Proximal Fifth Powers That Are Closer To A Cube Than To A Square

[Don Reble]

> Ed Pegg's A117594... When I first noticed the relative proximity of
> 14706104 and 14706146, which are both members of this sequence, I
> wondered if this was coincidental. I've now run across another
> example: 66430098 and 66430152.

    Those pairs are actually triples; but Ed removes the middle ones,
    because they're cubes:

        14706104, 14706125, 14706146
        66430098, 66430125, 66430152

> Might there be a ... explanation?

    Write f(a,x)=(a^3+3x)^(5/3) as Taylor series in x:
        f(a,x) = a^5 + (5 a^2) x + (5/a) x^2 + ...
    One can make the x^2 coefficient into a square, by letting a=5b^2:
        f(a,x) = (5^5 b^10) + (5^3 b^4) x + (1/b^2) x^2 + ...
    Which means that if x=-b or +b, f(a,x) is very nearly an integer;
    and (a^3+3x)^5 = f(a,x)^3 is nearly an integer's cube.

    Examples:

         b        f(5b^2,-b)          f(5b^2,b)
        --- ------------------- -------------------
         1         3001.0026882        3250.9973544
         2      3196001.0000834     3204000.9999167
         3    184497751.0000110   184558500.9999890
         4   3276672001.0000026  3276928000.9999973
         5  30517187501.0000009 30517968750.9999991

    The corresponding fifth-roots (a^3+-3b = 125b^6 +- 3b) are

         b
        --- -------- --------
         1       122      128
         2      7994     8006
         3     91116    91134
         4    511988   512012
         5   1953110  1953140
         6   5831982  5832018
         7  14706104 14706146
         8  32767976 32768024
         9  66430098 66430152

    If a fifth-power is not close to square, the fifth-root is in
    Ed's sequence.

-- 
Don Reble  djr at nk.ca


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