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

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Fermat's Little Theorem (FLT) is efficient for large numbers, particularly in modular exponentiation and primality testing, such as the Fermat primality test. In contrast, Wilson's Theorem is considered less useful, with limited applications primarily in analytic contexts. While FLT has practical uses in number theory, Wilson's Theorem does not significantly contribute to finding primes. The discussion highlights the efficiency of FLT compared to Wilson's Theorem in real-life applications. Overall, FLT is a valuable tool in number theory, while Wilson's Theorem has restricted relevance.
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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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matqkks said:
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.

FLT is pretty *damn* efficient for large numbers, look up modular exponentiation. I agree about Wilson's, though, I don't think there are too many applications to it, but FLT certainly has applications in primality testing (aka the Fermat primality test, which essentially is about applying the FLT to possible primes using random bases) and in theorem proving.
 
FlT is generally useful in big modulo reductions; some theoretical use can also be found, like FLT for n = 5. Wilson's theorem doesn't have a whole lot of applications, but they are generally used for analytic purposes.
 
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