Final answer:
The question pertains to proving the insignificance of the number of operations required to compute prime numbers less than a given number n in comparison to n itself, by using big O and little o notations.
Step-by-step explanation:
The question asks us to prove that the number of times we need to perform a specific operation to compute prime numbers smaller than a given number n is asymptotically negligible in comparison to n itself. This relates to the concept of big O notation in mathematics, which provides a way to describe the upper bound or limiting behavior of a function when the argument tends towards a particular value or infinity.
In the scenario provided, the number of prime numbers less than n that need to be calculated using the operation described is said to be fewer than √n. In mathematical analysis, an operation count that is less than √n means it will eventually become negligible as n gets larger and larger. This situation can be described using the little o notation (o(n)), indicating that the process of computing these primes grows at a slower rate than n itself, and thus the proportion of operation time compared to n diminishes as n increases.
If we denote the operation count as k(n), then because k(n) is less than √n, we can formally express that k(n) = o(n). This means that for any positive constant C, there exists a threshold N where for all n greater than N, k(n) < C · n. This succinctly shows that as n approaches infinity, the operation count becomes insignificant relative to n, which is what we desired to prove.