Dora Musielak
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Sophie Germain overcame gender stigmas and a lack of formal education yet proved, after Euler, that for all prime exponents n less than 100 Fermat’s Last Theorem holds. Hidden behind a man’s name, her brilliance as mathematician was first discovered by three of the greatest scholars of the eighteenth century¾ Lagrange, Gauss, and Legendre.
In Sophie’s Diary, Germain comes to life through a fictionalized journal that intertwines mathematics with historical descriptions of the brutal events that took place in Paris between 1789 and 1793. This format provides a plausible perspective of how a young Sophie could have learned mathematics on her own—both fascinated by numbers and eager to master tough subjects without a teacher’s guidance. Her passion for mathematics is integrated into her personal life as an escape from societal outrage. Sophie’s Diary is suitable for a variety of readers¾both young and old, mathematicians and novices¾who will be inspired and enlightened on a field of study made easy as is told through the intellectual and personal struggles of an exceptional young woman.
Dora Elia González y Musielak pursued advanced degrees in aerospace engineering stimulated by her love of mathematics. In addition to conducting research in space rocket propulsion, Dr. Musielak teaches science and mathematics courses at the University of Texas in Arlington.
Tuesday¾January 12, 1790
There is a special equation nobody knows how to solve. At first glance the equation appears very simple: xn + yn = zn. However, not even the best mathematicians in the world know how to solve it. This is a Diophantine equation, named after Diophantus of Alexandria. There is no general method for solving such equations. What makes the equation more fascinating is a mysterious note that a mathematician wrote in the margin of a book before he died. His name was Pierre de Fermat, a Frenchman who lived about one hundred and thirty-five years ago.
Monsieur de Fermat claimed that the equation has no non-zero integer solutions for x, y, and z, when n > 2. After his death, his son found his father’s Arithmetica, the book written by Diophantus hundreds of years ago¾a book I am studying. The son discovered that in the margin of one page Fermat had written: “To divide a cube into two other cubes, a fourth power or in general any power whatever, into two powers of the same denomination above the second is impossible, and I have assuredly found an admirable proof of this, but the margin is too narrow to contain it.”
Fermat meant that there are no whole number solutions for equations like these: x3 + y3 = z3, x4 + y4 = z4, x5 + y5 = z5, and so on. These are equations of the general form xn + yn = zn. I’ve solved the equation x2 + y2 = z2 many times. Using the Pythagorean Theorem to solve the triangle with two sides x and y equal to 3 and 4, respectively, yields the sum of perfect squares: 32 + 42 = 52. Well, this equation is easy.
But Fermat claimed that when the exponent n is greater than 2, the equation xn + yn = zn has no solutions! It is very difficult to prove this statement because there are an infinite number of equations and an infinite number of possible values for x, y, and z. A great Swiss mathematician named Leonhard Euler obtained only a partial proof for the case n = 3. A full proof would require the inclusion of all cases to n infinite.
Would it not be wonderful that I could prove one day Monsieur Fermat’s claim?
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