## A Fermat’s Last Theorem near-miss

By Fermat’s Last Theorem, the quartic equation,

$x^4+y^4 = z^4$

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

$x^4+y^4 = z^2$

So how do we explain the near-equalities,

$24576^4+48767^4 \approx (49535.000000000006\dots)^4$

$419904^4 + 1257767^4 \approx (126155.000000000000001\dots)^4$

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

$(192v^8-24v^4-1)^4+ (192v^7)^4= (192v^8+24v^4-1)^4+12(2v)^4$

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,

$(48v^8-12v^4-1)^4 + 2(48v^7)^4 = (48v^8+12v^4-1)^4 + 6(2v)^4$

$(12v^8-6v^4-1)^4 + 4(12v^7)^4 = (12v^8+6v^4-1)^4 + 3(2v)^4$

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

$b = \frac{8}{m},\;\; d = \frac{3m}{2}$

then,

$(3m^2 v^8-3m v^4-1)^4+b(3m^2 v^7)^4=(3m^2v^8+3mv^4-1)^4+d(2v)^4$

Note that,

$bd = 12$

In general, these are diagonal quartic surfaces of form,

$ax^4+by^4 = cz^4+dt^4$

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,

$x^4+2y^4 = z^4+4t^4$

has a non-trivial solution, then does

$x^4+y^4 = z^4+8t^4$

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.)