Using Elliptical Curves to Solve an Arithmetic SequenceDate: 05/02/2006 at 23:22:39 From: Ann Subject: 3 rational numbers in an arithmetic sequence I am looking for the first three terms of an arithmetic sequence. They are three rational numbers whose product is 11. Thanks for the help! I have found along with other math teachers that we are unable to come up with a rational solution for the common difference. We have come to conclude there is no solution. The terms are x, x+d, x+2d. The product of the terms is (x^3) + 3d(x^2) + 2(d^2)x = 11. According to the Rational Roots Theorem the only roots can be +-1, and +-11, and we have already tried all four of those solutions and none of them come out to be a rational common difference. Date: 05/03/2006 at 20:46:05 From: Doctor Vogler Subject: Re: 3 rational numbers in an arithmetic sequence Hi Ann, Thanks for writing to Dr. Math. You have proven that there is no solution when d is an integer. But there ARE solutions where d is not an integer, and then the Rational Roots Theorem does not apply (or at least not until you clear denominators). In fact, you have an elliptic curve. Are you familiar with elliptic curves? If you take u = -11/(x+d) v = 11d/(x+d) then this changes your equation into the Weierstrass form v^2 = u^3 + 121, for which you want to find rational points (u, v). It turns out that this elliptic curve has torsion group Z/3Z and rank one. That means that every solution is a multiple (in the elliptic curve group law) of the base point P = (u, v) = (12, 43), or such a multiple plus a torsion point T = (0, 11) -T = (0, -11) 3T = 0 For example, P + T = (-44/9, 55/27) P - T = (33/4, -209/8) 2P = (2280/1849, 881327/79507) 2P + T = (-7095/5776, -4791611/438976) 2P - T = (71896/225, -19277819/3375) 3P = (-440588231/99082116, 5672078027405/986263382664) etc. Yes, it is normal for the digits to grow quickly as n grows. The "height" of a point is roughly the number of digits in the numerator and the denominator of the u coordinate, and the height of nP is n^2 times the height of P. That means that the number of digits grows quite quickly. (Adding T doesn't change the height by much.) In any case, you can solve for x and d from u and v by x = (v-11)/u d = -v/u For example, doing this for P gets (x, d) = (8/3, -43/12) and so (x, x+d, x+2d) = (8/3, -11/12, -9/2). and you can check that the product of these three numbers is 11. Similarly, doing the same for P + T gets (x, d) = (11/6, 5/12) and P - T gives (x, d) = (-9/2, 19/6) while 2P gives (x, d) = (225/3268, -881327/98040. If you wonder about negative d's, it turns out that the negative of the point will always change the sign of v and leave u unchanged, which means that d will change its sign, and x will turn into x+2d, which just reverses the order of the three numbers in your arithmetic progression. If you're not sure what an elliptic curve is, what the group law is, or what in the world I'm talking about, please refer to Cubic Diophantine Equation in Three Variables http://mathforum.org/library/drmath/view/66650.html You can also learn more than you ever wanted to know about elliptic curves by searching the Internet. If you know about elliptic curves and you want to know how I found its rank and a generator, I did it the way nearly everyone does: By running John Cremona's program "mwrank" which you can find on and download from his home page at Home Page http://www.maths.nottingham.ac.uk/personal/jec/ mwrank Page http://www.maths.nottingham.ac.uk/personal/jec/mwrank/index.html Then you can use a math program to do simpler calculations like computing points on the curve. One of my favorites (especially since it's really good with elliptic curves) is GNU Pari, which you can download for free from http://pari.math.u-bordeaux.fr/ In Pari, you set up the elliptic curve with e = ellinit([0,0,0,0,121]) You can ask for its torsion group with elltors(e) and it will tell you that it has size 3 and is generated by (0,11). Our base point is p = [12,43] and our torsion point is t = [0,11] and then you can compute q = n*P, or q = n*P + T, or q = n*P - T by saying n = 1 q = ellpow(e, p, n) q = elladd(e, ellpow(e, p, n), t) q = elladd(e, ellpow(e, p, n), -t) and from q = (u, v), you can compute x and d by [x = (q[2]-11)/q[1], d = (-q[2])/q[1]] Then you can check: x*(x + d)*(x + 2*d) and so on. If you have any questions about this or need more help, please write back and show me what you have been able to do, and I will try to offer further suggestions. - Doctor Vogler, The Math Forum http://mathforum.org/dr.math/ |
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