Are Fermat's Little Theorem and Wilson's Theorems Useful in Number Theory?

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Fermat's Little Theorem and Wilson's Theorem are historically significant in number theory, providing foundational insights despite their inefficiency for large numbers. While they are not practical for finding primes in large datasets, they serve as valuable educational tools for understanding number theory concepts. These theorems are effective for simpler, homework-style problems, making them accessible for learners. Their real-life applications vary based on the context, as they may not be the most efficient methods for large-scale computations. Overall, their relevance is context-dependent, highlighting their role in the evolution of mathematical understanding.
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What use are Fermat’s Little Theorem and Wilson’s theorems in number theory? Do these theorems have any real life applications? We cannot use them to find primes as both are pretty inefficient for large numbers.
 
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First, they show that it is possible to answer some of these questions - this was historically important, as our knowledge is based upon the cumulative results of the past.

Second, these are easier to understand than many more recent results, and are useful when you are learning how to attack these problems. For homework-type problems they work quite well.

This is a partial answer to your first question ... as to useful applications today, it depends upon the application! If you always work with very large numbers, then they would not be the most efficient technique. But it depends upon your application.
 
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