Transcript Fermat’s Last Theorem can Decode Nazi military Ciphers

Fermat’s Last Theorem can
Decode Nazi military Ciphers
What does Euclid, Pythagoras, Pierre
de Fermat, Sophie Germain, and
Lame all have in common?
The connection between Fermat’s
Last Theorem and Nazi military
ciphers are:
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Cracking the World War II, circa mid 1940’s,
cipher’s used derivatives of mathematics
developed by number theory connected to
Fermat’s Last Theorem, which was x^3+y^3=z^3
developed in the 17th century.
Fermat’s theorem, however, borrowed the best
equation from the 6th century B.C. to measure a
triangle in the history of mathematics which was
Pythagoras’ geometric theorem a^2+b^2=c^2 as
his premise.
After Fermat developed his equation he began to
substitute the exponents from 3 to 4,5,6 on up, and
Fermat’s proof had to prove that these
substitutions had no solutions that existed
within this infinity of infinities.
 And even though the time period between
these 2 events are 302 years apart, this type
of logic parallels with the WWII Bletchley
Park military headquarters in the UK when
they were trying to crack the secret war
codes using some form of deductive
reasoning which stems from Euclid’s
geometric laws.
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What happens next historically in
Mathematics?
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Fermat declared that he had written his proof down for
this number theory, which was never found or proven.
So a new mathematician enters the male dominated
arena, in the early 19th century, by the name of Sophie
Germain. Sophie felt compelled that it was her
obligation to rediscover his proof.
She started working with prime numbers whose
numbers have no divisors, and began to develop pairs of
numbers like factorisation, I.e. [11=1x11] or
[(2x5)+1=11].
Miss Germain subtituted “n” as an exponent into
Fermat’s equation which became x^n+y^n=z^n.
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Due to tight restrictions of these prime numbers
for “n” forced her to prove there could be no
solutions which backed Fermat’s theory.
This is as far as her contributions went because
without Professor Carl Friedrich Guass, women
didn’t have any academic place in the mathematics
world.
See she was forced to take on the identity of a man
by using the name Monsieur LeBlanc to be able to
submit her mathematical number theories at
university level. Only then when Professor Guass
discovered him as a her did the submission of her
mathematical number theories enable her to grow
as a mathematician.
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At this juncture in time Professor Guass switches to
the astronomy department, and the lack of written
communication weakened the never married
Sophie’s confidence to continue her studies in pure
mathematics.
Since Guass was a great stepping stone and the fact
that the stating of values of “n” remained
intractable made her later concentrate on the
modern theory of elasticity.
Before Miss Germain died of breast cancer she
received an honorary award degree from Professor
Guass through Gottingen for her extensive
research. This is the same university where Sonia
Kovalevsky in 1874 was the first woman to receive
a Phd in mathematics.
Let’s Examine Military Ciphers
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The WWII Bletchley Park files concluded that the
British military chose people who had a good
sense of crosswords puzzles using a keen sense of
numerical patterns.
However, it was extremely difficult to examine
lines of letters that absolutely no sense.
The key ingredient here to decoding was the
typewriter machine called the Enigma which
translated and printed these random sequence
codes using a 3 wheel machine with 4 rotors.
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This machine had double indicators enciphering to
get this Morse code (radio transmissions) sent
abroad via German military.
Therefore, the only man to break such a code in
1944 was a German-Jewish mathematician by the
name Alan Turing, from Cambridge.
Turing used deducted reasoning substituting pairs
of numbers , 3 letters at a time, forming these
secret tables.
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In conjunction with the Lorenz’ machine(stolen
from the German navy by the British), the modulo
2 addition mathematical system which transmits a
string of letters which then becomes mixed up
during transmission, across seas, then prints the
code correctly after transmission.
This process was the breaking point for Turing’s
research.
This code is now referred to as algorithms in
mathematics.
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Algorithms as such are used in either 10^38 or 40128 bit encryptions for safe online shopping or
banking.
Bletchley Park would be considered the global
hackers of the 21st century sniffing for online
passwords and breaking into people’s email
accounts.
This type of coding could of even been helpful to
Mary Queen of Scots, who tried military
espionage on Queen Elizabeth. Mary eventually
became trapped by her own Beale cipher codes.
Such strategy later influenced numerical
strategies in WWI and WWII.
 Hence, Elizabeth had Mary killed because
the hidden location of the gold fortune
buried in Virginia, circa 19th century, was
never found and remains a mystery to date.
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Decoding of the Nazi Secrets.
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Our practice cipher is “Bo fbtz djqifs up csfbl!”
This phrase was a simple switching positions of
the alphabet letters, which was (n-1), therefore, it
would be the previous alphabet letter. This above
phrase translates to “An easy cipher to break!”
Such success of a translation entices one to
prepare for more challenging ciphers.
Now let’s tackle a 5x5 letter group divided into 7
columns, instead of one line of string. The
example is “GEGOH.” It identifies to a 5x5
matrices from pre-calculus, however what does a
mathematician do with 2 digit numbers
(G=6,E=5,G=6,*O=16,H=8) and how could one
Other examples of matrices used were 3x3’s
within a four to five sentence.
 Basically “cracking the ciphers” uses
applied random movement of how row
swapping solves algorithms producing
congruent answers. Geometric deductions
cribbed the dragging of digraphs, hence, the
pattern repeated itself 2 times with 3 pairs
between them.
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What to understand.
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Understanding such logic and number theory
connects to Euclid, Pythagoras, Fermat, Sophie
Germain, and Lame.
In number theory, from the equation a^2+b^2=c^2
to x^n+y^n=z^n, the prime number: 3(Euclid
proved cubes), 5(Sophie proved, as well as 2n+/1), 7(Lame proved) all worked except for the
exponent number 4(1x4,2x2) because it was not a
prime number.
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It was not a coincidence that the numbers 3,5,7
were all prime numbers or that they matched the
ciphers of 3x3’s, 5x5’s in 7 columns or the double
equation of the alphanumeric number minus 1.
All of this finally fits the rules of factorisation.
Gabriel Lame was an applied mathematician and
was led to Fermat’s Last Theorem like Sophie
Germain.
Lame made a substantial contribution to the
problem a=b is x^n+y^n=a^n by solving the case
n=7.
Although he believed he had solved the whole
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Lame did important contributions working on
differential geometry, number theory and showed
that the number of the divisions in Euclidean
algorithms never exceeded 5x the number of digits
in the smaller number.
In 1839, Lamed proved Fermat’s theorem that
n=7.
In 1753, Euclid proved p^2+3q^2 even though his
numbers didn’t behave, his case study n=3, did
work.
Sophie proved her case study that if n an 2n+1 are
primes then x^n+y^n=z^n implies one of the x,y,z
is divisible by n with numbers <100 and all
Her case study split Fermat’s theorem case #1 that
none of x, y, z is divisible by n. And case#2 that
1and only(prime number )1 of x,y,z is divisible by
n.
Sophie’s proof that the number 5 splits itself into
2(ie 2+2+1 or 2^2 or 1x5) shows that an even
number plus one divisible number by 5 are
distinct.
Euclid’s cube using the number 3 as his case study
proved that p=a^3-9ab^2, q=3(a^2b-b^3) then
p^2+3q^2=(a^2+3b^2)^3 is a cube an a+b exist as
p+q=a+(b times the square root of –3(complex
factorisation number of i). Therefore, that is how