Thank you, Fred! Very interesting! (It's probably high time I learned how to use PARI.) --Dan
On Jan 27, 2015, at 5:16 PM, Fred W. Helenius <fredh@ix.netcom.com> wrote:
On 1/26/2015 11:29 PM, Daniel Asimov wrote:
For integer n >= 0, we know
F(n) := Sum_{1 <= k <= n} k^2 = n(n+1)(2n+1)/6 (= n^3/3 + n^2/2 + n/6) .
It's also well known that if F(n) is an integer square, the only integer solutions are (n, F(n)) in {(0, 0), (1, 1), (24, 70^2)}.
Those are the solutions with n >= 0; there's also F(-1) = 0.
QUESTION: ---------
What are the rational solutions p/q, s/t to the equation
F(p/q) = (s/t)^2 ???
With a little assistance from PARI/GP, I find that the elliptic curve k^2 = n^3/3 + n^2/2 + n/6 is equivalent to the minimal model y^2 = x^3 - 36x via the substitution k = y/72, n = (x - 6)/12. This is curve 576h2 in Cremona's tables. It has rank 1 (with (x, y) = (-3, 9) as a generator, corresponding to (n, k) = (-3/4, 1/8)) and a four-element torsion subgroup (consisting of the points with y = 0 and the point at infinity). So there are infinitely many rational points on the curve; some others with small denominators are (n, k) = (1/2, 1/2), (-2/3, 1/9) and (-49/50, 7/125).
-- Fred W. Helenius fredh@ix.netcom.com
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