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Number Theory A problem on Diophantine Analysis.


Well-known member
MHB Math Helper
Mar 22, 2013
I was toying around recently with diophantine forms \(\displaystyle kX^2 + lY^3 + mZ^5 = 0\) over \(\displaystyle \mathbb{Q}^3\).

After a complex bit of elliptic modular methods applied at it, I managed to (partially) solve a special case (k, l, m) = (1, 1, -1728). The parameterization I found is a triple of 11, 20 and 30 degree homogeneous polynomials (\(\displaystyle u, v \in \mathbb{Q}\)):

\(\displaystyle X = u^{30} + v^{30} + 522(u^{25}v^5 - u^5v^{25}) - 10005(u^{20}v^{10} + u^{10}v^{20}) \\ Y = -u^{20} - v^{20} + 228(u^{15}v^5 - u^5v^{15}) - 494u^{10}v^{10} \\ Z = uv(u^{10} + 11 u^5 v^5 - v^{10}) \)

Indeed, I realized just a few hours ago, that this is the Icosahedral identity and the polynomials are, indeed the icoshardal invariants in homogeneous form. These are, of course, no coincidence which I understood after taking a look at my 9-page calculations.

My question is whether this particular parameterization is the general solution to the diophantine form of interest. I cannot seem to be able to extract that particular property from my calculations, even if it represents many interesting facts.

Thanks in advanced,