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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