Checking a Proof of a Theorem

Theorem 1.2 of Bennett and Skinner (Canad. J. Math., 2004) asserts that the Diophantine equation xp4yp=z2 is unsolvable for every prime p7.
The following is a possible proof (from an arXiv author) that Fermat’s Last Theorem is a consequence of this theorem, i.e., proof that if there exist integers x,y,z>0 such that (x,y)=1 and xp+yp=zp, then there exist integers a,b,c>0 such that (a,b)=1 and ap4bp=c2.

Take a prime
p7.
We will prove Fermat’s Last Theorem in the form:
Take integers x,y,z>0.
If
(x,y)=1,
then
xp+ypzp.

We argue by contradiction.
By the equation
xp+yp=zp
there is a rational
0<r<1
such that
xp=rzp  and  yp=(1r)zp,
so that
r2r+(xy)pz2p=0,
and hence
r=1+14(xy)p/z2p2  or  114(xy)p/z2p2.
Therefore,
the difference
14(xy)p/z2p0
is to be a perfect square.
But since
14(xy)pz2p=z2p4(xy)pz2p,
since
z2p
is a perfect square,
and since if z2p=4(xy)p
then from the equation
xp+yp=zp
we have
x=y
that leads to a contradiction,
so there is an integer
c>0
such that
z2p4(xy)p=c2.
On choosing
a:=z2  and  b:=xy
we have
a,b>0  and  ap4bp=c2.
Moreover,
because
(x,y)=1
and
xp+yp=zp,
we have
(x,y)=(y,z)=(x,z)=1,
whence
(a,b)=1.
But the existence of such
a,b,c
contradicts Theorem 1.2 of Bennett and Skinner [1].

Answer

Your proof looks OK to me. But don't rush to publish it seems to me that Theorem 1.2 (available at this link) depends on the Taniyama–Shimura–Weil conjecture (or the modularity theorem, as it should now be called), just like Wiles' proof of Fermat's Last Theorem did.

So your proof is roughly speaking using a consequence of Fermat's Last Theorem to prove Fermat's Last Theorem.

Attribution
Source : Link , Question Author : Megadeth , Answer Author : TonyK

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