It came up when I was trying to solve the equality $\sum_{i = 1}^{x}i^2 =n^2$ for integers $x$ and $i$. I've reduced it to the equation $2x^3+3x^2+x-6n^2 = 0$, which I don't know how to tackle. Is there some sort of method for solving a diophantine equation like this?
2026-04-07 01:03:56.1775523836
Solving the equation $2x^3+3x^2+x-6n^2 = 0$
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1
Alrighty; some elementary solutions were found in the 1980's. Henri Cohen summarizes these on pages 424 to 427 of
Number Theory: Volume I: Tools and Diophantine Equations, see
https://books.google.com/books?id=qxCWoFO-oxYC&pg=PA424&lpg=PA424&dq=lucas+square+pyramid&source=bl&ots=2DN6KgNBgA&sig=Gq__ESOX33BwpACDEWB6zEXLJ60&hl=en&sa=X&ei=S1nEVNmLN8SyoQS054GgCQ&ved=0CEEQ6AEwCjgK#v=onepage&q=lucas%20square%20pyramid&f=false