Update README.md

README.md

 # Twin Smooth Integers
 
 Sieving for large twin smooth integers using single solutions to the Prouhet-Tarry-Escott (PTE) problem.
-Optimized for single solutions by doing the modulo computation first. Implemented in Sage.
+Optimised for single solutions by doing the modulo computation first.
 
-The Sieve of Eratosthenes marks all integers in an interval as smooth (1) or non-smooth (0).
-The Naive Approach gives a list of integers l that produce twin smooth integers a(l)/C,b(l)/C by checking every smooth integer in the interval.
-The Optimised Approach gives a list of integers l that produce twin smooth integers a(l)/C,b(l)/C by checking only those integers l such that a(l) and b(l) are 0 modulo C.
+Ideal PTE solutions provide polynomials a(x) and b(x) such that |a(x) - b(x)| = C is an interger.
 
-More details can be found in the paper ***comming soon***.
+To search for twin smooth primes (i.e. p + 1 and p - 1 are smooth) one can define a search interval I and use one of the following approaches.
+
+Naive approach:
+    For every element l in I:
+        if a(l) and b(l) are smooth:
+            if a(l) / C and b(l) / C are integers:
+                print l
+
+The tree approach by Costello, Meyer and Naehrig combined many PTE solutions and used common factors of the polynomials.
+
+Optimised approach (for single solutions):
+    For every element l in I:
+        if a(l) / C and b(l) / C are integers:
+            if a(l) and b(l) are smooth:
+                print l
+This decreases the number of look-ups for smoothness and allows to do a large part of the "modulo C" calculations in a pre-computation, that can be used for all search intervals.
+
+There is a Sage implementation for explanatory reasons and two C implementations with maximal variable sizes of 64 and 128 bit respectively. For explanation of the individual steps please see the commentary in the code.
+
+More details and references can be found in the paper ***comming soon***.