Quick check on Korstelt's Criterion

  • Thread starter smithna1
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In summary, for a Carmichael number N that is square-free, if p is a prime that divides N, then (p-1) must divide (N-1), as shown by the fact that a^(N-1) ≡ 1 (mod p) for a generator a of Z*p.
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smithna1
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Homework Statement


N is a Carmichael number. It is known that N is square free, show tha tif p is prim and p|N, the (p-1)|(N-1).


Homework Equations


a^N ≡ a (mod N)



The Attempt at a Solution


I want to know if this is good enough or if there is a cleaner way to state it. Thanks!

Let a be a genrator of Z*p, so a has order (p-1). Now (p|N)|a(a^(N-1)-1) but not p|a, so a^(N-1)=1 mod p, hence (N-1) must be divisible by (p-1), the order of a mod p. QED
 
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  • #2



Your attempted solution is correct and concise. Here are some suggestions for improvement:

1. Use proper mathematical notation: Instead of using words like "divisible" and "mod," use the appropriate symbols (| and ≡) to make your solution clearer and more organized.

2. Define all symbols and variables: It would be helpful to define what N, p, a, and Z*p represent before using them in your solution. This will provide clarity and help the reader understand your thought process.

3. Provide a brief explanation: While your solution is correct, it may be helpful to provide a brief explanation of why N-1 must be divisible by p-1. This will help the reader understand the logic behind your solution.

Here is a revised version of your solution incorporating these suggestions:

Let N be a Carmichael number and p be a prime such that p|N. Let a be a generator of the multiplicative group Z*p. Since a has order (p-1), we have a^(p-1) ≡ 1 (mod p). Using the fact that N is square-free, we can write a^N ≡ a (mod N). Therefore, (a^(N-1)-1) ≡ 0 (mod p). However, since p does not divide a, we can cancel it from both sides to obtain a^(N-1) ≡ 1 (mod p). This implies that (N-1) is divisible by (p-1), the order of a in Z*p. Hence, (p-1)|(N-1). QED.
 

1. What is Korstelt's Criterion?

Korstelt's Criterion is a statistical test used to determine whether a set of data follows a normal distribution. It is based on the comparison of the sample kurtosis (measure of the "peakedness" of a distribution) to the kurtosis of a normal distribution.

2. How is Korstelt's Criterion calculated?

To calculate Korstelt's Criterion, the sample kurtosis is divided by the standard error of kurtosis, which is calculated using the sample size. The resulting value is then compared to a critical value determined by the sample size and desired level of significance.

3. What is the purpose of using Korstelt's Criterion?

The purpose of using Korstelt's Criterion is to determine whether a set of data can be considered normally distributed, which is important in many statistical analyses. If the calculated value is less than the critical value, the data can be assumed to follow a normal distribution.

4. What are the assumptions of Korstelt's Criterion?

The assumptions of Korstelt's Criterion include a random sample, a continuous variable, and a large enough sample size (usually at least 20). It also assumes that the data is normally distributed and that there are no outliers present.

5. What are the limitations of Korstelt's Criterion?

Korstelt's Criterion is sensitive to sample size and can give different results for different sample sizes. It also assumes that the data is normally distributed, which may not always be the case. Additionally, it does not detect all types of non-normality, such as skewness. Therefore, it should be used in conjunction with other tests to assess the normality of a distribution.

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