Mathematics and Computer Science
Arithmetic Progressions on Curves
This website was inspired by the website, A Collection of Algebraic Identities , maintained by Tito Piezas III; as well as Professor Edray Goins , who was my postdoctoral advisor. Please contact me if you have any questions, comments, suggestions regarding this website.
An arithmetic progression (AP) is a sequence of numbers such that the difference between any two consecutive numbers is constant. When we talk about an AP on a curve, we mean an AP in either the x or y coordinates.
Below we construct a table of curves with their longest known AP. Unless otherwise stated, we assume the curves are defined over the rationals.
I. Curves of the form \(y^2 = f(x) \)
Let \(f\) be a polynomial of degree \(d\) over the rationals. Given a curve of the form \(y^2 = f(x) \), we ask, what is the longest AP in the x or y coordinates?
| Curve | \( x \) or \( y \) AP | Longest Length for a Family |
Longest Length for an Example |
|---|---|---|---|
| \(y^2 = f(x), \deg f = 2 \) | \(x\)-AP | 8 (Allison) | 8 (Allison) |
| \(y^2 = f(x), \) \( \deg f = 3 \) | \(x\)-AP | 8 (Bremner; Campbell) | 8 (Bremner; Campbell) |
| \(y^2 = f(x), \) \( \deg f = 4 \) | \(x\)-AP | 12 (Ulas) | 14 (MacLeod; Alvarado) |
| \(y^2 = f(x), \) \( \deg f = 5 \) | \(x\)-AP | 12 (Alvarado) | 12 (Alvarado) |
| \(y^2 = f(x), \) \( \deg f = 6 \) | \(x\)-AP | 16 (Ulas) | 18 (Ulas) |
| \(y^2 = f(x), \deg f = d \) even | \( x \)-AP | at least \( 2d+2 \) (Alvarado) | at least \( 2d+2 \) (Alvarado) |
| \(y^2 = f(x), \deg f = d \) odd |
\( x \)-AP | at least \( 2d \) (Alvarado) | at least \( 2d \) (Alvarado) |
II. General Curves
| Curve | \( x \) or \( y \) AP | Longest Length for a Family |
Longest Length for an Example |
|---|---|---|---|
| \( x^2+y^2=1 \) | \( x \)-AP | 3 (Choudhry) | 3 (Choudhry) |
| Pellian equation: \( x^2-dy^2=m \) with positive integer \(d\) not a square and \( \gcd (m,d) \) square-free | \( y \)-AP | 4 (Dujella) | 7 (Dujella) |
| \( y=x^2 \) | \( y \)-AP | 3 (Fermat; Euler) | 3 (Fermat; Euler) |
| \( y=x^3 \) | \( y \)-AP | 3 (Legendre) | 3 (Legendre) |
| \( y=x^n \) | \( y \)-AP | 3 (Dénes; Darmon) | 3 (Dénes; Darmon) |
III. Elliptic Curves
| Curve | \( x \) or \( y \) AP | Longest Length for a Family |
Longest Length for an Example |
|---|---|---|---|
| \( y^2=x(x^2-n^2) \) where \( n \) is a squarefree integer | \( x \)-AP | 3 (Bremner) | 3 (Bremner) |
| Mordell curve: \( \quad y^2=x^3+k \) where \( k\neq 0 \) | \( x \)-AP | 4 (Lee & Velez; Alvarado) | 4 (Lee & Velez; Alvarado) |
| Mordell curve: \( \quad y^2=x^3+k \) where \( k\neq 0 \) | \( y \)-AP | 6 (Dey & Maji) | 6 (Lee & Velez; Alvarado) |
| Huff curve: \( \quad H_{a,b}: x(ay^2-1) = y(bx^2-1) \) with \( ab(a-b)\neq 0 \) | \( x \)-AP | 9 (Moody) | 9 (Moody) |
| Edwards curve: \( \quad E_d : x^2+y^2 = 1+dx^2+y^2 \) with \( d \neq 1 \) | \( x \)-AP | 9 (Moody) | 9 (Moody) |
| Hessian curve: \( \quad H_d : x^3+y^3-dxy+1 = 0 \) with \( d \in \mathbb{Z} \) | \( x \)-AP | 5 (Tengely) |
One can also talk about simultaneous arithmetic progressions (SAP's) on elliptic curves. A curve contains a simultaneous arithmetic progression of length \( n \) if the \( \{x_1, x_2, ..., x_n\} \) are in arithmetic progression and there exists a permutation \( \sigma \in S_n \) such that \( \{y_{\sigma(1)}, y_{\sigma(2)}, ..., y_{\sigma(n)} \} \) are in arithmetic progression.
| Curve | Longest Length for a Family |
Longest Length for an Example |
|---|---|---|
| elliptic curve | 6 (Garcia-Selfa) | |
| Mordell curve | 3 (Dey & Maji) | 3 (Dey & Maji) |
| \( y^2 +axy+by=x^3+cx^2+dx+e \) over \( \mathbb{R} \) | If a SAP exists of length \( k \), then \( k\leq 4319 \). (Schwartz) |
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