better description of why some primes dissappear under addition

papers/fermat/fermat.tex

@@ -100,26 +100,40 @@ this can be re-written as:
 %
 % ADDITION DESTROY UNCOMMON PRIME FACTORS
 %
-This means for $a^n+b^n$ the only prime factors guaranteed to be in $c^n$
-are are those common in $a$ and $b$.
-%
 Adding numbers creates a `dissolving' of  prime factors in the result:
 that is addition of numbers causes the uncommon prime factors to become
-lost, but increases the number other prime factors.
+lost.
 %
 Consider $43 +21 = 64$. These primes add up to a result with 
 a bag of six twos i.e. $bpf(64) = \{2,2,2,2,2,2\}$ or more conventionally $64=2^6$.
+%%
+Prime numbers are unique. Adding to them, or adding other prime numbers to them, takes that unique 
+property away.
 %
 If a prime is added to another prime number the result
 cannot be a prime number, simply because all prime numbers above two are odd; 
 the result of the addition must even and therefore have at least a prime factor of two.
 %
-Prime numbers are unique. Adding to them, or adding other prime numbers to them, takes that unique 
-property away.
+Further, if numbers are added, the prime  factors of the 
+result will not contain any of the uncommon primes.
+That is the only prime factors preserved in the result of addition of $a$ and $b$ 
+are the common ones, i.e. cbpf(a,b). 
+%
+%consider
 %
 Thus only common prime factors in $a$ and $b$ are   preserved 
 as a result of equation~\ref{eqn:primesexpanded1}.
+This is simply because in addition
+the common prime factors can be extracted, $a+b \equiv \prod bfp(a) + \prod bfp(b)$
+re-writing $\prod cbpf(a,b) \big( \prod ubpf(a) + \prod ubpf(b) \big)$:
+This means the uncommon prime factors of $\big( \prod ubpf(a) + \prod ubpf(b) \big)$
+are lost and the $\prod cbpf(a,b)$ preserved.
 %
+%
+%
+This means for $a+b$ and $a^n+b^n$ the only prime factors  preserved (i.e. in $c^n$)
+are those common to $a$ and $b$.
+
 
 \subsection{Conditions for having a integer root}