The document proposes a solution to Fermat's Last Theorem using a color analogy involving the CMYK color model. By treating colors as frequencies correlating to real numbers, the model asserts that the equation can be solved for any n greater than 2 through complementary colors. This approach presents a non-paradoxical interpretation of the theorem, suggesting a broader systems theory in mathematics and natural sciences.

1. A PROPOSED SOLUTION TO
FERMATS LAST THEOREM
USING A COLOUR ANALOGY
Andrew Hennessey
In number theory, Fermat's Last Theorem states that no three
positive integers a, b, and c can satisfy the equation an
+ bn
= cn
for
any integer value of n greater than two. This theorem was first
conjectured by Pierre de Fermat in 1637, famously in the margin of a
copy of Arithmetica where he claimed he had a proof that was too
large to fit in the margin. No successful proof was published until
1995 despite the efforts of many mathematicians. The unsolved
problem stimulated the development of algebraic number theory in
the 19th century and the proof of the modularity theorem in the 20th.
It is among the most famous theorems in the history of mathematics
and prior to its 1995 proof was in the Guinness Book of World
Records for "most difficult math problem".
Fermat's equation xn
+ yn
= zn
is an example of a Diophantine
equation.[6]
A Diophantine equation is a polynomial equation in
which the solutions are required to be integers.[7]
The integers (from the Latin integer, literally "untouched", hence
"whole": the word entire comes from the same origin, but via
French[1]
) are formed by the natural numbers including 0 (0, 1, 2, 3,
...) together with the negatives of the non-zero natural numbers (−1,

2. −2, −3, ...). Viewed as a subset of the real numbers, they are numbers
that can be written without a fractional or decimal component, and
fall within the set {... −2, −1, 0, 1, 2, ...}. For example, 65, 7, and −756
are integers; 1.6 and 1½ are not integers.
However, in this example of a model of a solution to Fermats
problem the numbers do not have to be of the set of integers but of
the set of CMYK colours as rendered into the Pantone colour
system.
We know though that colours are frequencies and frequencies are
real numbers – so there is a direct correlation between colours,
numbers and the mathematics of a solution to Fermats problem.
Colours chosen from the CMYK process are then converted into real
world applications in shops and commerce etc by the Pantone
colour system.
The maximum criteria for Fermats n is usually 2 in the set of integers
– but in the CMYK colour/maths system – n could be any real
number whatsoever and most are greater than 2.
In CMYK, The CMYK color model (process color, four color) is a
subtractive color model, used in color printing, and is also used to
describe the printing process itself. CMYK refers to the four inks
used in some color printing: cyan, magenta, yellow, and key black.
CMYK Blue number four [ultimately a commercial Pantone number]
added to CMYK Orange number 4 will give if mixed in equal
proportions, CMYK Grey number 4, where n could be literally
hundreds of real numbers that are part of the black weighting of K as
it mixes with CMY. Also a and b are self cancelling complementary
colours within the colour spectrum such as e.g. red and green, or
orange and blue, or purple and yellow, or black and white, [all of
these mix together to produce a grey] and c is always a grey of
black weighting n identical to the black weighting or K within the
selected a and b taken from the CMYK colour spectrum.
Thus an
+ bn
= cn
In the CMYK system when selecting the black weighting of a colour
e.g. n = 1 – 100 or 1-1000 for example is always a true and solvable
when applied to the Fermat equation. N isn’t restricted to Fermats 2
in the CMYK colour process
Where a and b are Complementary colours a c is always produced of
the same black weighting n in the Pantone series in the CMYK
colour system.
http://simple.wikipedia.org/wiki/Complementary_color

3. http://en.wikipedia.org/wiki/CMYK_color_model
http://en.wikipedia.org/wiki/Set_theory
Although all of these colours and processes going into the tins at
Walmart are natural substances, paradoxically such complimentary
colours, neither e.g. the blue or the orange used, could be called
mathematically ‘natural’ by Mr Fermat which seems counter-intuitive
and wrong.
We get the refutation of Fermats Last Theorem from this context
sensitive analogical colour model. Even if n is greater than 2 of the
K, black weighting in the CyanMagentaYellow [cmyk] colour system
- we still get a knowable and positive solution for any n.
Although Pantone numbers [commercial colour scale derived from
the CMYK colour set] are of a set of numbers in a colour system –
they do correspond with measurable empirical readings in the
natural world and its natural mathematics.
The real numbers that convert to their equivalent decimal places on
optometric equipment that calibrate the frequency of the light
emissions in the spectra of the blue and orange paint would have
been thought an impossible correlation by thousands of
mathematicians after the time of Fermat.
Fermats Last Theorem therefore has no paradox or problem within
the colour analogy of Complementary colours within the CMYK
colour system. Especially because colours are frequencies and
frequencies are (real) numbers.
That fact suggests that there is a general systems theory solution in
the natural world to Fermats paradoxical theorem using set theory
and real numbers - where a and b are complementary colours, c is
the resultant grey and n is the number equal to the k or black
weighting scale within the CMYK colour process.