14-term Arithmetic Progressions on Quartic Elliptic Curves

Allan J. MacLeod Department of Mathematics and Statistics University of Paisley High St. Paisley, Scotland PA1 2BE UK allan.macleod@paisley.ac.uk

Abstract Let P_{4}(x) be a rational quartic polynomial which is not the square of a quadratic. Both Campbell and Ulas considered the problem of finding an rational arithmetic progression x_{1}, x_{2}, …, x_{n}, with P_{4}(x_{i}) a rational square for 1 ≤ i ≤ n. They found examples with n = 10 and n = 12. By simplifying Ulas’ approach, we can derive more general parametric solutions for n = 10, which give a large number of examples with n = 12 and a few with n = 14.

1 Introduction

Let P_{4}(x) = ax^{4} + bx^{3} + cx^{2} + dx + e be a rational polynomial which is not the square of a quadratic. If P_{4}(s) = t^{2} for rational (s, t) then P_{4} is birationally equivalent to an elliptic curve. Campbell [1] investigated the possibility of such curves having a rational arithmetic progression x_{1}, x_{2}, …, x_{n} of arguments such that P_{4}(x_{i}) = y_{i}^{2}, i = 1, …, n, and provided an example with n = 12.

Ulas [3] approached the problem in a different manner and succeeded in deriving a parametric solution for n = 10, and used a specific elliptic curve to derive a family of solutions for n = 12. He also introduced the name quartic elliptic curve for such curves.

In this short note, we show that Ulas’ approach can be considerably simplified. This allows us to find a more general parametric solution for n = 10, which generates a large number of solutions for n = 12 and a few for n = 14, thus answering the open question at the end of Ulas’ paper.

2 Algebraic Formulation

Ulas considers the arithmetic progression ( AP for short ) {1, 2, 3, …, 9, 10}, and fits P_{4} at {1, 2, 3, 4, 5} to the set of values {p^{2}, q^{2}, r^{2}, s^{2}, t^{2}}. He then forces P_{4} at {6, 7, 8, 9, 10} to fit the set {t^{2}, s^{2}, r^{2}, q^{2}, p^{2}}.

It is a simple observation that this is equivalent to enforcing P_{4} to be symmetric about x = 5.5. But we can easily translate the x-arguments so that the quartic is symmetric about x = 0, which makes P_{4} an even function. Thus, the new approach is to assume P_{4}(x) = ax^{4} + bx^{2} + c.

Set P_{4}(1) = P_{4}(−1) = p^{2}, P_{4}(3) = P_{4}(−3) = q^{2}, and P_{4}(5) = P_{4}(−5) = r^{2}. It is standard linear algebra to find

a = \frac{2p^{2} − 3q^{2} + r^{2}}{384}

b = −\frac{34p^{2} − 39q^{2} + 5r^{2}}{192}

c = \frac{150p^{2} − 25q^{2} + 3r^{2}}{128}

Enforcing P_{4}(7) = P_{4}(−7) = s^{2} implies that we must have

s^{2} = 5p^{2} − 9q^{2} + 5r^{2}

Since (1, 1, 1, 1) is an obvious solution, we can parameterize as follows

p = −5u^{2} − 9v^{2} + 18uv + 5w^{2} − 10uw

q = 5u^{2} − 10uv + 9v^{2} − 10vw + 5w^{2}
r = 5u^{2} − 10uw − 9v^{2} + 18vw − 5w^{2}

Now, setting P_{4}(9) = P_{4}(−9) = t^{2}, gives

t^{2} = 25u^{4} + Eu^{3} + F u^{2} + Gu + H  (1)

with

(i) E = 40(15w − 7v),

(ii) F = 2(347v^{2} − 680vw + 25w^{2}),

(iii) G = −8(63v^{3} + 65v^{2}w − 235vw^{2} + 75w^{3}),

(iv) H = 81v^{4} + 1440v^{3}w − 2330v^{2}w^{2} + 800vw^{3} + 25w^{4}

We consider equation (1) as a quartic in u, with v, w as free parameters. Investigations show simple rational values of u which give rational t. One such is u = 9v/5 − w which gives t = 2(3v − 5w)(6v − 5w)/5. The existence of this point shows that the quartic is birationally equivalent to an elliptic curve.

Using the method described in Mordell [2], we find that the elliptic curve can be written in the form

J^{2} = K^{3} − (137v^{2} − 680vw + 475w^{2})K^{2}  (2)
+84(54v^{4} − 495v^{3}w + 1425v^{2}w^{2} − 1625vw^{3} + 625w^{4})K

with the transformation

u = \frac{2K(7v − 15w) + J − 42(27v^{3} − 195v^{2}w + 325vw^{2} − 125w^{3})}{5K − 210(3v^{2} − 20vw + 25w^{2})}  (3)

This elliptic curve has an obvious point of order 2, namely (0, 0). Numerical investigations suggest this is the only torsion point, but this would appear to be difficult to prove.

These numerical investigations, using small integer values of v and w, also indicate that the curve has rank at least 2, and often 3 or higher, with many integer points of small height.

Investigations found several algebraic formulae for points on the curves. Two of these are

( 2(3v − 5w)(6v − 5w) , ±10(v − 5w)(3v − 5w)(6v − 5w) )

and

( 6(3v − w)^{2} , ±6(3v − w)(3v^{2} − 106vw + 91w^{2}) )

It is possible to use these and the previous algebraic expressions to derive parametric formulae for quartic elliptic curves with 10-term APs, but these expressions will only cover a small fraction of possible curves. In the next section we employ a simple search procedure for 12-term and 14-term curves.

3 Curve Searching

As was stated before, the elliptic curves (2) contain a large number of integer points of small height. For a selected pair of (v, w) values, we searched for rational points with |K| < 10^{7}. From these points, we generated u, then (p, q, r) and finally (a, b, c).

For the quartic produced, we tested if P_{4}(11) was a square, giving a 12-term AP. We found several hundred such curves with (v, w) in the range |v| + |w| < 100.

For the successful curves, P_{4}(13) was tested to be square. This discovered a total of 4 quartics giving a 14-term AP. These are

(a) −17x^{4} + 3130x^{2} + 8551,

(b) 2002x^{4} − 226820x^{2} + 18168514,

(c) 3026x^{4} − 222836x^{2} + 3709234,

(d) 34255x^{4} − 1436006x^{2} + 447963175

Quartic (a) was derived from 3 different (v, w) pairs while (c) came from 2 different pairs. Given the paucity of 14-term curves compared to the abundance of 12-term curves, we would be very unlikely to find a 16-term curve with this method.

4 Acknowledgement

The author would like to express his sincere thanks to the anonymous referee for several very useful comments on the original submission.

References

[1] G. Campbell, A note on arithmetic progressions on elliptic curves, J. Integer Sequences, Paper 03.1.3, 2003.

[2] L. J. Mordell, Diophantine Equations, Academic Press, New York, 1969.

[3] M. Ulas, A note on arithmetic progressions on quartic elliptic curves, J. Integer Sequences, Paper 05.3.1, 2005.

2000 Mathematics Subject Classification: Primary 11G05; Secondary 11B25. Keywords: arithmetic progression, quartic, elliptic curve.

Received October 25 2005; revised version received November 18 2005. Published in Journal of Integer Sequences, November 18 2005.

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