The integer 2896753 can be factored using Pollard's p-1 method, but a large factor base, specifically primes up to 31, is necessary for effective results. The quadratic sieve is generally reliable for numbers of this size, indicating potential issues with the current implementation being used. It is recommended to perform trial division up to the first ten thousand primes before attempting more complex methods like ECM. The discussion emphasizes the importance of the right approach and tools for successful integer factorization. Proper implementation and method selection are crucial for effective factorization of integers.