warning, I used FIXED font to type the message, Hello, about those 2 numbers, we have here 45 digits of precision, which I used, pari-GP own LLL routine can crack these numbers easily since 1990's here is what maple (with an interface to pari-gp) gives,
0.54368901269207636157085597180174798652520330; 0.54368901269207636157085597180174798652520330
gp([%],4); 4 3 2 2 x + x + x - 3 x + 1
factor(%); 3 2 (2 x - 1) (x + x + x - 1)
solve(%); 1/2 1/3 1/2 1/3 (17 + 3 33 ) 2 (17 + 3 33 ) 1/2, ----------------- - ------------------- - 1/3, - ----------------- 3 1/2 1/3 6 3 (17 + 3 33 )
1 + --------------------- - 1/3 1/2 (1/3) 3 (17 + 3 33 ) / 1/2 1/3 \ 1/2 1/3 1/2 |(17 + 3 33 ) 2 | (17 + 3 33 ) + 1/2 I 3 |----------------- + -------------------|, - ----------------- | 3 1/2 1/3| 6 \ 3 (17 + 3 33 ) / 1 + --------------------- - 1/3 1/2 (1/3) 3 (17 + 3 33 ) / 1/2 1/3 \ 1/2 |(17 + 3 33 ) 2 | - 1/2 I 3 |----------------- + -------------------| | 3 1/2 1/3| \ 3 (17 + 3 33 ) /
0.64779887126104238549049829642576930694948772; 0.64779887126104238549049829642576930694948772
gp([%],4); 4 3 2 6 x - 10 x + 10 x - 7 x + 2
factor(%); 3 2 (3 x - 2) (2 x - 2 x + 2 x - 1)
solve(%); 1/2 1/3 1/2 1/3 (26 + 6 33 ) 4 (26 + 6 33 ) 2/3, ----------------- - ------------------- + 1/3, - ----------------- 6 1/2 1/3 12 3 (26 + 6 33 )
2 + ------------------- + 1/3 1/2 1/3 3 (26 + 6 33 ) / 1/2 1/3 \ 1/2 1/3 1/2 |(26 + 6 33 ) 4 | (26 + 6 33 ) + 1/2 I 3 |----------------- + -------------------|, - ----------------- | 6 1/2 1/3| 12 \ 3 (26 + 6 33 ) / 2 + ------------------- + 1/3 1/2 1/3 3 (26 + 6 33 ) / 1/2 1/3 \ 1/2 |(26 + 6 33 ) 4 | - 1/2 I 3 |----------------- + -------------------| | 6 1/2 1/3| \ 3 (26 + 6 33 ) / They are indeed algebraic, with small coefficients. Best regards, Simon Plouffe