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

Don Reble djr at nk.ca
Mon Apr 10 03:52:25 CEST 2006

```> 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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