Teaching Mathematics, History, and the History of Mathematics

Dedicated to the memory of Louise Karlquist

(who knew all the QA numbers by heart)

And thanks to

• The Ohio State University Libraries

particularly

• Danny Dotson

• Mary Scott

Fermat’s Last Theorem

1993

• Andrew Wiles lectures in Cambridge• “Modular forms, elliptic curves, and Galois

representations”• Concluded with Fermat’s last theorem• Xn + Yn = Zn is impossible in positive whole

numbers if n > 2• Flurry of email• Front page New York Times

Found a gap in the proof

• Proof withdrawn in December 1993• Wiles student Richard Taylor contributed• Proof complete by October 1994• Published April 1995, Annals of Mathematics

Greek Texts of Euclid

• J.L. Heiberg – 1883 – 1916• Sir Thomas L. Heath, Dover• Stamatis, 1974 (in library)• New (cheap) reprint with new translation by

Richard Fitzpatrick

OSU’s Euclid, 1570

• The elements of Geometrie of the most auncient Philosopher Euclid of Megara. Faithfully (now first) translated into the Englishe toung by H. Billingsley, Citizen of London. Whereunto are annexed certaine Scholies, Annotations, and Inuentions, of the best

Mathematicians both of time past, and in this our age. With a very fruitful Paeface made by M.I. Dee.

Al-Tusi

Plimpton 322

Pythagorean Triples in P-322

• 32 + 42 = 52

• 52 + 122 = 132

• 1192 + 1202 = 1692

• 652 + 722 = 972

• 5412 + 5462 = 7692

• 127092 + 135002 = 185412

Plimpton 322

• Otto Neugebauer: “The Exact Sciences in Antiquity”

• R.C. Buck: “Sherlock Holmes in Babylon”

• Eleanor Robson: Words and Pictures, New Light on Plimpton 322

Bill Casselman on Plimpton 322

Plimpton 322

Pythagorean Triples

• X2 + Y2 = Z2

• X = m2 – n2

• Y = 2mn• Z = m2 + n2

Diophantus of Alexandria 200-284

• Links to the MacTutor history of math site at St. Andrews, Scotland

Bachet’s Diophantus Cover Page

Pierre de Fermat 1601 -1665

Fermat’s Marginal Note

Cubem autem in duos cubos,

aut quadratoquadratum in duos quadratoquadratos,

et generaliter nullum in infinitum ultra quadratum potestatem

in duos eiusdem nominis fas est dividere:

cuius rei demonstrationem mirabilem sane detexi.

Hanc marginis exiguitas no caparet.

Fermat in the margin

Fermat’s Last Theorem

• NOVA page from PBS • MacTutor Page• Sophie Germain • Kummer and algebraic number theory

• Andrew Wiles, 1993

X4 + Y4 ≠ Z4

• Fermat really proved this case

• The “method of infinite descent”

Leonhard Euler 1707-1783

X3 + Y3 ≠ Z3

• Euler’s contribution 1770

• Small gap fixed by Gauss

Sophie Germain 1776-1831

• Pen-name letters to Gauss

• Sophie Germain primes

• p and 2p+ 1 both prime

• 3 (and 7); 5 (and 11); 11(and 23)

• Case I

• Xp + Yp ≠ Zp if p does not go into X,Y,Z

Gabriel Lame 1795-1870

• Cyclotomic integers

• ζ = cos (2π/p) + I sin(2π/p)

• Xp + Yp = (X+Y)(X + ζ Y)…(X+ ζp-1Y)

• Arithmetic in the ring Z[ζ]

• Unique factorization into primes??

• Nope, too bad.

Ernst Eduard Kummer 1810-1893

• Ideals and ideal numbers• Unique factorization into ideal factors• “class number” measures failure of prime

factorization of numbers• Regular Primes• Kummer: criterion for regular primes, FLT for

regular primes • 3,5,7,11,13,17,19,23,29,31, (not 37), 41, 43,53,

59, 61, (not 67), 71, 73, 79, . . .

So – FLT motivated a lot of algebraic number theory

A Century of Computation

• Wolfskehl prize • Flurry of wrong proofs• Regular primes not so hard• Irregular primes tough but possible• Exponent by exponent• Try out new computers! • Dead end?

What next?

• “Elliptic Curves”• Arose from integrals trying to measure the length

of an ellipse• Not an ellipse! Cubic• Group Structure• Really hot stuff starting in the 50s• Main line algebraic number theory• Andrew Wiles – dissertation at Cambridge

Taniyama – Shimura – Weil

• Technical conjecture describing elliptic curves • Frey curve 1984• TSW implies Fermat• Once again, FLT inspired main line math• Andrew Wiles started working (secretly) on TSW• Seven years in the attic

Andrew Wiles in Cambridge

Back to 1993

• Proof withdrawn in December 1993• Wiles student Richard Taylor contributed• Proof complete by October 1994• Published April 1995, Annals of Mathematics• Full force of Taniyama-Shimura-Weil now

proved

Maybe a Moral?

• Fermat’s Last Theorem easily understood and looks like just a puzzle

• Motivated a great deal of mathematics– Rings of Algebraic Integers– Applications of Elliptic Curves– Even more Galois Theory

• Mathematics swings from very concrete to the very abstract and back again.

What about Math 504?

• Required by State of Ohio for a secondary teaching license

• Strongly recommended by the College of Education for admission to the MSAT program

• Audience is mostly math majors who aspire to high school teaching

• Varying math skills, writing skills, history skills, geography skills, . . . . . . . .

What to emphasize??

• History vs. Heritage? – Grattan-Guinness

• Mathematics as a human endeavor? • Historical Approach?• Capstone for a Math major?• Using History to teach Math?

Third Writing Course

• Babcock Committee 1988/McHale Report 2006

• Book Review• Biography• Long Paper• Oral Presentation

But only ten weeks . . .