Homework Solution: (Number Theory) Assume that the Sieve of Eratosthenes needs roughly xln(ln(x)) "crossing out" operat…

    (Number Theory) Assume that the Sieve of Eratosthenes needs roughly xln(ln(x)) "crossing out" operations in order to generate a list of all primes less than x. Note that this is slightly different than the estimate shown in class. You have a powerful computer that can perform 9.2 trillion "crossing-out" operations per second. How many years does it take this computer to generate a list of primes less than 10^33? Be sure to fully explain your computations.

    Expert Answer

     
    The value of x = 10^{33}. Total number of "crossing out

    (Number Theory) Assume that the Sieve of Eratosthenes needs roughly xln(ln(x)) “crossing out” operations in command to breed a inventory of whole primes hither than x. Note that this is subordinately contrariant than the revere shown in assort.

    You possess a mighty computer that can transact 9.2 trillion “crossing-out” operations per promote. How divers years does it engage this computer to breed a inventory of primes hither than 10^33? Be secure to easily interpret your computations.

    Expert Vindication

     

    The treasure of x = 10^{33}.

    Total number of “crossing out” operations required to breed whole primes leq10^{33}

    x*ln(ln(x))
    10^{33}*ln(ln(10^{33}))
    4.330540007*10^{33}

    Since, Time needed to transact 9.2 trillion “crossing out” operations = 1 promotes

    So, Time needed to transact 1 “crossing out” operations

    = frac{1}{9.2 trillion} promotes =  frac{1}{9.2*10^{12}} promotes

    Thus, Time needed to transact 4.330540007*10^{33} “crossing out” operations

    = frac{4.330540007*10^{33}}{9.2*10^{12}} promotes = 4.707108703*10^{20} promotes
    = frac{4.707108703*10^{20}}{3600} hours     [1 hour = 3600 promotes]
    = frac{4.707108703*10^{20}}{24*3600} days     [1 day = 24 hours]
    = frac{4.707108703*10^{20}}{365*24*3600} years     [1 year = 365 days]
    = 1.492614378*10^{13} years