Fermat’s Last Theorem can Decode Nazi military Ciphers

1. Fermat’s Last Theorem can Decode Nazi military Ciphers What does Euclid, Pythagoras, Pierre de Fermat, Sophie Germain, and Lame all have in common?

2. The connection between Fermat’s Last Theorem and Nazi military ciphers are: • 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 continued to use the mathematical method called “trial and error.”

3. 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.

4. What happens next historically in Mathematics? • 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.

5. 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.

6. 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.

7. Let’s Examine Military Ciphers • 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.

8. 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.

9. 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.

10. Algorithms as such are used in either 10^38 or 40-128 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.

11. 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.

12. Decoding of the Nazi Secrets. • 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 possibly finish a 5x5 matrix within 2 steps?

13. 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.

14. What to understand. • 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.

15. 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 problem in one stage he later overlooked the lack of unique factorisation.

16. 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 numbers <197.

17. 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 Fermat’s Last Theorem bridges to decoding Nazi secrets.