Is Any Carmichael Number Divisible by a Perfect Square Greater Than 1?

  • Thread starter CornMuffin
  • Start date
  • Tags
    Square
In summary, the conversation discusses the task of proving or disproving that no Carmichael number is divisible by a perfect square greater than 1. It also mentions that a number is called a Carmichael number if it satisfies Fermat's little theorem, which applies to all prime numbers but can also apply to composite numbers known as Carmichael numbers. The suggested approach is to consider the simplest case where a=p^2 and try to find a counterexample.
  • #1
CornMuffin
55
5

Homework Statement


Prove or disprove (and salvage if possible):
No Carmichael Number is divisible by a perfect square > 1


Homework Equations


A composite number n is called a Carmichael number if and only if [tex]a^{n-1} \equiv 1 (mod \ n)[/tex] for all [tex]2\leq a \leq n-1[/tex] such that gcd(a,n) = 1

Carmichael numbers comes from fermat's little theorem that states that all prime numbers have this property. Carmichael numbers are numbers that have this property but are composite.


The Attempt at a Solution


I have been trying to figure out a way to do this problem for awhile, but no luck
 
Physics news on Phys.org
  • #2
What if there exists a prime p such that p2 divides a?

You might want to draw some inspiration from the simplest case (what if a=p2)
 

FAQ: Is Any Carmichael Number Divisible by a Perfect Square Greater Than 1?

What is a Carmichael number?

A Carmichael number is a positive composite integer that satisfies the condition that a^(n-1) ≡ 1 (mod n) for all integers a coprime to n, where n is the Carmichael number.

Why is it important to prove or disprove that no Carmichael number is divisible by a perfect square > 1?

This is important because Carmichael numbers have a unique property that makes them difficult to distinguish from prime numbers, which can lead to errors in certain mathematical computations. By proving or disproving this statement, we can better understand the properties of Carmichael numbers and improve our computational methods.

What is a perfect square?

A perfect square is an integer that is obtained by squaring another integer. For example, 25 is a perfect square because it is equal to 5².

How can we prove or disprove that no Carmichael number is divisible by a perfect square > 1?

This can be proven by using mathematical induction and the fundamental theorem of arithmetic. By assuming that there exists a Carmichael number that is divisible by a perfect square > 1, we can show that this leads to a contradiction, thus proving that no such Carmichael number exists.

Are there any known Carmichael numbers that are divisible by a perfect square > 1?

No, there are no known Carmichael numbers that are divisible by a perfect square > 1. This has been extensively studied and proven by mathematicians, and any potential counterexamples would have been discovered by now.

Similar threads

MHB Questions related to Carmichael number
Replies
1
Views
1K
Every integer n>1 is the product of a square-free integer?
Replies
5
Views
3K
The only prime p for which 3p+1 is a perfect square is p=5?
Replies
15
Views
3K
Easy (but not to me) Modern Algebra Proof
Replies
2
Views
2K
Prime factors of odd composites
Replies
6
Views
1K
I do not understand this proof
Replies
3
Views
2K
If ## p\neq5 ## is an odd prime, prove that either ## p^{2}-1 ## or....
Replies
15
Views
3K
Carmichael Numbers: Prove Product of Primes is a Carmichael Number
Replies
5
Views
2K
Proving 6601 is a Carmichael Number
Replies
1
Views
4K
Prove ##a \cdot 1 = a = 1 \cdot a## for ##a \in \mathbb{N}##
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top