Can $10^n+89$ ever be a Perfect Square for $n>3$?

There was a good point made by mathlove, the restriction of the base to a fixed number $(10)$ allows calculating three Mordell curves. The bad news is that neither Magma nor Sage is willing to do the final one. I will post the first two. Looking again, it appears the OP also was able to find solutions for the first two curves, but not the third. Looking through articles on computation written after 2005, say, it appears that most curves $y^2 = x^3 + k$ with $|k| \leq 100000$ have been done, but maybe not all, and in any case are not in any convenient websites. It would appear that $|k| > 100000$ makes it likely that nobody has finished the job.

Method: I had not written it down, it just looked correct. However, if $$ y^2 = 10^{6k+3} + 89, $$ take $x = 10^{2k+1}$ to get $$ y^2 = x^3 + 89. $$ If $$ u^2 = 10^{6k+1} + 89, $$ $$ 100u^2 = 10^{6k+3} + 8900, $$ take $y = 10 u, \; \;x = 10^{2k+1}$ to get $$ y^2 = x^3 + 8900. $$ If $$ v^2 = 10^{6k+5} + 89, $$ $$ 10000v^2 = 10^{6k+9} + 890000, $$ take $y = 100 v, \; \;x = 10^{2k+3}$ to get $$ y^2 = x^3 + 890000. $$

Meanwhile, note that there is an excellent selection of literature on Ramanujan-Nagell and Lebesgue-Nagell. https://en.wikipedia.org/wiki/Ramanujan%E2%80%93Nagell_equation The bad news is that the vast majority of explicit solutions is for $x^2 + C = y^n$ where $C>0.$ For those, many relevant articles can be downloaded for free.

TWO OUT OF THREE MORDELL CURVES

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http://tnt.math.se.tmu.ac.jp/simath/MORDELL/MORDELL+

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E_+00089: r = 2   t = 1   #III =  1
          E(Q) = <(-2, 9)> x <(-4, 5)>
          R =   1.2904854192
           8 integral points
            1. (-2, 9) = 1 * (-2, 9)
            2. (-2, -9) = -(-2, 9)
            3. (55, 408) = -1 * (-2, 9) + 1 * (-4, 5)
            4. (55, -408) = -(55, 408)
            5. (-4, 5) = 1 * (-4, 5)
            6. (-4, -5) = -(-4, 5)
            7. (10, 33) = -1 * (-2, 9) - 1 * (-4, 5)
            8. (10, -33) = -(10, 33)


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E_+08900: r = 3   t = 1   #III =  1
          E(Q) = <(-20, 30)> x <(40, 270)> x <(20, 130)>
          R =   2.4097371874
           24 integral points
             1. (5, 95) = 1 * (-20, 30) - 1 * (40, 270)
             2. (5, -95) = -(5, 95)
             3. (-20, 30) = 1 * (-20, 30)
             4. (-20, -30) = -(-20, 30)
             5. (440, 9230) =  -2 * (-20, 30)
             6. (440, -9230) = -(440, 9230)
             7. (40, 270) = 1 * (40, 270)
             8. (40, -270) = -(40, 270)
             9. (-4, 94) =  -1 * (-20, 30) - 1 * (40, 270)
            10. (-4, -94) = -(-4, 94)
            11. (340, 6270) =  -1 * (40, 270) + 1 * (20, 130)
            12. (340, -6270) = -(340, 6270)
            13. (23245, 3544005) = 2 * (-20, 30) - 1 * (40, 270) + 1 * (20, 130)
            14. (23245, -3544005) = -(23245, 3544005)
            15. (16, 114) = 1 * (-20, 30) - 1 * (20, 130)
            16. (16, -114) = -(16, 114)
            17. (20, 130) = 1 * (20, 130)
            18. (20, -130) = -(20, 130)
            19. (200, 2830) = 1 * (-20, 30) - 1 * (40, 270) - 1 * (20, 130)
            20. (200, -2830) = -(200, 2830)
            21. (-11, 87) = 1 * (40, 270) + 1 * (20, 130)
            22. (-11, -87) = -(-11, 87)
            23. (3685, 223695) = 1 * (-20, 30) - 2 * (20, 130)
            24. (3685, -223695) = -(3685, 223695)
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Let me throw in

Magma
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Elliptic Curve defined by y^2 = x^3 + 89 over Rational Field

    Torsion Subgroup is trivial
    Analytic rank = 2
    The 2-Selmer group has rank 2
    New point of infinite order (x = -4)
    New point of infinite order (x = -2)
    After 2-descent:
        2 <= Rank(E) <= 2
        Sha(E)[2] is trivial
    (Searched up to height 100 on the 2-coverings.)

[ (-4 : -5 : 1), (-2 : -9 : 1), (10 : 33 : 1), (55 : -408 : 1) ]

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Elliptic Curve defined by y^2 = x^3 + 8900 over Rational Field

    Torsion Subgroup is trivial
    Analytic rank = 3
    The 2-Selmer group has rank 3
    New point of infinite order (x = 16)
    New point of infinite order (x = 40)
    New point of infinite order (x = -20)
    After 2-descent:
        3 <= Rank(E) <= 3
        Sha(E)[2] is trivial
    (Searched up to height 100 on the 2-coverings.)

[ (-20 : 30 : 1), (-11 : -87 : 1), (-4 : -94 : 1), (5 : -95 : 1), (16 : 114 :
1), (20 : -130 : 1), (40 : 270 : 1), (200 : 2830 : 1), (340 : -6270 : 1), (440 :
-9230 : 1), (3685 : 223695 : 1), (23245 : -3544005 : 1) ]

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