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This paper explores the fascinating world of Diophantine equations, focusing on linear, quadratic, and trans-elliptic forms. It begins with foundational concepts like Linear Diophantine equations and progresses towards complex topics such as elliptic and hyperelliptic curves, Hilbert's tenth problem, and Thue's theorem. The work also presents various questions and challenges related to these equations, assessing their solvability and discussing their implications in number theory. Aimed at providing a comprehensive overview, the introduction delves into both elementary and advanced theories that underlie these equations.

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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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