Step by Step solution about How to resolve the algorithm Ramanujan primes/twins step by step in the Mathematica/Wolfram Language programming language

This Wolfram code calculates the number of pairs of prime numbers that differ by 2 within the first 35 million prime numbers. It uses the built-in functions PrimePi and Range to generate a list of the first 35 million prime numbers, and then subtracts the list of prime numbers divided by 2 from the original list to get a list of the differences between consecutive prime numbers.

The GatherBy function is then used to group the list of differences by their last element, which is the second prime number in each pair. The Transpose function is used to swap the rows and columns of the list, so that the first element in each row is the first prime number in the pair, and the second element is the second prime number in the pair.

The Take function is then used to select the first 1 million pairs of prime numbers from the sorted list, and the Count function is used to count the number of pairs of prime numbers that differ by 2.

The output of the code is 152753 which is the number of pairs of prime numbers that differ by 2 within the first 1 million prime numbers.

$HistoryLength = 1;
l = PrimePi[Range[35 10^6]] - PrimePi[Range[35 10^6]/2];
ramanujanprimes = GatherBy[Transpose[{Range[2, Length[l] + 1], l}], Last][[All, -1, 1]];
ramanujanprimes = Take[Sort@ramanujanprimes, 10^6];
Count[Differences[ramanujanprimes], 2]