one way as the repdigit sum of the four prime digits (2, 3, 5, 7), in each case with the summands concatenating to a prime in this digit order: 617 = 22 + 33 + 555 + 7 = 222 + 333 + 55 + 7 with 22335557 and 222333557 being primes. [Beedassy]
The largest of the RSA numbers has a length of 2048 bits (617 decimal digits). There was once a $200,000 prize offered for its factorization, but is now withdrawn. Anyway, RSA-2048 is unlikely to be factored anytime soon.
The largest multidigit prime that is exactly half of a number formed with distinct consecutive digits. [Dickman]
The smallest happy prime p for which the sum of the first p happy primes is a happy prime (10147001). [Gaydos]
The largest prime of form p^q-r^s, with prime bases, where bases and exponents are consecutive numbers, i.e., 617=5^4-2^3. [Loungrides]
Average gestation period (in days) of an Asian Elephant. [Bartholomy]