x↑↑x = -1

[sheldonison]

What are some possible answers to the equation x↑↑x = -1? Must a new type of number be conceptualized similar to the answer to the equation x*x = -1? Or can it be proved that this answer lies within the real and complex planes?

edit update. It turns out that at least one of the solutions to the Op's question, is in the range that I can deal with numerically. For x~=2.6918099719192 + 0.62660048483655i, sexp_x(x)~=-1.

The simplest algorithm I could come up with is to treat x^^x as an analytic function, and then iteratively use Newton's method to solve for x^^x=-1. For Newton's method to converge, it helps to get somewhere nearby first, which I did by hunt and peck, looking for bases where x^^x was near -1.

I used my complex base tetration program. I wrote a program that explored complex base tetration, link here, but it is limited in terms of what bases it works for, due to problems at both exp(1/e), and at the Shell Thron boundary, and problems due to figuring out a consistent logarithmic branch to use... The Taylor series below can accurately calculate z=tet(b-2). Then iterate z=b^z twice, to get -1.

Code:

base = 2.6918099719192010269 + 0.62660048483655230254*I;
{tet=    1.0000000000000000000
+x^ 1* ( 1.1102737979673991396 + 0.13444952486116713626*I)
+x^ 2* ( 0.28425372561063406460 + 0.20168515612498045279*I)
+x^ 3* ( 0.20142378046918946453 + 0.12287970799981736460*I)
+x^ 4* ( 0.056631358539842508401 + 0.082262149871280378960*I)
+x^ 5* ( 0.031626588237768922149 + 0.042664820582471964536*I)
+x^ 6* ( 0.0060514513770401941396 + 0.023085772637730528515*I)
+x^ 7* ( 0.0029937024980130884380 + 0.010708203875456837542*I)
+x^ 8* (-0.00043235417834169701508 + 0.0051416902992490886284*I)
+x^ 9* (-0.00017877732679188096499 + 0.0021743297255104601779*I)
+x^10* (-0.00042253710857487390916 + 0.00095593830941724313184*I)
+x^11* (-0.00016984009539643333991 + 0.00036797431151789026196*I)
+x^12* (-0.00013598497262565877056 + 0.00014934164052065933115*I)
+x^13* (-0.000050768369172865507021 + 0.000050790667372515377858*I)
+x^14* (-0.000032617521009029439342 + 0.000018777349770437327073*I)
+x^15* (-0.000011146610242226519358 + 0.0000050006190101552048081*I)
+x^16* (-0.0000066270855458264950034 + 0.0000015446107355450406250*I)
+x^17* (-0.0000020122113800518364907 + 0.000000068654818145175922999*I)
+x^18* (-0.0000011963396468454392777 - 0.000000050931639762404521931*I)
+x^19* (-0.00000030238855924614592898 - 0.00000012539726639748157461*I)
+x^20* (-0.00000019689839845097420283 - 0.000000057637500744298040916*I)
+x^21* (-0.000000035153434458995604130 - 0.000000041641278224258482194*I)
+x^22* (-0.000000030197104715047471141 - 0.000000015975908532546115455*I)
+x^23* (-0.0000000019166266427945385066 - 0.0000000095872390599983285641*I)
+x^24* (-0.0000000044548959632689572594 - 0.0000000032492643102921963649*I)
+x^25* ( 0.00000000051723734539716253138 - 0.0000000018638631462340237533*I)
+x^26* (-0.00000000067021649772138737120 - 0.00000000054652627375747029680*I)
+x^27* ( 0.00000000024760346059064288247 - 0.00000000032605780533942069098*I)
+x^28* (-1.1228394170747083309 E-10 - 7.6537918519681082716 E-11*I)
+x^29* ( 7.1268952289713966228 E-11 - 5.3189341538283539220 E-11*I)
+x^30* (-2.2369708781921881068 E-11 - 8.0625867943287913385 E-12*I)
+x^31* ( 1.7517436693666143597 E-11 - 8.3668572196268659348 E-12*I)
+x^32* (-5.1777236965884120904 E-12 - 2.5187031197558940487 E-13*I)
+x^33* ( 4.0336539933777435031 E-12 - 1.3265926389475846122 E-12*I)
+x^34* (-1.2900297482939488422 E-12 + 1.8193947317104394501 E-13*I)
+x^35* ( 9.0792695347242281651 E-13 - 2.2446482170229203341 E-13*I)
+x^36* (-3.2656612422347855209 E-13 + 7.4097030312727709098 E-14*I)
+x^37* ( 2.0437092576263104990 E-13 - 4.2548590115227614080 E-14*I)
+x^38* (-8.1938431592790255774 E-14 + 2.0924876810646870153 E-14*I)
+x^39* ( 4.6518922470943806673 E-14 - 9.0908340973673717718 E-15*I)
+x^40* (-2.0262143014199117716 E-14 + 5.2297859307372668790 E-15*I)
+x^41* ( 1.0743467122240305126 E-14 - 2.1121380807455872169 E-15*I)
+x^42* (-4.9476280628213598972 E-15 + 1.2446927622317468161 E-15*I)
+x^43* ( 2.5145731619654922803 E-15 - 5.1105870278086611369 E-16*I)
+x^44* (-1.1976391479208810646 E-15 + 2.9141474535196681368 E-16*I)
+x^45* ( 5.9468603024404147363 E-16 - 1.2519578605776942842 E-16*I)
+x^46* (-2.8845734316656989916 E-16 + 6.8168421844852586540 E-17*I)
+x^47* ( 1.4166399462777367792 E-16 - 3.0654575399052642127 E-17*I)
+x^48* (-6.9322821884199600055 E-17 + 1.6036264175328628801 E-17*I)
+x^49* ( 3.3908470426705261004 E-17 - 7.4731310274704053355 E-18*I)
+x^50* (-1.6653357776046968983 E-17 + 3.7997687551447424638 E-18*I)
+x^51* ( 8.1416335448137121457 E-18 - 1.8140427842389372934 E-18*I)
+x^52* (-4.0032414617874811484 E-18 + 9.0620279617655147611 E-19*I)
+x^53* ( 1.9590024443115400557 E-18 - 4.3905474199233700091 E-19*I)
+x^54* (-9.6344349916460340144 E-19 + 2.1720467909141856696 E-19*I)
+x^55* ( 4.7210606401205367545 E-19 - 1.0610621969521044634 E-19*I)
+x^56* (-2.3217937825693896625 E-19 + 5.2248674247269975946 E-20*I)
+x^57* ( 1.1392068860099503909 E-19 - 2.5632761528515730279 E-20*I)
+x^58* (-5.6028249121768971146 E-20 + 1.2600230734782307737 E-20*I)
+x^59* ( 2.7520884846767807057 E-20 - 6.1943235049722404547 E-21*I)
+x^60* (-1.3537732547522302047 E-20 + 3.0441749210418233788 E-21*I)
}