Magic trick

Leo, the magician asks a member of the audience to write down at most seven distinct prime numbers (\(>2\)) without showing them to Leo. Then the chosen member multiply each number with itself, and tells the amount of the perfect squares to Leo. From that in a few secundum (without using calculator) Leo finds out how many numbers did the member write down.

For example if Sarah was the chosen member, and she writes down the numbers \(3, 13, 23, 29,\) then she tells the number \(3^2+13^2+23^2+29^2=1548\).

Imagine that now you are the magician and you do the same trick (as Leo did) to your friend, Bob. Bob tells you the number \[550632557\]

How many numbers did Bob write down?

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