Classroom Worksheet on Sophie Germain’s Work Related to Fermat’s Last Theorem and Her Prime Numbers


by Katy Rountree and Melissa Shoaf



 

"Algebra is but written geometry and geometry is but figured algebra."

 

Sophie Germain

(1776-1831)

 

The Life of Sophie Germain

Sophie Germain was a French mathematician born in 1776, in Paris. She was the daughter of an upper-class family, but lived an austere personal life and never married.  As a child she became enthralled by mathematics after reading Histoire des mathématiques by J. F. Montucla.  It is said that she was fascinated by the idea that Archimedes was so engrossed in a geometric shape in the sand that he did not hear the approach of the Roman army.  Therefore she resolved to devote her life to the study of mathematics.  She taught herself Latin, Greek, and basic mathematics.

Her parents were originally opposed to her mathematical studies.  They even went so far as to take her clothes and remove all fire and light from her bedroom to prevent her studies.  However, she had a secret cache of candles that she used to study by.  She would pretend to be asleep, then get up, and wrap herself in quilts and blankets to keep warm, and study, even when her ink froze. 

Because she was a woman, Germain was denied entrance into educational and scientific institutions.  However, she gained lecture notes and problems from the mathematical courses at the Ecole Polytechnique through the assumed name of Monsieur LeBlanc.  She was also able to correspond with renowned mathematicians such as Joseph Louis Legrange, Karl Friedrich Gauss, and Adrien Marie Legendre.  

Sometimes referred to as "the Hypatia of the 19th Century", Germain made notable discoveries in number theory, acoustics, and the theory of elasticity.  Her most notable work was on Fermat’s Last Theorem. The work that has been and continues to be done on this very challenging proof is based on the work Germain did. 

Germain never published any of her work on Fermat’s Last Theorem, but she did publish some of her work on elastic surfaces: Recherche sur la théorie des surfaces élastiques (1821), Remarques sur la nature, les bornes et l’étendue de la question des surfaces (1826), and “Examen des principes qui peuvent conduire à la connaissance des lois de l’equilibre et du movements des solides élastiques,” Annales de Chimie 38 (1828).

Her honors and awards include the grand prize from the French Academy for a paper describing her findings on the mathematics of elasticity.  The true prize in this achievement is she was the first woman to receive the award.  In addition, she was to be awarded an honorary doctorate from the University of Gottingen, where Gauss worked, but she died of breast cancer a month before she could receive the degree.   

Sophie Germain’s Work

Fermat’s Last Theorem is one of the most challenging mathematical problems in history.  Germain was the first person to generalize a way to prove Fermat’s Last Theorem.  Fermat’s Last Theorem states:

            xn + yn = zn has no non-zero integer solutions, for n>2.

For n=2, there exist many solutions, as seen most often in right triangles.  One solution is x = 3, y = 4, and z = 5.  This is true because 32 + 42 = 52, 9 + 16 = 25.  Find at least one more solution to x2 + y2 = z2.

 

 

Fermat knew there were many solutions to the equation with n = 2, but he was interested in n > 2.  Before Germain began working on Fermat’s Last Theorem, Euler, Fermat, and Legendre found solutions for n=3, n=4, and n=5, respectively.  Germain sought a general approach to the proof, not just one for a specific number.  In order to prove Fermat’s Last Theorem, a proof showing there are no solutions for powers that are prime is all that was needed, because all other cases are divisors of primes.  Germain worked on solutions for p<100, where p is no non-zero integer and p is an odd prime that does not divide x, y, or z.  Germain’s explanation of her approach is a modular arithmetic approach.  

According to Germain, “If the Fermat equation for exponent p prime has a solution, and if t is a prime number with no non-zero consecutive pth powers modulo t, then t must divide one of the numbers x, y, or z.”  Note:  for any integer a, a mod n, read “a mod n” or “a modulo n” is the remainder of a divided by n.  The values of p must be equal to Germain numbers in order to show that there are no solutions to Fermat’s Last Theorem, xp + yp = zp. 

A Germain prime is a prime, p, such that 2p+1 is also a prime number.

Here is a list of prime numbers.

19, 11, 17, 53, 3, 5, 41, 59, 2, 13, 7, 23, 37, 29, 31

Find those that are Germain primes.  (Hint: There are 8 Germain primes in the list.)

 

 

 

 

An essential part of Germain’s proof attempt relied on t, a prime number, with no non-zero consecutive pth powers modulo t.  Here is an example and a non-example for you to see what she was intending.

Let p = 3.  This means 2p + 1 = 7 (2*3 + 1= 7).  Therefore t = 7.  Check the remainders of x3 mod 7 to make sure there are no consecutive answers.

To check mod 7:

1.     Cube the integer.

2.     Divide that answer by 7, taking the answer in the form of a whole number divisor and a remainder.

3.     Multiply the whole number answer for step 2 by the integer.

4.     Subtract the answer for step 3 from the answer of step 1.

5.     This answer will be the remainder.

Example: 43 = 64.  64/7 = 9 with a remainder.  7*9 = 63.  64 – 63 =1.  1 is the remainder.  Find the other answers.

13 mod 7 =

23 mod 7 =

33 mod 7 =

43 mod 7 =

53 mod 7 =

63 mod 7 =

 

Recall that Germain’s proof needed a prime p with no consecutive answers mod 2p + 1.  We will see that p = 7 does not satisfy this condition.  Notice that 2p + 1 = 2*7 + 1 = 15.  15 is not prime and thus 7 is not a Sophie Germain prime.  Use your calculator, following the steps above (except change cubing the integer to raising the integer to the 7th power and divide by 15 instead of 7) to find the remainder of:

47 mod 15 =

57 mod 15 =

67 mod 15 =

Are these answers consecutive?

 

Germain never succeeded in proving Fermat’s Last Theorem for a single exponent p, because she could not find an infinite number of primes t satisfying t has no non-zero consecutive pth powers mod t.  However she did design a method that would produce many primes t.  In addition, her method illustrated the fact that solutions to Fermat’s Last Theorem would have to be quite large, at least 30 decimal digits in size.

Today her prime numbers are still of interest.  The large Sophie Germain primes, those that are at least 150 digits long, are used in coding theory.  The largest Sophie Germain prime number, found in 2001, is p = 10943307 * 266452 – 1, with 20013 digits.

References

1.     http://www.chaucer.ac.uk/subjects/mathemat/mathematician.htm

2.     Dr. Sarah’s Maple Demo on Sophie Germain’s Modular Arithmetic Work on Fermat’s Last Theorem.

3.     The Biographical Dictionary of Women in Science: Pioneering Lives from Ancient Times to the Mid-20th Century. Harvey, Joy and Ogilvie, Marilyn.  Vol 1, A-K.  Routledge, New York, NY.