diff --git a/papers/fermat/fermat.tex b/papers/fermat/fermat.tex index 2fc11c8..6ec928b 100644 --- a/papers/fermat/fermat.tex +++ b/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}