How Noncomputability Relates To Fermat's Last Theorem (revised)
How
Noncomputability
Relates
To
Fermat's
Last
Theorem
By David Williams
Abstract
Noncomputable
functions
can
seem
so
quixotically
remote
from
everyday mathematics
that
we
may
imagine
they
have
little
relevance
to
more
familiar domains
of
application.
That
this
isnt
true
will
be
demonstrated
here
using
the proofs
set
forward
by
Tibor
!ado
in
his
"#$%
article
in
The
Bell
&ystem Technical
'ournal

On
Noncomputable
functions.
(
will
apply
his
proofs
to
the
principles
governing
the
generation
of
)ythagorean triples
to
show
how
noncomputable
functions*
indeed
noncomputability
per se*
is
essential
to
the
logical
rigour
underpinning
+ermats
,ast
Theorem* proved
by
ndrew
Wiles
in
"##*
which states that
there
are
no
whole
number solutions
to
the
formula/
for
all
integer values
of
n0%
A
Brief
Summary
!ado
proved
that
there
are
welldefined
functions
which
grow
so
quic1ly
that
no explicit
formulation
of
them
is
possible.
To
aid
him
he
relied
upon
the
design
of a
rudimentary
Turing
2achine*
which
would
read
and
write
a
"
or
a
3
on
a potentially
infinite
tape*
then
shift
one
step
to
the
left
or
right
before
continuing or
stopping.
The
behaviour
of
each
machine
is
governed
by
a
number
of
states which
each
have
basic
rules*
e.g.
“
)rint
a
"
if
tape
read
3*
or
else
leave
it
and then
go
to
state
4
”
.
&tate
4
would
have
its
own
rules
which
could
differ arbitrarily
from
its
predecessor.+or
a
Turing
machine
with
n
states*
it
can
be
found
that
the
number
of
possible permutations
of
instruction
sets
is/
5"6+or
example
a
single
state
can
ma1e
$7
different
machines
5or
programmes
in modern
terms6*
two
states
%389$
machines
etc.
Notice
the
number
of permutations
rises
extremely
fast
but
is
still
calculable
using
5"6.
(f
we
decide
to run
every
possible
machine
with
a
particular
number
of
states
we
will
find
that some
will
do
nothing*
some
will
loop
infinitely*
and
some
will
run
for
a
time
and then
stop.
:ach
of
them
will
also
print
any
number
of
"s
and
3
from
3
to
infinity depending
on
the
instruction
set
being
followed.
;f
interest
to
!ado
was
the
calculation
of
the
highest
possible
number
of
"s achievable
with
each
set
of
states.
<e
defined
a
function
that
would
return
the highest
finite
score
attainable
whilst
ignoring
those
that
looped
forever.
This
is the
Busy
Beaver
function*
more
formally
1nown
as
∑
5n6*
and
its
noncomputability
was
proved
by
!ado
in
steps
which
will
be
omitted
here
as their conclusions need only be outlined in sufficient detail for our purposes. They are as follows/56
+or
every
explicit*
monotonically
rising
function
f5n6*
the
function
∑
5n6
will always
increase
at
a
faster
rate
and
therefore
can
not
be
explicitly
defined.5B6
+or
every
score
in
∑
5n6
the
number
of
steps
that
the
winning
machine ma1es
will
always
be
either
equal
to
or
greater
than
∑
5n6.
With
5
6and
5B6
in
mind
we
will
now
loo1
at
the
set
containing
those
Turing machines
which
halt
and
refer
to
its
cardinality
as
N5n6.
(t
has
also
been
proven by
!ado
that
this
set*
though
not
incalculably
large*
is
still
noncomputable
on account
of
the
halting
problem
proved
by
Turing
in
his
"#98
paper
On computable numbers, with an application to the Entscheidungsproblem
. Turing
showed
that
it
is
impossible
to
generally
determine
which
machine
will either
halt
or
continue
indefinitely*
and
!ados
later
wor1
clearly
establishes
an equivalence
between
the
halting
problem
and
the
noncomputability
of
N5n6.The
proof
of
the
noncomputability
of
N5n6
goes
thus/
we
assume
that
it
is possible
to
run
all
the
machines
with
a
certain
number
of
states
and
count
each one
that
stops
after
"
step*
%
steps*
etc.
until
we
have
accounted
for
every machine
with
n
number
of
states.
;nce
counted
this
way
we
find
we
can
obtain N5n6
for
any
n
we
choose.
=nfortunately
this
cannot
be
the
case
as
the
count must
include
the
number
of
steps
ta1en
by
the
highest
scoring
machine
which* as
we
1now
from
and
B*
is
impossible
to
calculate.
Therefore
N5n6
is
noncomputable.
Pythagorean
Triples
We
return
now
to
+ermats
,ast
Theorem
and
relate
it
to
all
of
the
above.
+irstly we
arrange
5"6
in
the
following
form
for
when
n>"/
5%6Notice
now
that
the
formula
for
listing
all
possible
Turing
machines
with
n>" states
has
been
expressed
as
a
possible
)ythagorean
triple
in
terms
of
the
sum of
those
machines
which
have
no
halting
instruction
5the
second
term6
and
a remainder*
)?%n*
which
represents
the
nontrivial
sum
of
machines
with
halting codes
which
either
halt
after
a
finite
number
of
steps*
or
loop
forever.
(t
is
clear
that
as 5%6
applies
to
machines
with
one
state
then
in
this
instance
it has
no
whole
number
solution.
<owever
if
we
allow
n
to
increase
inside
the brac1eted
terms
without
a
corresponding
increase
in
the
outer
powers
of
n
we find
we
can
generate
triples
of
the
form/
596
576(mportantly
in
596
and
576
the
)
terms
are
also
expressible
as
the
sum
of
two squares/
56
5$6
Nonomputable
Triples
,et
us
now
imagine
a
consequence
of
assuming
+ermats
,ast
Theorem
to
be false.
(t
would
be
reasonable
to
assume
given
5"6
that
5%6
would
be
soluble
for many
triples
of
the
form/
+or all n0% 586(t
would
also
be
reasonable
to
assume
that
at
least
one
value
for
)?%n
could
be expressible
as
the
sum
of
two
smaller numbers of power %n
as
in
examples
56 and
5$6.
(f
this
were
the
case
we
would
be
in
a
position
to
describe the
number of
halters
and
loopers*
which
have
halt
codes*
as
a
sum
of
two %npower whole numbers.
&uch
an
equation
would
equate
the
welldefined
formula
for
the
set
of halters
with
one
of
the two terms*
so
if
we
wished
to
establish
the
cardinality
of the
set
of
halters
N5n6
we
need
only
to
run
the
machines
associated
with
)
and count
those which halt.
(f
the
count
exceeds
the
lower
of
the
two terms
then
we may
assume
it
is
equal
to
the
larger term
and
thus
also deduce
the
cardinality of
the
set
of
nontrivially
infinite
loopers.(n
this
way
we
could
solve
many
open
problems
in
number
theory
by
running their
equivalent
Turing
machines
and
seeing
which
cardinality
they
possess.
;f course
this
approach
is
impossible
as
establishing
the
set
of
halters
N5n6
is equivalent
to
counting
the
set
of
infinite
loopers
which
we
1now
cannot
be done.
onclusion
(f
+ermats
,ast
Theorem
had been proved false
it
would
have
implied
the constructibility
of
an
ndimensional
shape
with
sides
corresponding*
with
welldefined
functionality*
with
the
cardinalities
of
of
the
sets
of
nontrivial
Turing halters
and
loopers* as in 586.
(t
would
also
have
implied
the
nonconstructibility of
noncomputable
functions
that
are
welldefined.
;pen
questions
in mathematics
would
be
decidable
and
the
halting
problem
would
be
crac1able.
This
is
not
so
of
course*
and
+ermats
,ast
Theorem
appears
to
support
the principle
of
noncomputability.
@onversely*
noncomputability
may
serve
as
a supporting
proof
of
+ermats
,ast
Theorem.