The Primality of $2n^2 – 1$

Consider numbers $t(n)$ of the form $t(n) = 2n^2 – 1$ with $n \gt 1$.
The first such numbers are $7, 17, 31, 49, 71, 97, 127$ and $161$.
It turns out that only $49 = 7 \cdot 7$ and $161 = 7 \cdot 23$ are not prime.
For $n \le 10000$ there are $2202$ numbers $t(n)$ that are prime.
How many numbers $t(n)$ are prime for $n \le 50\,000\,000$?

The problem actually requires application of the concept of primality testing to the numbers of the form t(n) = 2n^2 – 1. The specific answer to this question would require a significant amount of computational power and extensive programming knowledge to generate; it’s not something that one could solve purely by hand. But I will tell you how you could go about solving this.

> Steps:

1. Write a program in a language like Python or C++ that can handle large quantities of data and is efficient for numerical computations.

2. This program will need to do two main things:

a. Generate all the numbers of the form t(n) = 2n^2 – 1 for n less than or equal to 50,000,000.

b. Test each of these numbers for primality.

3. Primality testing can be done in a few different ways, and for a problem of this scale, it would be best to use a method that is both accurate and optimized for large numbers. One such method is the Miller-Rabin primality test.

4. For every number generated from the formula t(n) = 2n^2 – 1, the program will apply the Miller-Rabin primality test to it. If the number passes the test, it is likely prime (the Miller-Rabin test is a probabilistic test, but it can be made deterministic for all numbers less than 3,317,044,065,679 by testing against the base values 2, 3, 5, 7, and 11), and the program will increment a count of prime numbers.

5. At the end of the execution, the program will return the count of prime numbers, which is the answer to the problem.

> Note:

The Miller-Rabin test isn’t fully accurate for all large numbers, but with the right measures (as described above), it can function quite well for a problem of this magnitude. There are other tests for large prime numbers, like the AKS primality test, but they are quite complicated to implement and not necessarily faster.

However, it’s worth noting that for a number as large as 50,000,000, any calculation involving prime numbers will take a significant amount of computation time and are typically conducted with parallel computing or on high-performance hardware.

General algorithms and programmatic concepts are commonly tackled in computer science courses, so if this is a field you’re hoping to go into, problems like these are great practice for building efficiency and problem-solving skills. But if you’re looking for the specific number of primes which satisfy the given condition for n ≤ 50,000,000, you would need a high computational computer to perform the task.

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