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Robin P. Clark 2015-06-29 11:42:58 +01:00
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22
papers/fermat/fermat.tex
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@ -64,9 +64,9 @@ The numbers $a$ and $b$ may have common and uncommon prime factors; these can b
three 'bags', those only in $a$; $ubpf(a)$, those only in $b$; $ubpf(b)$ and those common to both; $cbpf(a,b)$.
A `Set' in mathematics is a collection of objects that may have only one of each type of element.
A `bag' is similar to a Set, except that it may have duplicates.
Thus the number $32$ is represented as the product of a bag of prime numbers thus: $\prod \{2,2,2,2,2\}$ i.e. $2^5 = 32$.
The number $32$ is represented as the product of a bag of prime numbers thus: $\prod \{2,2,2,2,2\}$ i.e. $2^5 = 32$.
Viewing the addition of $a^n +b^n$ as products of bag of prime factors:
Viewing the addition of $a^n +b^n$ as products of bags of common and uncommon~prime~factors:
\begin{equation}
\label{eqn:primesexpanded0}
\prod{cbpf(a,b)}^n \prod{ubpf(a)^n} + \prod{cbpf(a,b)}^n \prod{ubpf(b)^n} = c^n \; ,
@ -84,15 +84,14 @@ this can be re-written as:
\section{Properties of numbers viewed as products of bags of prime factors}
\subsection{ Primes are guaranteed not preserved in addition }
Only common prime factors are guaranteed preserved as a result of equation~\ref{eqn:primesexpanded1}.
%
%
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 the prime factors in the result.
Addition of primes causes the highest prime factors to become
lost, but increases the number of smaller prime factors.
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.
%
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 $64=2^6$.
@ -100,7 +99,10 @@ a bag of six twos i.e. $bpf(64) = \{2,2,2,2,2,2\}$ or more conventionally $64=2^
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.
%
Thus only common prime factors in $a$ and $b$ are preserved
as a result of equation~\ref{eqn:primesexpanded1}.
%
\subsection{conditions for having a integer root}
@ -197,7 +199,7 @@ a power $p^t$ by addition is to add proportions that add up to one $p^t$.
This means both a and b must contain this prime factor {\em in some proportion}
so that $a p^{t+n} + b p^{t+n} = p^{t+n} $ satisfy the highest prime in $c$.
In order for this to be true $a$ and $b$ must both fractions of a whole number.
In order for this to be true $a$ and $b$ must both be fractions of a whole number.
\subsection{trivial case}
@ -209,7 +211,7 @@ $$ a^n + b^n = 7^n = 49 $$
In order to get the prime factor 7 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 result.
Any other number will not give a 7 in the bag of prime numbers representation of the result.