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There are a few disadvantages of the Fibonacci search: It can be slower than other search algorithms if the data is not sorted. It can be less accurate than other search algorithms if the data is not sorted. It can be more difficult to implement than other search algorithms.
The first Fibonacci number greater than 23 is 34.
Other than the fact that much of mathematics is about patterns, NO.
It is not particularly significant, other than being a prime.
pure mathematics is mathematics studied and used reasons other than application. So some algebra certainly might be, but not all algebra. Modern or abstract algebra would fit better in that category.