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User Generated
Subject
MathLab
School
University of Alabama at Birmingham
Type
Homework
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Question 1:
Problem Statement:
Submit a positive integer less than 1000… yes, just one whole number
greater than zero and less than 1000! The person who submits the lowest
unique number (X) among the entire class receives an extra (X/Z)*1000
points on this HW! (Z is the sum of all numbers submitted by the class)
Explanation:
It’s a game theory problem.
Assume that there is only one student in the class, the total score student
will have is 1000. In case when the student increase to 2, probability of
winning each of them is reduces to ½ and the resultant score varies based
on the value each of the student submitted. So, if Student one put the value
as 1 while the second student submit the unique number as 2, then 1
st
student will have 333 score and the student 2 will not achieve any score.
Meanwhile, if student try to improve results by increasing the value of the
number than the overall sum of number also increases which leads to
decrement in overall value. Consider in case of 3 student, realizing all three
cant submit the lowest value of 1 so if one student intentionally increase
the value to 2 instead of 1 than that student is only helping the opponent
to win. Perhap’s the best strategy is to go with Prisoner’s Dilemmawhich
means all should submit 1 and all gets nothing. It’s a discussion, I chose
my number as 12.
Question 2:
Problem Statement:
One formula that can be used to generate prime numbers (many, but not
all) is
    where x is a prime number and n is an integer.
Plugging in consecutive integers 1, 2, 3...etc., up to a point, will generate

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prime numbers. Give the smallest value of n where this equation fails to
produce a prime number. For the next 100 values of n, does the formula
continue to produce primes? Use Matlab to automate the process and check
for primes. Explain your methods.
Explanation and results:
The given equation
    is a famous Eulers equation to predict
the prime number. As per the prime number theorem, for a large enough
value of N, the probability that a random number chosen in the interval
between 1 to N is

. Additionally, the frequency of availability of prime
numbers reduces as the value of N increases.
The Eulers Equation predicts the prime number successfully for every value
of N in the interval 0 to 39. Thus, it produces 40 prime numbers successfully
before give a composite number at N=40 which is 1681 = 41 x 41.
In MATLAB, the Euler’s equation in tested using MATLAB’s inbuilt function
isprime (). isprime() returns binary output as 1 if the number is prime
number else returns 0. For example :
  
   

  
 Moreover,the
equation is further tested using for loop for 100 succeeding numbers. The
results shows that out of 100 instances the equation correctly predicts 74
further instances and fail to predicts the accurate prime number in 26
instances. The values are as shown:
The detected prime numbers are: 1847 1933 2111 2203 2297 2393 2591
2693 2797 2903 3011 3121 3347 3463 3581 3701 3823 3947 4073 4201
4463 4597 4733 4871 5011 5153 5297 5443 5591 5741 6047 6203 6361
6521 7013 7351 7523 7873 8231 8597 8783 8971 9161 9547 9743 9941
10141 10343 10753 11171 11383 11597 11813 12251 12473 12697 12923

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Question 1: Problem Statement: Submit a positive integer less than 1000… yes, just one whole number greater than zero and less than 1000! The person who submits the lowest unique number (X) among the entire class receives an extra (X/Z)*1000 points on this HW! (Z is the sum of all numbers submitted by the class) Explanation: It’s a game theory problem. Assume that there is only one student in the class, the total score student will have is 1000. In case when the student increase to 2, probability of winning each of them is reduces to ½ and the resultant score varies based on the value each of the student submitted. So, if Student one put the value as 1 while the second student submit the unique number as 2, then 1st student will have 333 score and the student 2 will not achieve any score. Meanwhile, if student try to improve results by increasing the value of the number than the overall sum of number also increases which leads to decrement in overall value. Consider in case of 3 student, realizing all three cant submit the lowest value of 1 so if one student intentionally increase the value to 2 instead of 1 than that student is only helping the opponent to win. Perhap’s the best strategy is to go with Prisoner’s Dilemmawhich means all should submit 1 and all gets nothing. It’s a discussion, I chose my number as 12. Question 2: Problem Statement: One formula that can be used to generate prime numbers (many, but not all) is 𝑥 = 𝑛2 + 𝑛 + 41 where x is a pr ...
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