Today’s class revolved around conjectures.

Goldbach’s Conjecture: any even integer greater or less than 2 that can be written as the sum of two primes.

      ex: 16 = 13+3 and 11+5

Mersenne Numbers: any number that can be written as 2p-1

M=2n-1 (n=root of 2)

# ___ Mp

1           1

2           3

3           7

4           15

5            31

6            63

7            127

Prime number: a number that can only be divided by 1 and by itself (number must be a whole, real or natural #)

Perfect number: number whose product or sum of its factors are divisible and equal to itself

           ex. 6=2×3, 1×6 & 2+3+1

           First 4 perfect #s: 6, 28, 496, 8,128…

Conjecture: referral that there aren’t any ‘odd’ prime numbers

Goldbach’s conjecture: theorem that every even number n greater than or equal to 4 can be written on the sum of two primes;

4=2+2, 6=3+3, 8=5+3, 10=7+3; 5+5, etc.

Mersenne Primes:

p       Mp       p#

2       3          yes

3       7          yes

5       31        yes  (2x2x2x2x2)-1

7       127      yes

11     2047    no  (divides by 23)

Primes 2,3,5 and 7 were found by a Greek mathematician in the 5th century

Prime 13 was found in 1456 A.D.

17 was found in the 1500s

French monk/mathematician saw a similarity in gaps between primes, and would later be names in his honor as Mersenne Primes; from this, the theorem is that if n is an even perfect number, it can be written as:

2 (rooted prime)-1=Mp

P          Mp         2(p-1)

2          3           6

3          7           28

5          31         496

7          127       8,128

x3_=2 (mod p)

       From this, x3 would leave a remainder of 2 when divided by prime “p”

x4_=2 (mod p)

       x4 would leave a remainder of 2 when divided by prime “p”

ex. 5_=2 mod (3), as 5 divided by 3 = 1 (with a remainder of 2)

ex. 5_=5 mod (7), as 5 divided by 7 equals 0 (with a remainder of 5)

A majority of Sophie Germain’s work reflected on applications relevant to engineering. Her studies included vibration of plates in sand and soap film patterns, along with her work on number theory and the use of integers.

Born in Paris, Sophie was raised in a wealthy but working class family. Different from most of women mathematicians we have studied, Sophie was more of less discouraged from all directions to learn. Society looked down on such activities for women, and her parents attempted to punish her from trying to study. Both would prove to no avail, and Sophie would become a prominent figure as a female mathematician.

Her father would eventually be promoted from his courtly duties to running the Bank of France. As such, her family had the instruments necessary for such a person to take up learning. Outside of tutoring, which would only teach her minimally or not at all in intellectual studies, Sophie studied on her own in the family library. Mathematicans such as Archimedes intrigued Sophie, and she became interested in how mathematics could shape the world. Her parents couldn’t provide her with all the books necessary, so Sophie learned differential calculus on her own.

WIth the founding of the Ecole Polytechnic, the most prominent technical school in France at the time, mathematicians around the known world were flocking to Paris to take part in the new hub for these studies. However, women were not allowed to attend, and when Sophie was eighteen befriended students and teachers alike, allowing her access to the lectures given. Using the surname “M. LeBlanc,” she submitted a paper on analysis to a Professor LeGrange who was impressed by his student’s thoughts. When he was introduced to the “Monsieur LeBlanc,” he was overtaken and even more impressed that the analytic report was by a female, nevertheless a female who didn’t physically attend any of his lectures. He would later introduce Sophie to his colleagues.

In 1801, Sophie would write to a Prof. Gauss, a German mathematician, on her results involving number theory. Gauss was just as impressed as LeGrange was, yet wouldn’t ever meet Sophie.

In 1807, the French occupied Hanover, the town in which Gauss lived. Sophie wrote to the French commander in charge, who was a friend of her family, and helped get Gauss to safety. Years later, Gauss would recommend Sophie Germain for an honorary degree from the University of Gottingen in 1831, but she never made it. She would pass away before recieving the honor and meeting Gauss, but would be remembered for many of her achievements, most of which revolved around number theory.

Born in Italy in 1718, Maria Agnesi would live a life the complete opposite of Emilie du Chatelet, yet be considered influential in the world of mathematics for her work on algebra and analytic responses to problems of infinite quantity.

In Italy, only the wealthy socialites would have access to books. Furthermore, women even in the upper-class were discouraged from reading and writing. Instead, women were suggested to learning embroidery, all while taking part in their religion, etiquite and other activities in reservation. 

One of twenty-one siblings, Maria Agnesi would have initially stayed within these boundaries. Her father, whose parents were wealthy merchants and himself a mathematician, would host parties with colleagues. Noticing some of his children’s attributes, he allowed for behavior not common for young girls and women, such as Maria Gaetana Agnesia’s sister Maria Theresa Agnesi, who would eventually become a famous Italian musician. Maria Gaetana Agnesi’s father noticed she had an impeccable memory, so he allowed for her to study and learn. By nine, Maria knew several languages fluently, and by her early teens knew Greek, Hebrew, French, English, Latin along with her native Italian. From this, her father allowed a young Maria to socialize with his guests, along with conducting speeches with his colleagues on matters pertaining to mathematics, philosophy, language and more.

Though mathematics would only take part of twenty or so years in Maria’s life, her most important transcripts of her studies is the two-volume Analytic Institutions, recieving praise from Issac Newton and others, as the assertion of calculus was relatively new and Maria’s work on this new form of mathematics showed her determination at clarifying this form of mathematics. From this, Maria proved that women could surpass these traditional views of women, as she excelled in abstract reasoning of problems and was interested in such works.

Born in Korea, she was enticed by mathematics and recieved her high school degree in the mathematics field. She recieved her B.S. at the Seoul National University in Mathematics in 1975. Upon moving to the U.S., she recieved her Masters in Pure Mathematics at Brown University in 1985, and then her Ph.D. in Communitive Algebra from the University of Illinois (at Urbana-Champagne) in 1988.

At this time, she had three girls (two whom were twins) and eventually moved back to the New England area and worked for Brown as an engineer. In 1990, she began teaching at UMass Dartmouth as an associate professor in mathematics.

She also spoke of her interests in mathematics, specifically the concentration of Applied Mathematics, in which she is currently teaching. The use of inverse problems and computational mathematics is currently her field of expertise, and noted that it was used in scientific training on computers.

Further, she showed how her field of mathematics applies to cryptology, and how codes and ciphers are cracked using these concepts.