Seeking Efficient Prime Identification Methods for Constrained Numerical Ranges

With precision in my inquiry, I am interested to learn whether there is any mathematical framework, method, or algorithm capable of efficiently identifying all prime numbers around a specific numerical value, such as 125,468,923, within a predetermined range of [-10,000, +10,000]. This approach would ideally bypass the exhaustive testing of every candidate within this interval, utilizing a more strategic mechanism for isolating and confirming primes. Insights into such methods are earnestly sought.

 

Over the past two months, I've developed a method for efficiently identifying prime numbers of any size within any specified range. This method draws inspiration from a novel interpretation of Fermat's Last Theorem, offering a fresh perspective on identifying primes. I invite anyone interested to verify the effectiveness of this approach by testing the numbers generated. In addition to this framework, I've created two other distinct methods tailored to different prime-searching needs—whether it's locating primes within a specific range or predicting subsequent primes. Based on my research and understanding, these methods are pivotal for advancing our ability to study prime numbers.
In recent explorations around Mersenne primes, I focused on M(4423), which is a prime number with 1332 digits. My method targets a range from -10004 to +10005 around the given Mersenne prime. The chosen range and method, although not elaborated here due to its complexity, demonstrate a unique approach to prime number discovery that no traditional sieve method can achieve in a short time frame of 270 seconds.

Enter a value for a: [Mersenne prime value with 1332 digits]
Enter a value for b: 2
Enter a value for n: 1
Enter a value for m: 1

The computed value is: [computed value]
[c] is not a prime number. Checking for near primes...
Total primes found: 14
Execution time: 269.095 seconds.
This demonstration showcases the capability of my method to efficiently identify prime numbers in a large-scale setting. This innovative methodology holds significant potential for computational prime number research, offering rapid insights into the prime distribution near colossal numbers, such as Mersenne primes. With sufficient computational power, this approach can be applied universally to any number, without exception. This universality underscores its potential to revolutionize our understanding of prime distribution.
P.S: Unlike traditional sieves or other exhaustive methods, this technique does not test every number in the vicinity. It strategically isolates prime candidates, enhancing efficiency significantly.
If anyone is interested in testing this method, please provide a random number with up to 1000 digits. Using the specified range, I will compute and provide the execution time along with the output of the primes found. This will allow you to verify the efficiency and accuracy of the method firsthand.

 

One efficient method for identifying prime numbers within a specific numerical range is the Sieve of Eratosthenes. Although traditionally used to find all primes up to a certain limit, it can be adapted to find primes within a given interval by starting the sieve at the lower bound of the interval and marking multiples of primes up to the upper bound.

Here's a basic outline of how you can adapt the Sieve of Eratosthenes for this purpose:

Create a boolean array of size equal to the range you want to examine (in this case, 20,001 elements for the range [-10,000, +10,000]).

Initialize all elements of the array to true, indicating that all numbers are initially considered potential primes.

Start with the first prime number, 2, and mark all of its multiples within the specified range as non-prime.

Move to the next number (3), if it's marked as prime, mark all of its multiples as non-prime.

Repeat step 4 until you've processed all numbers up to the square root of the upper bound of your range.

The remaining unmarked numbers within your specified range are prime.

This method significantly reduces the number of divisions needed compared to traditional trial division, making it more efficient for larger ranges. However, note that for very large ranges, more advanced algorithms like the Sieve of Atkin or the Sieve of Sundaram might be more suitable.


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