By *Fermat’s Last Theorem*, the quartic equation,

has no non-trivial rational solutions. In fact, the same can be said for the less strict,

So how do we explain the near-equalities,

A search for others with *z* < 8,000,000 will not yield better approximations. Noam Elkies showed that an identity is behind it, namely,

Since the second term on the RHS is small compared to the others, this gives an excellent near-miss to Fermat’s Last Theorem. Inspired by Elkies’ quartic identity, Seiji Tomita found similar ones as,

It can be shown all three belong to the same family. For arbitrary *m* let,

then,

Note that,

In general, these are diagonal quartic surfaces of form,

The case {*a, b, c, d*} = {1, 2, 1, 4}, or the *Swinnerton-Dyer quartic surface*, was discussed in the previous post and, in the link above, Elsenhans gives first-known but large solutions to other *positive* {*a, b, c, d*}. From Elkies’ and Tomita’s results, it is tempting to speculate that if,

has a non-trivial solution, then does

have one as well? Unfortunately, this particular form does not seem to be in Elsenhans’ list. (For positive {*a, b, c, d*}, the only other case I know of that has an infinite number of solutions is *a = b = c = d* = 1.)

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