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Robin P. Clark 2015-07-07 12:00:17 +01:00
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papers/fermat/fermat.tex
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@ -132,7 +132,7 @@ 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 $a^n+b^n = c^n$.
% %
\subsubsection{trivial example, single prime factor preserved} \subsubsection{Trivial example, single prime factor preserved}
% %
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
so adding them should result in a number which when factored contains none of the primes in $182$ and $2365$. so adding them should result in a number which when factored contains none of the primes in $182$ and $2365$.
@ -157,7 +157,7 @@ So, $ 322 + 49665 = 49987$: $bpf(49987) = \{7,37,193\}$.
As expected the common prime factor,7, exists in the result of the addition As expected the common prime factor,7, exists in the result of the addition
but the uncommon prime factors have disappeared. but the uncommon prime factors have disappeared.
\subsubsection{trivial example, multiple prime factor preserved} \subsubsection{Trivial example, multiple prime factor preserved}
Consider a repeated prime factor (i.e. a prime $p$ to the power $t$ $p^t$). The same rules apply. Consider a repeated prime factor (i.e. a prime $p$ to the power $t$ $p^t$). The same rules apply.
% %
@ -357,7 +357,7 @@ $$\prod \{2,2\} + \prod \{3,3,5,5,13,13,17,17\} = \{10989229\};$$
%$ans = 10989229 %$ans = 10989229
gives a larger prime number 10989229 and so on. gives a larger prime number 10989229 and so on.
\subsection{fishing with cubic primes} \subsection{Fishing with cubic primes}
Fishing with primes cubed reveals larger primes quicker: take Fishing with primes cubed reveals larger primes quicker: take
$$ \prod \{2,2,2,5,5,5,11,11,11\} + \prod \{3,3,3,7,7,7,13,13,13\} = \prod \{ 383 , 56599 \} \; ,$$ $$ \prod \{2,2,2,5,5,5,11,11,11\} + \prod \{3,3,3,7,7,7,13,13,13\} = \prod \{ 383 , 56599 \} \; ,$$
Adding 56599 as a single uncommon factor; Adding 56599 as a single uncommon factor;