Ultimate Theorem

Sexy Primes, What are they? They are prime numbers that differ by 6. For example, 5 & 11,  7 & 13. The name comes from the Latin word for six; sex. What we are going to do is, write a program to generate all the sexy prime pairs within a given interval of natural numbers.  Sexy Prime Pairs This is a program to generate all the sexy prime pairs below 10000. Program def check_prime(n):     for i in range(2,n//2+1):         if n%i == 0:     return False     return True def primes_list(a,b):     primes_list = []     for i in range(a,b):         if check_prime(i):     primes_list.append(i)     return (primes_list) def sexy_list(primes_list):     sexy_list = []     for i in primes_list:         for j in primes_list:     if j-i == 6:         sexy_list.append((i,j))     return(sexy_...

 Monte Carlo Integration Monte Carlo integration is a  powerful method of computing the value of complex integrals using probabilistic techniques. This technique uses random numbers to compute the definite integral of a function. Here we are going to use the python programs written in the  previous post  to generate pseudorandom numbers and approximate the value of the definite integral of a function. Consider a function to be integrated, as shown below: That is we need to evaluate the definite integral $\int_{a}^{b} f(x)\mathrm{d}x$. The integral is just the area under the curve. The width of the interval $(b- a)$ times the average value of the function is also the value of the integral, that is, $\int_{a}^{b}f(x)\mathrm{d}x = (b - a)f_{average} = (b-a)\langle f \rangle$ So if we had some independent way of calculating the average value of the integrand, then we could evaluate the integral. This is where random numbers come in. Imagine that we had a list of random n...