How to motivate the study of Fermat's Little Theorem

In summary, the conversation is about introducing Fermat's Little Theorem (FLT) to first year undergraduate students who are taking an elementary number theory course. The speaker is looking for a way to motivate students to learn this theorem and is also interested in learning about the applications of FLT. They inquire about resources and mention the use of FLT in RSA and Shor's algorithm. The conversation also touches on the importance of large primes in RSA for security purposes.
  • #1
matqkks
285
5
What is the best way to introduce Fermat’s Little Theorem (FLT) to students?

What can I use as an opening paragraph which will motivate and have an impact on why students should learn this theorem and what are the applications of FLT? Are there any good resources on this topic?
 
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  • #2
Who are the students and what do they study? This term is not very specific in American English.
 
  • #3
First year undergraduate doing an elementary number theory course and this is there first proof based course.
 
  • #4
How about a test for prime numbers needed for RSA? Class field theory would probably be a bit early.
 
  • #5
Okay I will have to look up RSA. The only thing I was aware of was that it used to large primes to make a product n=pq and security is dependent on factorising this large product. Thanks for this.
 
  • #6
matqkks said:
Okay I will have to look up RSA. The only thing I was aware of was that it used to large primes to make a product n=pq and security is dependent on factorising this large product. Thanks for this.
I think Shor's algorithm uses little Fermat and yes, RSA needs large primes, so I'm sure Shor is already too slow. But it's a start and a reason for why the primes have to be large! Probability algorithms based on ERH are likely a bit over the edge.
 

1. What is Fermat's Little Theorem?

Fermat's Little Theorem is a mathematical theorem that states that if p is a prime number, then for any integer a, a^p - a is divisible by p. In other words, it shows a special relationship between prime numbers and their powers.

2. Why is it important to study Fermat's Little Theorem?

Studying Fermat's Little Theorem can help us understand the properties of prime numbers and their relationship with other numbers. It also has many applications in cryptography and number theory, making it an important concept to understand for further research in these fields.

3. How can Fermat's Little Theorem be applied in real life?

Fermat's Little Theorem has many real-world applications, particularly in cryptography. It is used in encryption algorithms to ensure the security of data and information. It also has applications in coding theory, which is used in telecommunications and computer science.

4. What are some common misconceptions about Fermat's Little Theorem?

One common misconception is that Fermat's Little Theorem can be applied to all numbers, when in fact it only applies to prime numbers. Another misconception is that it can be used to prove the primality of a number, when in reality it is just a necessary but not sufficient condition for primality.

5. How can one motivate the study of Fermat's Little Theorem?

One way to motivate the study of Fermat's Little Theorem is by highlighting its practical applications in cryptography and coding theory. Another approach is to emphasize its historical significance and the contributions of mathematicians like Pierre de Fermat in developing this theorem. Additionally, exploring the beauty and elegance of this mathematical concept can also serve as a motivation for studying it.

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