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A Naïve Introduction to Trans-Elliptic Diophantine Equations Donald E. Hooley Bluffton University Bluffton, Ohio. Outline. Linear Diophantine Equations Quadratic Diophantine Equations Hilbert’s 10th Problem Thue’s Theorem Elliptic Curves Hyperelliptic Curves Superelliptic Curves

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  1. A Naïve Introduction to Trans-Elliptic Diophantine EquationsDonald E. HooleyBluffton UniversityBluffton, Ohio

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

  3. Linear DiophantineEquations Q1) How many beetles and spiders are in a box containing 46 legs?

  4. Linear DiophantineEquations Q1) How many beetles and spiders are in a box containing 46 legs? 6x + 8y = 46

  5. Quadratic DiophantineEquations Q2) x2 + y2 = z2 Q3) In 1066 Harold of Saxon claimed 61 squares of men. When he added himself they formed one mighty square.

  6. Quadratic DiophantineEquations Q2) x2 + y2 = z2 Q3) In 1066 Harold of Saxon claimed 61 squares of men. When he added himself they formed one mighty square. x2 – 61y2 = 1

  7. Question Q) For which N does x2 – Ny2 = 1 have positive solutions?

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

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

  10. Elliptic Curves y2 = p(x) where deg(p) = 3 or 4

  11. y2 = x3 - x

  12. y2 = x3 – x + 1

  13. Hyperelliptic Curves y2 = p(x) where deg(p) > 4

  14. y2 = x5 – 5x - 1

  15. Superelliptic Curves y3 = p(x) where deg(p) > 3

  16. y3 = x4 – x - 1

  17. y3 = x5 – 5x – 1

  18. Trans-Elliptic Equations y5 = x4 – 3x – 3

  19. y5 = x5 – 5x - 1

  20. Wolfram’s Challenge Equation y3 = x4 + xy + a

  21. y3 = x4 + xy + 5

  22. y3 = x4 + xy + 5 y =

  23. Questions Q0) Find distinct positive integers x, y, z so that x3 + y3 = z4. Q1) The trans-elliptic Diophantine equation y3 = x4 + xy + 5 has solutions (1, 2) and (2, 3). Does it have any more solutions?

  24. More Questions Q2) The trans-elliptic Diophantine equation y3 = x4 + xy + 59 has solutions (1, 4), (4, 7) and (5, 9). Does it have any more solutions? Q3) For which integers a does the Diophantine equation y3 = x4 + xy + a have multiple solutions?

  25. 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, McGraw-Hill Book Co., Inc., 1939. S. Wolfram, A New Kind of Science, Wolfram Media (2002) 1164.

  26. Solutions S1) 6x + 8y = 46 Sol. 3x + 4y = 23 3x 3 mod 4 x 1 mod 4 x = 1 + 4t so x = 1, 5, … 3(1 + 4t) + 4y = 23 3 + 12t + 4y = 23 y = (23 – 3 – 12t) / 4 = 5 – 3t so y = 5, 2, …

  27. x2 – 61y2 = 1 S2) 1,766,319,0492 – 61.226,153,9802 = 1 x2 = 3,119,882,982,860,264,401

  28. x3 + y3 = z4 S3) No sol. to x3 + y3 = z3 by Fermat. 33 + 53 = 152 1523.33 + 1523.53 = 1523.152 4563 + 7603 = 1524

  29. y3 = x4 + xy + 5 S4) Methods: 1) Modular arithmetic If x = y = 0 mod 2 then y3 = 0 mod 2 but x4 + xy + 5 = 1 mod 2

  30. y3 = x4 + xy + 5 2) Convergents of continued fractions 3) Fermat’s method of descent 4) Bound and search Check y3 - x4 – xy = 5 No other solutions for -10,000,000 < x < 10,000,000

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