code in ada with Ada.Numerics.Generic_Elementary_Functions;package body Number_Theory is -- Instantiate the library for floating point math using Floating_Type. package Floating_Functions is new Ada.Numerics.Generic_Elementary_Functions(Floating_Type); use Floating_Functions; function Factorial(N : in Factorial_Argument_Type) return Positive is begin -- TODO: Finish me! -- -- 0! is 1 -- N! is N * (N-1) * (N-2) * ... * 1 return 1; end Factorial; function Is_Prime(N : in Prime_Argument_Type) return Boolean is Upper_Bound : Prime_Argument_Type; Current_Divisor : Prime_Argument_Type; begin -- Handle 2 as a special case. if N = 2 then return True; end if; Upper_Bound := N - 1; Current_Divisor := 2; while Current_Divisor < Upper_Bound loop if N rem Current_Divisor = 0 then return False; end if; Upper_Bound := N / Current_Divisor; end loop; return True; end Is_Prime; function Prime_Counting(N : in Prime_Argument_Type) return Natural is begin -- TODO: Finish me! -- -- See the lab page for more information. return 0; end Prime_Counting; function Logarithmic_Integral(N : in Prime_Argument_Type) return Floating_Type is begin -- TODO: Finish me! -- -- See the lab page for more information. return 1.0; end Logarithmic_Integral;end Number_Theory; 2. Number theory is a branch of mathematics that concerns itself with the properties of the integers. One function of considerable interest in number theory is the prime counting function. It is traditionally given the name n (but it hasnothing to do with circles). For example n(6) = 3 because there are three prime numbers less than or equal to 6 (namely, 2, 3, and 5).Start by adding a function to package Number_Theory that computes n(n) for positive values n greater than or equal to 2. Note that the package already has a function Is_Prime that you will no doubt find useful. You might also want tomake use of the defined subtype Prime_Argument_Type. Ada allows you to use Greek letters in variable names, but I suggest using the name Prime_Counting for n instead.3. Modify the main file main.adb to exercise your function (ask the user to input a value n and then output (n)). Here are some values of r(n) you can check.n(n)104100251_0001681_2299 59278_498664_579100_000_000 5_761_4551_000_000_000 50_847_53410_000100_0001_000_00010_000_0004. Computing r(n) exactly can be time-consuming, especially for large values of n (note: you should be able to double the speed of your implementation by skipping even numbers.. don't forget to handle 2 as a special case). It turns outthere is an approximation formula for computing n(n) that is much faster to calculate. It entails evaluating an infinite series that uses the natural logarithm function. Here is the series:v + In(1n(n)) + In²(n)/(1*1!) + 1n²(n)/(2*2!) + 1n³(n)/(3*3!) + ...Here y (gamma) is Euler's constant and has the value (approximately):v = 0.57721_56649_01532_86060_65120_90082_48243_10421_59335_93992smaller. This series isAlthough the series has infinitely many terms you don't need to add them all because the terms get smalleroughly equal to li(n).I the "logarithmic integral function" and goes by the name li(n). It is an amazing fact that n(n) isAdd a function to your package Number_Theory that computes li(n). Call your function Logarithmic_Integral.The package Ada. Numerics.Generic_Elementary_Functions contains a function Log which computes the natural logarithm (shown as 'In' in the formula above). Note that the logarithmic integral is computed using floating pointnumbers, so you'll need some type conversions to go back and forth between integers and floating point values. For example if N is an integer of some kind, you can convert its value to the floating point type declared in Number_Theoryby doing Floating_Type(N). This package contains subprograms for doing number theoretic computations.It is used in VTC's CIS-2730, Lab #2..M4 v package Number_Theory isConstant Definitions..7Gamma : constant := 0.57721_56649_01532_86060_65120_90082_40243_10421_59335_93992;1011Type Definitions1213The range of values for which N! can be computed without overflow.subtype Factorial_Argument_Type is Integer range 0 .. 12;141516-- The range of values that might meaningfully be asked: are you prime?subtype Prime_Argument_Type is Integer range 2 .. Integer'Last;171819A floating point type with at least 15 significant decimal digits.type Floating_Type is digits 15;2021222324Subprogram Declarations252627Returns N!28function Factorial(N : Factorial_Argument_Type) return Positive;29Returns True if N is prime; False otherwise.function Is_Prime(N : in Prime_Argument_Type) return Boolean;32Returns the number of prime numbers less than or equal to N.function Prime_Counting(N : in Prime_Argument_Type) return Natural;333435-- The logarithmic integral function, which is an approximation of the prime counting function.function Logarithmic_Integral(N : in Prime_Argument_Type) return Floating_Type;36373839end Number_Theory;123 tn ONmmm mmm mm mm

with Ada.Numerics.Generic_Elementary_Functions; package body Number_Theory is -- Instantiate the library for floating point math using Floating_Type. package Floating_Functions is new Ada.Numerics.Generic_Elementary_Functions(Floating_Type); use Floating_Functions; function Factorial(N : in Factorial_Argument_Type) return Positive is begin -- TODO: Finish me! -- -- 0! is 1 -- N! is N * (N-1) * (N-2) * ... * 1 return 1; end Factorial; function Is_Prime(N : in Prime_Argument_Type) return Boolean is Upper_Bound : Prime_Argument_Type; Current_Divisor : Prime_Argument_Type; begin -- Handle 2 as a special case. if N = 2 then return True; end if; Upper_Bound := N - 1; Current_Divisor := 2; while Current_Divisor < Upper_Bound loop if N rem Current_Divisor = 0 then return False; end if; Upper_Bound := N / Current_Divisor; end loop; return True; end Is_Prime; function Prime_Counting(N : in Prime_Argument_Type) return Natural is begin -- TODO: Finish me! -- -- See the lab page for more information. return 0; end Prime_Counting; function Logarithmic_Integral(N : in Prime_Argument_Type) return Floating_Type is begin -- TODO: Finish me! -- -- See the lab page for more information. return 1.0; end Logarithmic_Integral; end Number_Theory;

7th Edition

ISBN:9780133594140

Author:James Kurose, Keith Ross

Publisher:James Kurose, Keith Ross

Chapter1: Computer Networks And The Internet

Section: Chapter Questions

Problem R1RQ: What is the difference between a host and an end system? List several different types of end...