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Using Elliptical Curves to Solve an Arithmetic Sequence

Date: 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

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

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 

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 

  mwrank Page 

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 

In Pari, you set up the elliptic curve with

  e = ellinit([0,0,0,0,121])

You can ask for its torsion group with


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 
Associated Topics:
College Number Theory

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