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Niel Harding talked about the classic right angle triangle 3^2+4^2=5^2

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Robin P. Clark 2015-07-07 15:12:35 +01:00
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papers/fermat/fermat.tex
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@ -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 $a^n+b^n = c^n$.
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%%% NIEL HARDINGS BIT
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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
$\prod \{3,3\} + \prod\{2,2,2,2\} = \prod \{5,5\}$.
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\subsubsection{Trivial example, single prime factor preserved}
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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:
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$$ a^n + b^n = 7^n = 49 \; . $$
$$ a^n + b^n = 7^n = 343 \; . $$
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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.