A Nave Introduction to TransElliptic Diophantine Equations Donald

A Naïve Introduction to Trans-Elliptic Diophantine Equations Donald E. Hooley Bluffton University Bluffton, Ohio

Outline • • • Linear Diophantine Equations Quadratic Diophantine Equations Hilbert’s 10 th Problem Thue’s Theorem Elliptic Curves Hyperelliptic Curves Superelliptic Curves Trans-elliptic Diophantine Equations Wolfram’s Challenge Equation

Linear Diophantine Equations Q 1) How many beetles and spiders are in a box containing 46 legs?

Linear Diophantine Equations Q 1) How many beetles and spiders are in a box containing 46 legs? 6 x + 8 y = 46

Quadratic Diophantine Equations Q 2) x 2 + y 2 = z 2 Q 3) In 1066 Harold of Saxon claimed 61 squares of men. When he added himself they formed one mighty square.

Quadratic Diophantine Equations Q 2) x 2 + y 2 = z 2 Q 3) In 1066 Harold of Saxon claimed 61 squares of men. When he added himself they formed one mighty square. x 2 – 61 y 2 = 1

Question Q) For which N does x 2 – Ny 2 = 1 have positive solutions?

Hilbert’s Tenth Problem Is there a general algorithm to decide whether a given polynomial Diophantine equation with integer coefficients has a solution?

Thue’s Theorem A polynomial function F(x, y) = a with deg(F) > 2 has only a finite number of solutions.

Elliptic Curves y 2 = p(x) where deg(p) = 3 or 4

y 2 = x 3 - x

Hyperelliptic Curves y 2 = p(x) where deg(p) > 4

y 2 = x 5 – 5 x - 1

Superelliptic Curves y 3 = p(x) where deg(p) > 3

Trans-Elliptic Equations y 5 = x 4 – 3 x – 3

y 5 = x 5 – 5 x - 1

Wolfram’s Challenge Equation y 3 = x 4 + xy + a

y 3 = x 4 + xy + 5

y 3 = x 4 + xy + 5 y=

Questions Q 0) Find distinct positive integers x, y, z so that x 3 + y 3 = z 4. Q 1) The trans-elliptic Diophantine equation y 3 = x 4 + xy + 5 has solutions (1, 2) and (2, 3). Does it have any more solutions?

More Questions Q 2) The trans-elliptic Diophantine equation y 3 = x 4 + xy + 59 has solutions (1, 4), (4, 7) and (5, 9). Does it have any more solutions? Q 3) For which integers a does the Diophantine equation y 3 = x 4 + xy + a have multiple solutions?

References A. H. Beiler, Recreations in the Theory of Numbers – The Queen of Mathematics Entertains, Dover Pub. , Inc. , 1964. Y. Bilu and G. Hanrot, Solving superelliptic Diophantine equations by Baker's method, Compositio Math. 112 (1998) 273 -312. U. Dudley, Elementary Number Theory, W. H. Freeman and Co. , San Francisco, 1969. J. W. Lee, Isomorphism Classes of Picard Curves over Finite Fields, http: //eprint. iacr. org/2003/060. pdf (accessed August 2007). R. J. Stroeker and B. M. M. De Weger, Solving elliptic Diophantine equations: the general cubic case, Acta Arithmetica, LXXXVII. 4 (1999) 339 -365. J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, Mc. Graw-Hill Book Co. , Inc. , 1939. S. Wolfram, A New Kind of Science, Wolfram Media (2002) 1164.

Solutions S 1) 6 x + 8 y = 46 Sol. 3 x + 4 y = 23 3 x 3 mod 4 x 1 mod 4 x = 1 + 4 t so x = 1, 5, … 3(1 + 4 t) + 4 y = 23 3 + 12 t + 4 y = 23 y = (23 – 12 t) / 4 = 5 – 3 t so y = 5, 2, …

x 2 – 61 y 2 = 1 S 2) 1, 766, 319, 0492 – 61. 226, 153, 9802 = 1 x 2 = 3, 119, 882, 982, 860, 264, 401

x 3 + y 3 = z 4 S 3) No sol. to x 3 + y 3 = z 3 by Fermat. 33 + 53 = 1523. 33 + 1523. 53 = 1523. 152 4563 + 7603 = 1524

y 3 = x 4 + xy + 5 S 4) Methods: 1) Modular arithmetic If x = y = 0 mod 2 then y 3 = 0 mod 2 but x 4 + xy + 5 = 1 mod 2

y 3 = x 4 + xy + 5 2) Convergents of continued fractions 3) Fermat’s method of descent 4) Bound and search Check y 3 - x 4 – xy = 5 No other solutions for -10, 000 < x < 10, 000