Implement the formulas using Java Language

CS285

Discrete Mathematics Applications

Project  1

Consider a polynomial F(x)=x²+x+41. Let’s check its values for a few first integers: 

F(1)=43 which is prime. 

F(2)=47 which is also prime. F(3)=53, F(4)=61, F(5)=71, F(6)=83, F(7)=97, F(8)=113, F(9)=131, all of which are prime.

Is it right to conclude that F(x) is a prime for all integer x?

Let’s check a couple more values: F(10)=151 is a prime; F(11)=173 and F(12)=197 are both prime. However, it’s wrong to conclude that F(x) is prime for all integer x. In fact, F(40)=40⋅40+40+41=40⋅(40+1)+41=412. 

Still, it’s interesting that F(x) is prime for all integers from 1 through 39.

G(x)=x²−x+41 is prime for x from 0 through 40, and H(x)=x²−79x+1601 is prime for x from 1through 80. H(81)=41⋅43. 80 is a long run of primes indeed.

Objective:

• Perform a research on the most recent prime numbers generator formulas (at least 7 formulas)
• Implement the formulas using Java Language
• Test your formulas for the first 10 000 Numbers
• Show your results as follow:
  • Which formulas generate most of prime numbers
  • Which formulas provide the first non-prime number
  • Time to compute all prime numbers (up to 10 000 first numbers)
  •  

Your program should provide possibility to call via menu any of the above mentioned formulas, execute on the given datasets and provide:

What to submit:

• Report (4 pages maximum) containing results of different formulas on each datasets (template will be provided later on LMS)
  • Description of each formulas and it’s reference
  • Description of each code (details of class, methods, ..)
• Your Java code (documented code!!!!)
• PPT presentation  (no more than 6 slides)

How to work:

2 students per project

When to submit:

Date of submission:   6^th  December 2021

Date of presentation:  6^th  December 2021