Grace Chisholm Young was born in England in 1868.  By the age of 13, she showed mathematical skills.  At 17, Grace passed the entrance exams at Cambridge, but because she was a woman, she was unable to study medicine.  She ended up enrolling in Girton College of Education at Cambridge, where she met her future husband.  After graduation, she was unable to further her education in Grad school because she was a woman.  Grace then attended Gottigen University, which was in Germany; this is where she received her PhD.  After receiving her PhD, Grace decided to move back to England.  She married the afore mentioned ‘future husband’, William Young and the two had six children together; Grace was a very committed mother and wife.  He was also a mathematician and the couple collaborated to produce some groundbreaking work and books.  (Before Grace, William really didn’t have any spectacular finds)  Grace’s main field of interest were computative alegebra.  Grace passed away in 1944.

Sonia Kovalevskaya was born in Moscow in 1850.  Both her grandfather and great-grandfather were mathematicians.  At the age of 15, she became fascinated by calculus when she read her families textbook.  Sonia ended up marrying Vladimir O Kovalevskaya, mostly out of convenience so that she could travel outside of Russia.  She moved to Heidelberg, Germany with her sister Anyuta to attend class, however, she found out that, because she was a woman, she couldn’t ‘officially’ attend class.  She remained an ‘unofficial’ student for two years, but then left to study with Carl Weiertraff. She wrote three compositions by 1874, which should have earned her a doctorate of philosophy in math, but she wasn’t allowed.  Eventually, she was awarded tenureship and was invited to be a professor at Stockholm University in 1889 and she did get her doctorate eventually, making her the first woman in Europe to do so.   Sonia passed away in 1891.
 

Emmy Noether was born in 1882 in Sweden.  Her father Max was a mathematician, although Emmy was not allowed to go to school as a child.  She assisted her father with research and sometimes he allowed her to teach, although this was not a paying job.  Her main focus was in abstract algebra and the Noetherian ring.  She wrote her thesis, but had a heck of a time finding a university that would allow her to teach/give her a job.  She applied for a job in Gottingen University in 1915, but she wasn’t accepted.  She ended up re-applying in 1919 and David Hilbert tried to help her out.  He made the comment “I don’t see why the sex of the candidate is relevant – this is an academic institution, not a bath house.”  They still wouldn’t give her a job, although she was allowed to teach, unofficially, and without pay.  She died in the U.S. after fleeing Nazi Germany (she was Jewish), where she passed away due to complications after a ‘simple’ surger in 1935.

It was a fairly relevant puzzle as it related to the years of Mary Somerville’s birth and death.

_ _ _ _ – _ _ _ _

The first 2 digits of her birth year were the 7^th prime number (17)

The last 2 digits of her birth year are equal to the sum of the first 2 numbers 10(7+1) = 80.  (Birth year = 1780)

The first 2 digits of her death year were the reverse of her birth year (187)

The last 2 digits of her death year are the sum of the first two digits multiplied by the second digit (1+8)8 = 72 (death year, 1872)

Pretty wild, huh?

Then we discussed Chladni Diagrams, which are diagrams that deal with the

vibration of sand patterns; this was very interesting to Sophie Germain.

            Basically, you spread sand over glass plates and run a violin bow over the edges to observe the patterns created by the vibration

We had a brief discussion of Trigonometry as well.

Professor Kim gave us an interesting way to remember Sin, Cos, and Tan, with the sentence “Some Old Hag Caught A Hippie Tripping On Acid”.

Sin(Θ) = O/H

Cos(Θ) = A/H

Tan(Θ) = O/A

(Θ is theta)

Finally we discussed a sequence involving one of my favorite fuzzy animals… The Bunny.

Fibonacci Rabbit Sequence – Single pair of rabbits (1 male, 1 female) born at the beginning of a year; not fertile until they reach the first month of life, cannot give birth until the second month.  At the end of each month, they give birth to another pair (1 male, 1 female).  If no rabbits die, by the end of one year, how many rabbits will have been born?

Mary Fairfax Greig Somerville was born on December 26, 1780 in Scotland to a fairly well-respected family.  She had six siblings.  Like many female academics of the time, Mary didn’t receive formal training and was self-taught, but she was taught domestic skills and how to read the bible.  Mary was sent to a boarding school in Edinburgh at age ten, where she taught herself to read Latin.  She discovered math through an article for a magazine for women, and asked her brother’s tutor to buy her a book so she could understand the mathematical symbols.

She married two cousins: Samuel Greg (she gave him two children) who passed away and William Somerville (she gave him four children).  Of the two husbands, William was more accepting of an educated wife and allowed her to continue her studies.  In fact, Mary was allowed to attend lectures at the Royal Institute.  Mary was astronomer and mathematician.  She wrote The Magnetic Properties of the Violet Rays of the Solar Spectrum (1826) and The Connection of the Physical Sciences (1835).  In 1835 shortly after her second book was published she was admitted to the Royal Astronomical Society.  In 1870 she received the Vitroai Hold Medal from the Royal Geographic Society.  Oxford even named one of their colleges after her.  She died at the age of 92.

Ada Byron Lovelace was the daughter of poet Lord Byron.  She was the protégé of Mary Somerville and she worked with Charles Babbage, who was a math professor at Cambridge University.  Lovelace and Babbage created the first computer problem.  Ada also worked on the Tower of Hanoi problem.  She passed away at 37.

The Tower of Hanoi problem is a problem that consists of 3 rods and a number of different sized disks that can slide onto any rod.  All disks start stacked neatly (in order of size; it sort of resembles a cone) on one rod.

Rules: 1. Only one disk can move at a time

            2. each move consists of taking the upper disk from one of the pegs and sliding it onto another rod, on top of the other disks that may already be on the rod

            3. No disk can be placed on a smaller disk

Today we talked about how something called a ‘one-way function’ is often referred to as a ‘humpty dumpty function, due to the fact that they are difficult to reverse, or reconstruct.

We went over the formula x3 mod P=2.  It is pretty difficult to solve and has multiple solutions.

We also dicussed cryprography, which is the practice and study of hiding information, and is used in sending and receiving info over the internet using mod formulas.

Modular (mod) Arithmetic is a system of arithmetic for integers where numbers wrap around after they reach a certain value (the ‘modulus’); created by K.F. Gauss (Sophie Germain’s buddy)

For example:

Alice and Bob want to communicate, but they don’t want it overheard.

They agree to use a prime number (p=23) and base (g=5).  Alice chooses a ‘secret’ integer (a=6) and sends Bob (g^a mod p)… 5⁶ mod 23 = 8.

Bob then chooses a ‘secret’ integer (b=15) and sends Alice (g^b mod p)… 5¹⁵ mod 23 = 19.

Alice then computes (g^b mod p)^a mod p… 19⁶ mod 23=2

Bob then computes (g^a mod p)^b mod  p…8¹⁵ mod  23 = 2

Confused?  I know I am.

 We also learned about Sophie Germain primes.

A prime number is a Germain Prime if 2p-1 is also prime.

The first few Sophie Germain primes are 2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131

For example:

23 is a Germain prime because it is prime AND 2×23+1 = 47

Germain Primes relates to Fermat’s Last Theorem. 

Fermat’s Last Theorem states that xⁿ + yⁿ = zⁿ has no positive integer solutions for x,y,z when n>2.

Sophie Germain proved that Case I holds when n and 2n+1 are both prime. 

mathworld.wolfram.com/SophieGermainPrime.html

http://www.agnesscott.edu/Lriddle/WOMEN/germain-FLT/SGandFLT.htm

Prime numbers are numbers that can only be divided by 1 and itself (ex. 1,3,5,7,11…)

Perfect numbers, on the other hand, are numbers whose sum of its factors are equal to itself.  Perfect numbers are pretty rare… 6, followed by 496, followed by 8128, and there are no odd perfect numbers (this hasn’t officially been proven, it’s only a conjecture at this point.

            (ex. 6×1=6, 2+3+1=6, 2×3=6)

Number theory is the the study of whole, real integers.

Goldbach was really interested in Prime Numbers and, in his studies, came up with what we know as Goldbach’s Conjecture.

Goldbach’s Conjecture states that every even number that is great or equal to 4 can be written as a sum of 2 primes.

For example:

2+2 = 4

3+3 = 6

5+3 = 8

7+3 & 5+5 = 10

7+5 = 12

7+7 & 11+3 = 13

A 17th century French scholar by the name of Mersennae was also pretty interested in Prime numbers and created a list that we now refer to as Mersenne Primes.

Mersenne Primes:

            Let P be a prime number: Mp=2p-1

                        Prime numbers 2, 3, 5, 7, 13

                        Corresponding Mersenne’s 3, 7, 31, 127, 9191 (11 and 2047 are not; 2047 is not prime)

Theorem – N is an even perfect number if and only if it has the form 2p-1(2p-1)

            Ex. 2 is prime 3, Mersenne prime is 2(2-1) = 6

This comic expresses how I feel…

            Sophie Germain was born in 1776 in France, to a fairly financially comfortable working class family; her father was a merchant banker.  At the age of thirteen, her family moved to a rural part of Paris in order to keep the family safe from what was happening to the wealthy during the French Revolution.

            Out of boredom, Sophie began to read her father’s books.  Because an educated woman was kind of a taboo subject at the time, her parents were pretty upset.  Her parents made several attempts, all to no avail, to stop her from learning… they were pretty convinced that all that information she was putting in her head would be the end of her.  Eventually, they just gave up.  Nothing would deter Sophie from pursuing education.

            Around the time the Revolution was tapering off, Sophie became interested in pursuing higher education.  Unfortunately, high learning was off-limits to women, so Sophie continued to self-teach, using lecture notes given to her by male friends.  She ended up writing a paper and submitting it using a male name, but the professor eventually discovered that a woman had submitted it.  This turned out to be a blessing for Sophie, because the professor began to mentor her. 

           At the age of twenty-five, Sophie began to write to a German Professor of Mathematics by the name of Gauss and the two became friends.  In 1807, France was occupying Hanover (that’s where Gauss was) and Sophie became concerned with his safety.  She sent Jean Baptiste Fournier, a French commander and family friend, to take care of him.

             She submitted an experiment that dealt with sand patterns to the French Academy of Sciences, but her experiment was returned to her twice, due to a ‘flawed solution’.  Finally in 1816, her re-submitted problem won her the French Academy of Sciences prize.  After this, she became interested in Fermat’s theorem and experimented with soap film patterns and applied mathematics.