May 21st, 2015, 09:43 AM  #21 
Senior Member Joined: May 2013 Posts: 118 Thanks: 10 
First you can take for granted that x,y,z are pairwise coprime and that every prime factor of x+y is also a prime factior of z. The equations in question were given by Barlow in 1810. He proved that if $\displaystyle x^n+y^n=z^n$ (1), then the equations all hold.The opposite is true too and much easier to prove.So the equations are a necessary and sufficient condition for (1).So don't waste your time trying to prove or disprove them As for the $\displaystyle n^mx+yz$ stuff, that's what we're working on,with limited succes, we must confess 
May 26th, 2015, 09:23 AM  #22  
Senior Member Joined: Apr 2015 From: Barto PA Posts: 170 Thanks: 18  Quote:
be considered coprime. Here is the issue: If _every_ prime factor of x + y is also a prime factor of z then x + y is clearly some factor of z; hence z = g(x + y), where g > 0 is the product of all other  if any  factors of z. Now we can write x^n + y^n = z^n = [g(x + y)]^n. Claim 1 proved this is a contradiction if n > 1 even if g = 1. Claim 1 did not depend on x, y, z being coprime  that is totally irrelevant. Here is what is relevant: A proved claim can NOT be contradicted. Let's examine Barlow's equations based on your statement that every prime factor of x + y must also be a prime factor of z: x + y = p^n, Q(x,y) = q^n, z = pq. According to your claim, if every prime factor of x + y is nothing more than p and q in this case then x + y = pq, and if x + y = p^n then p^n = pq, from which it follows that q must divide p. And if p and q are supposed primes Then p = q is necessary. Then z = p^2 = x + y. Then z^n = p^(2n) = (x + y)^n, again contradicting Claim 1. Also, the exponent of z is now some even integer, and the exponent of both x and y must necessarily be the same integer. The argument considers just odd integers > 1. x + y can not be factored from x^n + y^n if n is even. The only conclusion is that the equations can not hold simultaneously as you claimed if your assertion that every prime factor of x + y is also a prime factor of z is true.  
May 26th, 2015, 01:59 PM  #23 
Senior Member Joined: May 2013 Posts: 118 Thanks: 10 
You say: 1."It was never an issue whether x, y, z can be considered coprime." If we talk about the equations,they were proven on the assumption x,y,z being coprime,as you have hinted in #20 post.If you assume that two of x,y,z share a common factor,that leads to contradiction in seconds 2."If _every_ prime factor of x + y is also a prime factor of z then x + y is clearly some factor of z " Not necessarily.Take,e.g.$\displaystyle x+y=2^2*3$ and z=2*3*5 3."Let's examine Barlow's equations based on your statement that every prime factor of x + y must also be a prime factor of z" It's not my claim.It's true because $\displaystyle z^n=(x+y)Q(x,y)$ .So what holds is that x+y is a factor of $\displaystyle z^n$ ,not z. 4."And if p and q are supposed primes" Arbitrary. p and q are just coprime 

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