added iterative fishing for primes method based on the addition of

uncommon factors removing primes in the result.

papers/fermat/fermat.tex

@@ -325,8 +325,46 @@ 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.
 
 
+\section{Further work}
 
+Using the rule of adding uncommon factors, an iterative `fishing' method for finding prime numbers can be applied.
+An addition is made,  and the lower prime factors of the result are added as uncommon factors in a subsequent addition
+until finally a larger single prime number is found.
+This can be done with single primes, or for finding larger primes in less iterations, with multiple powers of primes.
+For example consider
+%
+starting with the first two uncommon primes squared,
+$$\prod \{2,2\} + \prod \{3,3\} = \{13\};$$
+this gives the prime number 13.
+%octave:26> factor(2^2+3^2)
+%ans =  13
+%
+%
+%octave:28> factor(2^2+3^2*13^2)
+%ans =
+By placing 13 squared as an uncommon factor, more primes are shaken out:
+$$\prod \{2,2\} + \prod \{3,3,13,13\} = \prod \{5,5,61\}.$$
+% 
+Making the prime number 5 uncommon in the addition gives:
+$$\prod \{2,2\} + \prod \{3,3,5,5,13,13\} = \prod \{17,2237\};$$
+%octave:29> factor(2^2+3^2*13^2*5^2)
+%ans =
+%
+removing the 17 by making it an uncommon factor in the addition:
+   %  17   2237
+$$\prod \{2,2\} + \prod \{3,3,5,5,13,13,17,17\} = \{10989229\};$$
+%octave:30> factor(2^2+3^2*13^2*5^2*17^2)
+%$ans =  10989229
+gives a larger prime number 10989229 and so on.
 
+\subsection{fishing with cubic primes}
+Fishing with primes cubes reveals larger primes quicker: take
+$$ \prod \{2,2,2,5,5,5,11,11,11\} + \prod \{3,3,3,7,7,7,13,13,13\} = \prod \{ 383 ,  56599 \} \; ,$$
+Adding 56599 as a single uncommon factor;
+$$ \prod \{2,2,2,5,5,5,11,11,11\} + \prod \{3,3,3,7,7,7,13,13,13,56599\} = \prod \{ 151, 359, 1553, 13679 \} \; , $$
+Removing 359 reveals a large prime:
+$$ \prod \{2,2,2,5,5,5,11,11,11\} + \prod \{3,3,3,7,7,7,13,13,13,56599,359\} = \{413419682557097\} $$
+ 
 % 
 % \begin{equation}
 % \label{eqn:primesexpanded1}