Six days ago I wrote: On Mon, Dec 24, 2018 at 5:42 PM Elkies, Noam <elkies@math.harvard.edu> wrote: < 5441 arises for {a,b,c} = {1960, 2183, 2360} (with square factor 39^2). < An overnight search roughly tripled the number of examples up to 10^6; < the new ones start 201, 889, 5441, 5601, 6969, 7761, 8049. < No two-digit solutions turned up, but it may be reasonable to guess that < every squarefree 8k+1 arises eventually, though it may take a long time. < Indeed I might venture to call this guess a conjecture once it's known that < there's no "Brauer-Manin obstructuion". < Trying all a,b,c up to N takes time about N^3; that's basically what I did, < though with some tricks [ . . . ] Update: I coded this more carefully to make sure that every primitive solution of a^4+b^4+c^4 = x^2 appears as long as (a,b,c) is in the closed ball of radius R about the origin (i.e. a^2+b^2+c^2 <= R^2), and extended the search to R = 2^20 (so a bit over one million). This took about two weeks of CPU time, which came to about one day on 19 processors. That's probably about as far as it makes sense to take this method, because the run time is proportional to R^3, and the R^2 techniques, albeit with a much larger constant, are surely faster by now. For R = 2^20, there are 15838 primitive solutions with 0 <= a <= b <= c, and as expected the count grows about linearly in R -- e.g. 7981 of the 15838 solutions are in the R=2^19 ball. Also as expected, the squarefree part 1 (re)appeared for (a,b,c) = (95800, 217519, 414560). It was not quite the first duplicate, though: x = 53781801801 works for both (a,b,c) = (38960, 123140, 227107) and (a,b,c) = (6365, 87224, 230740). This x is squarefree (3*7*13*7417*26561); the next duplicate is not: 313049121489 = 3^4 13 269 1105177 for both (a,b,c) = (49600, 91319, 559400) and (a,b,c) = (91319, 426200, 504800). There were no further duplicates even when each x was reduced to its squarefree part. No two-digit squarefree parts turned up; indeed the smallest new squarefree part is 24401 = 13*1877, for (a,b,c) = (52360, 133145, 139384) with x = 1041^2 * 24401. Happy 1^4 + 2^4 + 3^4 + 5^4 + 6^4, --NDE