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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., Y. Bilu and G. Hanrot, Solving superelliptic Diophantine equations by Baker's method, Compositio Math. 112 (1998) U. Dudley, Elementary Number Theory, W. H. Freeman and Co., San Francisco, J. W. Lee, Isomorphism Classes of Picard Curves over Finite Fields, (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) J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill Book Co., Inc., 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 = t + 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 – ,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 = = = 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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