"Gringotts is the safest place in the world fer anything yeh want ter keep safe -- 'cept maybe Hogwarts."
--Rubeus Hagrid (PS5)
After entering the huge, grand bank and explaining what they want to a goblin, Hagrid and Harry are taken by Griphook down to the vaults on a speeding cart, which makes Hagrid sick. First they stop at the Potter vault, then continue to the higher security Vault 713, where Hagrid gets the Philosopher’s Stone.
This takes place on Harry's 11th birthday.
The Philosopher's Stone was being stored in Vault 713. The numbers 7 and 13 are both associated with magic. Specifically, 7 is considered a lucky number and 13 is considered unlucky. 7 and 13 tend to be prominent in magic because they are relatively small prime numbers. 7 is especially appears as an important motif throughout the Harry Potter series. -BB
Sirius' vault is numbered 711 (PA22). This number is not far off from 713. Perhaps the vaults are located fairly close to one another.
Additionally, 7+1+3=11. -BB
The vaults themselves vary in size and security. The largest, most well protected vaults belong to the oldest wizarding families and lie deepest beneath the surface (DH25). Those vaults closer to the surface seem to be smaller and have fewer security precautions surrounding them - they use keys, for example, rather than requiring the touch of a goblin to gain access. The Potter family's vault, which requires a key for entry and seems to be of moderate depth - the cart travels for a while to get there, but it's not as far down as vault 713, where the Philosopher's Stone was held (PS5). Vault 713, which stored only the Philosopher's Stone when Harry visited with Hagrid. The vault was very deep and required a goblin to stroke the door with his finger in order to gain entry (PS5).
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