diff --git a/papers/fermat/fermat.tex b/papers/fermat/fermat.tex
index 3da77e9..5fc5c13 100644
--- a/papers/fermat/fermat.tex
+++ b/papers/fermat/fermat.tex
@@ -132,6 +132,12 @@ are lost and the $\prod cbpf(a,b)$ preserved.
 Because of this property of addition of numbers in relation to preserved
 prime factors, it can be used to make inferences on the equation $a^n+b^n = c^n$.
 %
+%%% NIEL HARDINGS BIT
+%
+For the classic right angle triangle $3^2+4^2=5^2$ the prime numbers in the 
+addition are not preserved. As products of bags of prime numbers this is 
+$\prod \{3,3\} + \prod\{2,2,2,2\} = \prod \{5,5\}$.
+%
 \subsubsection{Trivial example, single prime factor preserved}
 %
 Consider $bpf(182)=\{2,7,13\}$ and $bpf(2365)=\{5,11,43\}$ these have no common prime factors
@@ -315,11 +321,11 @@ Thus where $a$ and $b$ are $ > 1$; $a^n + b^n \neq c^n$ for whole numbers.
 
 \subsection{trivial case}
 
-Take the trivial case where $n=2$ and  $c$ has the prime number 7 as one of its prime~factors:
+Take the trivial case where $n=3$ and  $c$ has the prime number 7 as one of its prime~factors:
 %
-$$ a^n + b^n = 7^n = 49 \; . $$
+$$ a^n + b^n = 7^n = 343 \; . $$
 %
-In order to get the prime factor 7 in the result both a and b must have the prime number 7 in them.
+In order to get the prime factor $7^3$ in the result both a and b must have the prime number 7 in them.
 That is the numbers $a$ and $b$ must both have the number 7 as a common prime factor
 to get seven as a prime factor in the result.
 Any other number will not give a 7 in the bag of prime numbers representation of the result.