How Does Prime Number Distribution Aid in Integer Factorization?

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Understanding prime number distribution can enhance integer factorization by providing insights into algorithm efficiency and reliability. While it may not directly aid in the factorization process, it can help establish the effectiveness of certain algorithms, ensuring they work consistently rather than sporadically. Additionally, knowledge of prime distribution can lead to improved bounds on algorithm runtime, potentially optimizing performance. This relationship highlights the importance of prime numbers in computational number theory. Overall, prime distribution plays a crucial role in refining factorization techniques and algorithm development.
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I've been Googleing for days now and haven't found a suitable answer to a question I have so I'll try it here. How exactly would knowing the distribution of prime numbers assist one in integer factorization?
 
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IIRC, knowing the distribution wouldn't help one factor integers; it would allow one to prove that certain algorithms will work all of the time instead of a lot of the time, or it might allow one to devise a better bound on how long an algorithm has to run... things like these.
 
Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?

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