# Excerpts from Sophie’s Diary: A Mathematical Novel

*Today we enjoy a bit of math, as told in Sophie’s Diary: A Mathematical Novel. Written by Dr. Dora Musilek, this novel was inspired by the French mathematician Sophie Germain, an important contributor to number theory and mathematical physics. Her correspondence with some of history’s great mathematicians such as Lagrange, Legendre, and Gauss are known while her life prior to that is shrouded in the unknown. Dr. Musilek explored Germain’s early life and uses the concept of an adolescent’s diary to discuss how Germain may have taught herself math while dealing with the social upheaval of the French Revolution, which occurred at this time in her life. Read on for an engaging lesson in math.*

*n*belongs to one of four different categories:

*n*= 4

*k*

*n*= 4

*k*+ 1

*n*= 4

*k*+ 2

*n*= 4

*k*+ 3

*k*= 3, 5, 6, and 7, I write:

*n*= 4(3) = 12, and

*n*= 4(6) = 24; or

*n*= 4(5) + 2 = 22, and

*n*= 4(7) + 2 = 30. The resulting numbers clearly are not primes. Thus, I can categorically say that prime numbers cannot be written as

*n*= 4

*k*, or

*n*= 4

*k*+ 2. That leaves the other two categories.

*n*= 4

*k*+ 1, or

*n*= 4

*k*+ 3. For example, for

*k*= 1 it yields

*n*= 4(1) + 1 = 5, and

*n*= 4(1) + 3 = 7, both are indeed primes. Does this apply for any

*k*? Can I ﬁnd primes by using this relation? Take another value such as

*k*= 11, so

*n*= 4(11) + 1 = 45, and

*n*= 4(11) + 3 = 47. Are 45 and 47 prime numbers? Well, I know 47 is a prime number, but 45 is not because it is a whole number that can be written as the product of 9 and 5. So, the relation

*n*= 4

*k*+ 1 will not produce prime numbers all the time.

Over a hundred years ago Pierre de Fermat concluded that “odd numbers of the form

*n*= 4

*k*+ 3 cannot be written as a sum of two perfect squares.” He asserted simply that

*n*= 4

*k*+ 3 ≠a

^{2}+ b

^{2}. For example, for

*k*= 6,

*n*= 4(6) + 3 = 27, and clearly 27 cannot be written as the sum of two perfect squares. I can verify this with any other value of

*k*. But that would not be necessary.

*And now we skip ahead to another excerpt where differential calculus is described.*

*Inﬁniment petits*. I went back to the basic deﬁnition: “a derivative of a function represents an inﬁnitesimal change in the function with respect to whatever parameters it may have.” The simple derivative of a function

*f*with respect to

*x*is denoted by

*f’*(

*x*), which is the same as

*df/dx*. Newton used ﬂuxions notation

*dz/dt*=

*ż*, but it means the same, so I will use

*f’*or

*df/dx*from now on. Well, I can now take the derivative of certain classes of functions because I just follow certain rules.

*x*, I use the fact that

^{n}*d/dx*(

*x*) =

^{n}*nx*

^{n}^{−1}.

^{}So, if I have

*f*(

*x*) =

*x*

^{5}, its derivative should be 5

*x*

^{4}. This is easy. If I

^{}analyze trigonometric functions such as sin

*x*and cos

*x*, then I use the

^{}derivatives

*d/dx*(sin

*x*) = cos

*x*, and

*d/dx*(cos

*x*) = − sin

*x*.

^{ }

or a bit more complicated such as the linear differential equation:

or even a nonlinear equation such as this:

*y’*for the

*dy/dx*derivative, or

*y’’*for

*d*

^{2}

*y*/

*dx*

^{2}, and so forth. Thus,

^{}the previous linear equation would be written as (

*x*

^{2}+ 1)

*y’*+ 3

*xy*= 6

*x*. I need to keep these differences of notation in mind, since I am studying from ﬁve different books.

*P*(

*t*) represents the population change in time (

*t)*, I write

where the rate *k* is constant. I observe that if *k* > 0, the equation describes growth, and if *k* < 0, it models decay. The exponential equation is linear with a solution *P*(*t*) = *P*_{0}*e ^{kt}*, where

*P*

_{0}is the initial population, i.e.,

*P*(

*t*= 0) =

*P*

_{0}.

*k*> 0, then the population grows and continues to expand to inﬁnity. On the other hand, if

*k*< 0, then the population will shrink and tend to 0. Clearly, the ﬁrst case,

*k*> 0, is not realistic. Population growth is eventually limited by some factor, like war or disease. When a population is far from its limits of expansion, it can grow exponentially. However, when nearing its limits, the population size can ﬂuctuate. Well, I think that the equation I use to predict the rate of change of population can be modiﬁed to include these factors to obtain a result closer to reality.

*Dr. Dora Musilek is a research scientist and also lectures on the role and contributions of women scientists and mathematicians. She holds a Ph.D. in aerospace engineering. You can learn more about Dr. Musilek and her novel at sophiesdiary.net You can learn more about Dr. Musilek’s writing process at MAAA Books Blog.*

*These views are the opinion of the author and do not necessarily either reflect or disagree with those of the DXS editorial team.*