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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 Because of this property of addition of numbers in relation to preserved
prime factors, it can be used to make inferences on the equation $a^n+b^n = c^n$. 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} \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 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} \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 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. 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. Any other number will not give a 7 in the bag of prime numbers representation of the result.