Short & Sweet Math Challenge #21: Powers that be

[J-F Garnier]

(11-09-2016 11:53 PM)Valentin Albillo Wrote:   
>RUN

? 8
Sort by C,P ? C

1.32471795724 x^3-x-1
1.38027756910 x^4-x^3-1
...
1.99196419661 x^7-x^6-x^5-x^4-x^3-x^2-x-1
1.99603117974 x^8-x^7-x^6-x^5-x^4-x^3-x^2-x-1


i.e.: it found 41 constants in 239.96 seconds, just shy of 4 min.

It was an interesting and fun challenge, and the opportunity for me to learn something in math.

I have two comments:

1) It seems that your program doesn't explore all the polynomials. It uses basically the same algorithm that I used in my previous thread (based on Paul's idea), and I got not 41 but 48 constants.
I had a look at your program, but had some difficulties to understand the logic of the polynomial scanning. However, I was able to hack your program to display all the tested polynomials, and indeed some are missing:
? 4
Sort by C,P ? p
1.00000000000 1
1.00000000000 x-1
1.00000000000 x^2+1
1.00000000000 x^2-1
1.00000000000 x^2-x+1
1.00000000000 x^2-x-1
1.00000000000 x^3+1
1.00000000000 x^3+x+1
1.00000000000 x^3+x-1
1.00000000000 x^3-1
1.00000000000 x^3-x+1
1.00000000000 x^3-x-1
1.00000000000 x^3-x^2+1
1.00000000000 x^3-x^2+x+1
1.00000000000 x^3-x^2+x-1
1.00000000000 x^3-x^2-1
1.00000000000 x^3-x^2-x+1
1.00000000000 x^3-x^2-x-1
1.00000000000 x^4+x-1
1.00000000000 x^4+x^2+x-1
1.00000000000 x^4+x^2-1
1.00000000000 x^4+x^2-x-1
1.00000000000 x^4-1
1.00000000000 x^4-x-1
1.00000000000 x^4-x^2+x-1
1.00000000000 x^4-x^2-1
1.00000000000 x^4-x^2-x-1
1.00000000000 x^4-x^3+x-1
1.00000000000 x^4-x^3+x^2+x-1
1.00000000000 x^4-x^3+x^2-1
1.00000000000 x^4-x^3+x^2-x-1
1.00000000000 x^4-x^3-1
1.00000000000 x^4-x^3-x-1
1.00000000000 x^4-x^3-x^2+x-1
1.00000000000 x^4-x^3-x^2-1
1.00000000000 x^4-x^3-x^2-x-1
36

For instance, the polynomials x^2+x+1 and x^2+x-1 are not tested, as well as the x^3+x^2... polynomials.

2) my second comment is what I already mentioned in my previous message: the value 1.75487766625, root of x^4-x^3-x^2-1, is not found by this algorithm, although its powers clearly tend to an integer (quite quickly actually):
x^10 276.992792634
x^11 486.088465506
...
x^21 134643.001528
x^22 236281.996298
...
x^27 3932464.99924
x^28 6900995.00047

The roots of the polynomial x^4-x^3-x^2-1 are:
(.122561166877,-.74486176662)
(.122561166877,.74486176662)
(-1,0)
(1.75487766625,0)
Clearly, the roots 1 or -1 should not cause to reject the polynomial under test, since powers of -1 or +1 will not impact the decimals of the powers.

Thanks again for this challenge!

J-F