What steps can I take to optimize this code for determining the greatest prime factor of 600851475143?

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Start by using trial division to find the smallest prime factor of the number, then divide the number by this factor and repeat until you reach the largest prime factor.

Use a more efficient algorithm for finding prime factors, such as the Pollard-Rho algorithm or the Sieve of Eratosthenes.

Optimize the loop for finding prime factors by only checking odd numbers, since even numbers cannot be prime except for 2.

Implement memoization to store previously computed values and avoid redundant calculations.

Use GMP (GNU Multiple Precision Arithmetic Library) to handle large numbers efficiently.

Parallelize the algorithm to take advantage of multi-core CPUs or distributed computing.

Consider using a specialized hardware accelerator, such as a Field-Programmable Gate Array (FPGA) or Graphics Processing Unit (GPU).

Rewrite the algorithm in a lower-level language such as C or assembly for more efficient execution.