Calculating the largest Mersenne prime: there is no largest. A prime number can only be divided by 1 and itself. 3, 5, 11, and 1301 are prime. 27, 48, and 1568 are not. To get a Mersenne you multiply 2 by itself (powers of two) a whole bunch of times then subtract 1. If the result is prime it is a Mersenne. However, a limited number of Mersenne Primes have been conclusively calculated, so they were likely referring to the largest confirmed number.
Flynn dons the 'Cap of Invisibility', a winged helmet in operatic (as opposed to historicaly accurate) viking style. It is the Tarnhelm, invented in the 1800s by Richard Wagner for his Ring Cycle operas. He needed his characters to be able to disappear and shapeshift so he wrote in the Tarnhelm.
The way the librarians defeat Apep has a hint of The Wizard of Oz to it. Ezekiel gives Apep a love potion, a heart (Tinman), Cassandra transports thought, a brain (Scarecrow) and Jacob bestows the gift of inner soul which could be interpreted as "courage" (The Lion).
Jenkins is turned to stone by Medusa's head, ironically Flynn later uses the Cap of Invisibility, which Perseus used in his defeat of Medusa.
Ezekiel says "Garnet crystals have an extremely high level of light refraction". Even amateur gemologists use the term 'refractive index'. All the garnets, like the vast majority of gemstones (and leaded glass), have a refractive index of 1.7 to 1.8. Distilled water has a refractive index of 1.33; diamond has 2.4.