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

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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

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Linear Diophantine Equations Q1) How many beetles and spiders are in a box containing 46 legs?

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

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Quadratic Diophantine Equations Q2) x 2 + y 2 = z 2 Q3) In 1066 Harold of Saxon claimed 61 squares of men. When he added himself they formed one mighty square.

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Quadratic Diophantine Equations Q2) x 2 + y 2 = z 2 Q3) In 1066 Harold of Saxon claimed 61 squares of men. When he added himself they formed one mighty square. x 2 – 61y 2 = 1

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

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

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

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

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y 2 = x 3 - x

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y 2 = x 3 – x + 1

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

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y 2 = x 5 – 5x - 1

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

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y 3 = x 4 – x - 1

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y 3 = x 5 – 5x – 1

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

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y 5 = x 5 – 5x - 1

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

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y 3 = x 4 + xy + 5

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y =

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Questions Q0) Find distinct positive integers x, y, z so that x 3 + y 3 = z 4. Q1) The trans-elliptic Diophantine equation y 3 = x 4 + xy + 5 has solutions (1, 2) and (2, 3). Does it have any more solutions?

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More Questions Q2) 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? Q3) For which integers a does the Diophantine equation y 3 = x 4 + xy + a have multiple solutions?

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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).http://eprint.iacr.org/2003/060.pdf 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.

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

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x 2 – 61y 2 = 1 S2) 1,766,319,049 2 – 61. 226,153,980 2 = 1 x 2 = 3,119,882,982,860,264,401

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x 3 + y 3 = z 4 S3) No sol. to x 3 + y 3 = z 3 by Fermat. 3 3 + 5 3 = 152 152 3. 3 3 + 152 3. 5 3 = 152 3. 152 456 3 + 760 3 = 152 4

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y 3 = x 4 + xy + 5 S4) Methods: 1) Modular arithmetic If x = y = 0 mod 2 then y 3 = 0 mod 2 but x 4 + xy + 5 = 1 mod 2

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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 -10,000,000 < x < 10,000,000

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