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
Trans-elliptic Diophantine Equations
Wolfram’s Challenge Equation
Linear Diophantine
Equations
Q1) How many beetles and spiders are in
a box containing 46 legs?
Linear Diophantine
Equations
Q1) How many beetles and spiders are in
a box containing 46 legs?
6x + 8y = 46
Quadratic Diophantine
Equations
Q2) x2 + y2 = z2
Q3) In 1066 Harold of Saxon claimed 61
squares of men. When he added
himself they formed one mighty square.
Quadratic Diophantine
Equations
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
Question
Q) For which N does
x2 – Ny2 = 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
y2 = p(x) where deg(p) = 3 or 4
y2 = x3 - x
1.5
1
0.5
-1
-0.5
0.5
-0.5
-1
-1.5
1
1.5
2
y2 = x3 – x + 1
1.5
1
0.5
-1
-0.5
0.5
-0.5
-1
-1.5
1
1.5
2
Hyperelliptic Curves
y2 = p(x) where deg(p) > 4
y2 = x5 – 5x - 1
3
2
1
-1.5
-1
-0.5
0.5
-1
-2
-3
1
1.5
2
Superelliptic Curves
y3 = p(x) where deg(p) > 3
y3 = x4 – x - 1
3
2
1
-2
-1
1
-1
-2
-3
2
y3 = x5 – 5x – 1
3
2
1
-2
-1
1
-1
-2
-3
2
Trans-Elliptic Equations
y5 = x4 – 3x – 3
3
2
1
-2
-1
1
-1
-2
-3
2
y5 = x5 – 5x - 1
3
2
1
-2
-1
1
-1
-2
-3
2
Wolfram’s Challenge Equation
y3 = x4 + xy + a
y3 = x4 + xy + 5
4
3.5
3
2.5
2
1.5
1
0.5
-3
-2
-1
1
2
3
y3 = x4 + xy + 5
y=
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?
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?
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.
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, …
x2 – 61y2 = 1
S2)
1,766,319,0492 – 61.226,153,9802 = 1
x2 = 3,119,882,982,860,264,401
x3 + y3 = z4
S3)
No sol. to x3 + y3 = z3 by Fermat.
33 + 53 =
152
1523.33 + 1523.53 = 1523.152
4563 + 7603 = 1524
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
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