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rigour: exact complete: 'no'- complete-note: it holds the minimal polynomials for the first $50$ Pisot numbers- in $(1,\varphi)$, matching the range in HREF{Pisot_numbers_less_than_the_golden_ratio}[the- root table] exactly+ complete-note: it holds the minimal polynomials for $\theta_1$ to $\theta_{79}$,+ matching the range in HREF{Pisot_numbers_less_than_the_golden_ratio}[the root+ table] exactly rigour details: The generator forms the exact integer polynomials given in Formula CITE{formula-families} and factors them over $\mathbb Q$. It selects the irreducible
number-header: $m_r(x)$ Numbers:-- comment: $P_{2}(x)=(x-1)\,m_{1}(x)$, and the root of $m_{1}$ in $(1,\varphi)$ is- HREF{Pisot_numbers_less_than_the_golden_ratio#1}[$\theta_{1}$], the plastic ratio- and the smallest Pisot number.- params:- r: '1'- number: x^3 - x - 1-- comment: $m_{2}(x)=Q_{2}(x)$, and the root of $m_{2}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#2}[$\theta_{2}$].- params:- r: '2'- number: x^4 - x^3 - 1-- comment: $m_{3}(x)=Q_{3}(x)$, and the root of $m_{3}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#3}[$\theta_{3}$].- params:- r: '3'- number: x^5 - x^4 - x^3 + x^2 - 1-- comment: $P_{3}(x)=(x^2-1)\,m_{4}(x)$, and the root of $m_{4}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#4}[$\theta_{4}$], the supergolden- ratio.- params:- r: '4'- number: x^3 - x^2 - 1-- comment: $m_{5}(x)=Q_{4}(x)$, and the root of $m_{5}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#5}[$\theta_{5}$].- params:- r: '5'- number: x^6 - x^5 - x^4 + x^2 - 1-- comment: $P_{4}(x)=(x-1)\,m_{6}(x)$, and the root of $m_{6}$ in $(1,\varphi)$ is- HREF{Pisot_numbers_less_than_the_golden_ratio#6}[$\theta_{6}$].- params:- r: '6'- number: x^5 - x^3 - x^2 - x - 1-- comment: $m_{7}(x)=Q_{5}(x)$, and the root of $m_{7}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#7}[$\theta_{7}$].- params:- r: '7'- number: x^7 - x^6 - x^5 + x^2 - 1-- comment: $m_{8}(x)=E(x)$, and the root of $m_{8}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#8}[$\theta_{8}$].- params:- r: '8'- number: x^6 - 2*x^5 + x^4 - x^2 + x - 1-- comment: $P_{5}(x)=(x^2-1)\,m_{9}(x)$, and the root of $m_{9}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#9}[$\theta_{9}$].- params:- r: '9'- number: x^5 - x^4 - x^2 - 1-- comment: $m_{10}(x)=Q_{6}(x)$, and the root of $m_{10}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#10}[$\theta_{10}$].- params:- r: '10'- number: x^8 - x^7 - x^6 + x^2 - 1-- comment: $P_{6}(x)=(x-1)\,m_{11}(x)$, and the root of $m_{11}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#11}[$\theta_{11}$].- params:- r: '11'- number: x^7 - x^5 - x^4 - x^3 - x^2 - x - 1-- comment: $m_{12}(x)=Q_{7}(x)$, and the root of $m_{12}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#12}[$\theta_{12}$].- params:- r: '12'- number: x^9 - x^8 - x^7 + x^2 - 1-- comment: $P_{7}(x)=(x^2-1)\,m_{13}(x)$, and the root of $m_{13}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#13}[$\theta_{13}$].- params:- r: '13'- number: x^7 - x^6 - x^4 - x^2 - 1-- comment: $m_{14}(x)=Q_{8}(x)$, and the root of $m_{14}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#14}[$\theta_{14}$].- params:- r: '14'- number: x^10 - x^9 - x^8 + x^2 - 1-- comment: $P_{8}(x)=(x-1)\,m_{15}(x)$, and the root of $m_{15}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#15}[$\theta_{15}$].- params:- r: '15'- number: x^9 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1-- comment: $m_{16}(x)=Q_{9}(x)$, and the root of $m_{16}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#16}[$\theta_{16}$].- params:- r: '16'- number: x^11 - x^10 - x^9 + x^2 - 1-- comment: $P_{9}(x)=(x^2-1)\,m_{17}(x)$, and the root of $m_{17}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#17}[$\theta_{17}$].- params:- r: '17'- number: x^9 - x^8 - x^6 - x^4 - x^2 - 1-- comment: $m_{18}(x)=Q_{10}(x)$, and the root of $m_{18}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#18}[$\theta_{18}$].- params:- r: '18'- number: x^12 - x^11 - x^10 + x^2 - 1-- comment: $P_{10}(x)=(x-1)\,m_{19}(x)$, and the root of $m_{19}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#19}[$\theta_{19}$].- params:- r: '19'- number: x^11 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1-- comment: $m_{20}(x)=Q_{11}(x)$, and the root of $m_{20}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#20}[$\theta_{20}$].- params:- r: '20'- number: x^13 - x^12 - x^11 + x^2 - 1-- comment: $P_{11}(x)=(x^2-1)\,m_{21}(x)$, and the root of $m_{21}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#21}[$\theta_{21}$].- params:- r: '21'- number: x^11 - x^10 - x^8 - x^6 - x^4 - x^2 - 1-- comment: $m_{22}(x)=Q_{12}(x)$, and the root of $m_{22}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#22}[$\theta_{22}$].- params:- r: '22'- number: x^14 - x^13 - x^12 + x^2 - 1-- comment: $P_{12}(x)=(x-1)\,m_{23}(x)$, and the root of $m_{23}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#23}[$\theta_{23}$].- params:- r: '23'- number: x^13 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x -- 1-- comment: $m_{24}(x)=Q_{13}(x)$, and the root of $m_{24}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#24}[$\theta_{24}$].- params:- r: '24'- number: x^15 - x^14 - x^13 + x^2 - 1-- comment: $P_{13}(x)=(x^2-1)\,m_{25}(x)$, and the root of $m_{25}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#25}[$\theta_{25}$].- params:- r: '25'- number: x^13 - x^12 - x^10 - x^8 - x^6 - x^4 - x^2 - 1-- comment: $m_{26}(x)=Q_{14}(x)$, and the root of $m_{26}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#26}[$\theta_{26}$].- params:- r: '26'- number: x^16 - x^15 - x^14 + x^2 - 1-- comment: $P_{14}(x)=(x-1)\,m_{27}(x)$, and the root of $m_{27}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#27}[$\theta_{27}$].- params:- r: '27'- number: x^15 - x^13 - x^12 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3- - x^2 - x - 1-- comment: $m_{28}(x)=Q_{15}(x)$, and the root of $m_{28}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#28}[$\theta_{28}$].- params:- r: '28'- number: x^17 - x^16 - x^15 + x^2 - 1-- comment: $P_{15}(x)=(x^2-1)\,m_{29}(x)$, and the root of $m_{29}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#29}[$\theta_{29}$].- params:- r: '29'- number: x^15 - x^14 - x^12 - x^10 - x^8 - x^6 - x^4 - x^2 - 1-- comment: $m_{30}(x)=Q_{16}(x)$, and the root of $m_{30}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#30}[$\theta_{30}$].- params:- r: '30'- number: x^18 - x^17 - x^16 + x^2 - 1-- comment: $P_{16}(x)=(x-1)\,m_{31}(x)$, and the root of $m_{31}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#31}[$\theta_{31}$].- params:- r: '31'- number: x^17 - x^15 - x^14 - x^13 - x^12 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6 -- x^5 - x^4 - x^3 - x^2 - x - 1-- comment: $m_{32}(x)=Q_{17}(x)$, and the root of $m_{32}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#32}[$\theta_{32}$].- params:- r: '32'- number: x^19 - x^18 - x^17 + x^2 - 1-- comment: $P_{17}(x)=(x^2-1)\,m_{33}(x)$, and the root of $m_{33}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#33}[$\theta_{33}$].- params:- r: '33'- number: x^17 - x^16 - x^14 - x^12 - x^10 - x^8 - x^6 - x^4 - x^2 - 1-- comment: $m_{34}(x)=Q_{18}(x)$, and the root of $m_{34}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#34}[$\theta_{34}$].- params:- r: '34'- number: x^20 - x^19 - x^18 + x^2 - 1-- comment: $P_{18}(x)=(x-1)\,m_{35}(x)$, and the root of $m_{35}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#35}[$\theta_{35}$].- params:- r: '35'- number: x^19 - x^17 - x^16 - x^15 - x^14 - x^13 - x^12 - x^11 - x^10 - x^9 - x^8- - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1-- comment: $m_{36}(x)=Q_{19}(x)$, and the root of $m_{36}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#36}[$\theta_{36}$].- params:- r: '36'- number: x^21 - x^20 - x^19 + x^2 - 1-- comment: $P_{19}(x)=(x^2-1)\,m_{37}(x)$, and the root of $m_{37}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#37}[$\theta_{37}$].- params:- r: '37'- number: x^19 - x^18 - x^16 - x^14 - x^12 - x^10 - x^8 - x^6 - x^4 - x^2 - 1-- comment: $m_{38}(x)=Q_{20}(x)$, and the root of $m_{38}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#38}[$\theta_{38}$].- params:- r: '38'- number: x^22 - x^21 - x^20 + x^2 - 1-- comment: $P_{20}(x)=(x-1)\,m_{39}(x)$, and the root of $m_{39}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#39}[$\theta_{39}$].- params:- r: '39'- number: x^21 - x^19 - x^18 - x^17 - x^16 - x^15 - x^14 - x^13 - x^12 - x^11 - x^10- - x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1-- comment: $m_{40}(x)=Q_{21}(x)$, and the root of $m_{40}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#40}[$\theta_{40}$].- params:- r: '40'- number: x^23 - x^22 - x^21 + x^2 - 1-- comment: $P_{21}(x)=(x^2-1)\,m_{41}(x)$, and the root of $m_{41}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#41}[$\theta_{41}$].- params:- r: '41'- number: x^21 - x^20 - x^18 - x^16 - x^14 - x^12 - x^10 - x^8 - x^6 - x^4 - x^2 -- 1-- comment: $m_{42}(x)=Q_{22}(x)$, and the root of $m_{42}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#42}[$\theta_{42}$].- params:- r: '42'- number: x^24 - x^23 - x^22 + x^2 - 1-- comment: $P_{22}(x)=(x-1)\,m_{43}(x)$, and the root of $m_{43}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#43}[$\theta_{43}$].- params:- r: '43'- number: x^23 - x^21 - x^20 - x^19 - x^18 - x^17 - x^16 - x^15 - x^14 - x^13 - x^12- - x^11 - x^10 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1-- comment: $m_{44}(x)=Q_{23}(x)$, and the root of $m_{44}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#44}[$\theta_{44}$].- params:- r: '44'- number: x^25 - x^24 - x^23 + x^2 - 1-- comment: $P_{23}(x)=(x^2-1)\,m_{45}(x)$, and the root of $m_{45}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#45}[$\theta_{45}$].- params:- r: '45'- number: x^23 - x^22 - x^20 - x^18 - x^16 - x^14 - x^12 - x^10 - x^8 - x^6 - x^4- - x^2 - 1-- comment: $m_{46}(x)=Q_{24}(x)$, and the root of $m_{46}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#46}[$\theta_{46}$].- params:- r: '46'- number: x^26 - x^25 - x^24 + x^2 - 1-- comment: $P_{24}(x)=(x-1)\,m_{47}(x)$, and the root of $m_{47}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#47}[$\theta_{47}$].- params:- r: '47'- number: x^25 - x^23 - x^22 - x^21 - x^20 - x^19 - x^18 - x^17 - x^16 - x^15 - x^14- - x^13 - x^12 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 -- x - 1-- comment: $m_{48}(x)=Q_{25}(x)$, and the root of $m_{48}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#48}[$\theta_{48}$].- params:- r: '48'- number: x^27 - x^26 - x^25 + x^2 - 1-- comment: $P_{25}(x)=(x^2-1)\,m_{49}(x)$, and the root of $m_{49}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#49}[$\theta_{49}$].- params:- r: '49'- number: x^25 - x^24 - x^22 - x^20 - x^18 - x^16 - x^14 - x^12 - x^10 - x^8 - x^6- - x^4 - x^2 - 1-- comment: $m_{50}(x)=Q_{26}(x)$, and the root of $m_{50}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#50}[$\theta_{50}$].- params:- r: '50'- number: x^28 - x^27 - x^26 + x^2 - 1-- params:- r: '51'- number: x^27 - x^25 - x^24 - x^23 - x^22 - x^21 - x^20 - x^19 - x^18 - x^17 - x^16- - x^15 - x^14 - x^13 - x^12 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4- - x^3 - x^2 - x - 1- comment: $P_{26}(x)=(x-1)\,m_{51}(x)$, and the root of $m_{51}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#51}[$\theta_{51}$].-- params:- r: '52'- number: x^29 - x^28 - x^27 + x^2 - 1- comment: $m_{52}(x)=Q_{27}(x)$, and the root of $m_{52}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#52}[$\theta_{52}$].-- params:- r: '53'- number: x^27 - x^26 - x^24 - x^22 - x^20 - x^18 - x^16 - x^14 - x^12 - x^10 - x^8- - x^6 - x^4 - x^2 - 1- comment: $P_{27}(x)=(x^2-1)\,m_{53}(x)$, and the root of $m_{53}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#53}[$\theta_{53}$].-- params:- r: '54'- number: x^30 - x^29 - x^28 + x^2 - 1- comment: $m_{54}(x)=Q_{28}(x)$, and the root of $m_{54}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#54}[$\theta_{54}$].-- params:- r: '55'- number: x^29 - x^27 - x^26 - x^25 - x^24 - x^23 - x^22 - x^21 - x^20 - x^19 - x^18- - x^17 - x^16 - x^15 - x^14 - x^13 - x^12 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6- - x^5 - x^4 - x^3 - x^2 - x - 1- comment: $P_{28}(x)=(x-1)\,m_{55}(x)$, and the root of $m_{55}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#55}[$\theta_{55}$].-- params:- r: '56'- number: x^31 - x^30 - x^29 + x^2 - 1- comment: $m_{56}(x)=Q_{29}(x)$, and the root of $m_{56}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#56}[$\theta_{56}$].-- params:- r: '57'- number: x^29 - x^28 - x^26 - x^24 - x^22 - x^20 - x^18 - x^16 - x^14 - x^12 - x^10- - x^8 - x^6 - x^4 - x^2 - 1- comment: $P_{29}(x)=(x^2-1)\,m_{57}(x)$, and the root of $m_{57}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#57}[$\theta_{57}$].-- params:- r: '58'- number: x^32 - x^31 - x^30 + x^2 - 1- comment: $m_{58}(x)=Q_{30}(x)$, and the root of $m_{58}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#58}[$\theta_{58}$].-- params:- r: '59'- number: x^31 - x^29 - x^28 - x^27 - x^26 - x^25 - x^24 - x^23 - x^22 - x^21 - x^20- - x^19 - x^18 - x^17 - x^16 - x^15 - x^14 - x^13 - x^12 - x^11 - x^10 - x^9 -- x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1- comment: $P_{30}(x)=(x-1)\,m_{59}(x)$, and the root of $m_{59}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#59}[$\theta_{59}$].-- params:- r: '60'- number: x^33 - x^32 - x^31 + x^2 - 1- comment: $m_{60}(x)=Q_{31}(x)$, and the root of $m_{60}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#60}[$\theta_{60}$].-- params:- r: '61'- number: x^31 - x^30 - x^28 - x^26 - x^24 - x^22 - x^20 - x^18 - x^16 - x^14 - x^12- - x^10 - x^8 - x^6 - x^4 - x^2 - 1- comment: $P_{31}(x)=(x^2-1)\,m_{61}(x)$, and the root of $m_{61}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#61}[$\theta_{61}$].-- params:- r: '62'- number: x^34 - x^33 - x^32 + x^2 - 1- comment: $m_{62}(x)=Q_{32}(x)$, and the root of $m_{62}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#62}[$\theta_{62}$].-- params:- r: '63'- number: x^33 - x^31 - x^30 - x^29 - x^28 - x^27 - x^26 - x^25 - x^24 - x^23 - x^22- - x^21 - x^20 - x^19 - x^18 - x^17 - x^16 - x^15 - x^14 - x^13 - x^12 - x^11 -- x^10 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1- comment: $P_{32}(x)=(x-1)\,m_{63}(x)$, and the root of $m_{63}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#63}[$\theta_{63}$].-- params:- r: '64'- number: x^35 - x^34 - x^33 + x^2 - 1- comment: $m_{64}(x)=Q_{33}(x)$, and the root of $m_{64}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#64}[$\theta_{64}$].-- params:- r: '65'- number: x^33 - x^32 - x^30 - x^28 - x^26 - x^24 - x^22 - x^20 - x^18 - x^16 - x^14- - x^12 - x^10 - x^8 - x^6 - x^4 - x^2 - 1- comment: $P_{33}(x)=(x^2-1)\,m_{65}(x)$, and the root of $m_{65}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#65}[$\theta_{65}$].-- params:- r: '66'- number: x^36 - x^35 - x^34 + x^2 - 1- comment: $m_{66}(x)=Q_{34}(x)$, and the root of $m_{66}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#66}[$\theta_{66}$].-- params:- r: '67'- number: x^35 - x^33 - x^32 - x^31 - x^30 - x^29 - x^28 - x^27 - x^26 - x^25 - x^24- - x^23 - x^22 - x^21 - x^20 - x^19 - x^18 - x^17 - x^16 - x^15 - x^14 - x^13 -- x^12 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1- comment: $P_{34}(x)=(x-1)\,m_{67}(x)$, and the root of $m_{67}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#67}[$\theta_{67}$].-- params:- r: '68'- number: x^37 - x^36 - x^35 + x^2 - 1- comment: $m_{68}(x)=Q_{35}(x)$, and the root of $m_{68}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#68}[$\theta_{68}$].-- params:- r: '69'- number: x^35 - x^34 - x^32 - x^30 - x^28 - x^26 - x^24 - x^22 - x^20 - x^18 - x^16- - x^14 - x^12 - x^10 - x^8 - x^6 - x^4 - x^2 - 1- comment: $P_{35}(x)=(x^2-1)\,m_{69}(x)$, and the root of $m_{69}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#69}[$\theta_{69}$].-- params:- r: '70'- number: x^38 - x^37 - x^36 + x^2 - 1- comment: $m_{70}(x)=Q_{36}(x)$, and the root of $m_{70}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#70}[$\theta_{70}$].-- params:- r: '71'- number: x^37 - x^35 - x^34 - x^33 - x^32 - x^31 - x^30 - x^29 - x^28 - x^27 - x^26- - x^25 - x^24 - x^23 - x^22 - x^21 - x^20 - x^19 - x^18 - x^17 - x^16 - x^15 -- x^14 - x^13 - x^12 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2- - x - 1- comment: $P_{36}(x)=(x-1)\,m_{71}(x)$, and the root of $m_{71}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#71}[$\theta_{71}$].-- params:- r: '72'- number: x^39 - x^38 - x^37 + x^2 - 1- comment: $m_{72}(x)=Q_{37}(x)$, and the root of $m_{72}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#72}[$\theta_{72}$].-- params:- r: '73'- number: x^37 - x^36 - x^34 - x^32 - x^30 - x^28 - x^26 - x^24 - x^22 - x^20 - x^18- - x^16 - x^14 - x^12 - x^10 - x^8 - x^6 - x^4 - x^2 - 1- comment: $P_{37}(x)=(x^2-1)\,m_{73}(x)$, and the root of $m_{73}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#73}[$\theta_{73}$].-- params:- r: '74'- number: x^40 - x^39 - x^38 + x^2 - 1- comment: $m_{74}(x)=Q_{38}(x)$, and the root of $m_{74}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#74}[$\theta_{74}$].-- params:- r: '75'- number: x^39 - x^37 - x^36 - x^35 - x^34 - x^33 - x^32 - x^31 - x^30 - x^29 - x^28- - x^27 - x^26 - x^25 - x^24 - x^23 - x^22 - x^21 - x^20 - x^19 - x^18 - x^17 -- x^16 - x^15 - x^14 - x^13 - x^12 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6 - x^5 -- x^4 - x^3 - x^2 - x - 1- comment: $P_{38}(x)=(x-1)\,m_{75}(x)$, and the root of $m_{75}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#75}[$\theta_{75}$].-- params:- r: '76'- number: x^41 - x^40 - x^39 + x^2 - 1- comment: $m_{76}(x)=Q_{39}(x)$, and the root of $m_{76}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#76}[$\theta_{76}$].-- params:- r: '77'- number: x^39 - x^38 - x^36 - x^34 - x^32 - x^30 - x^28 - x^26 - x^24 - x^22 - x^20- - x^18 - x^16 - x^14 - x^12 - x^10 - x^8 - x^6 - x^4 - x^2 - 1- comment: $P_{39}(x)=(x^2-1)\,m_{77}(x)$, and the root of $m_{77}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#77}[$\theta_{77}$].-- params:- r: '78'- number: x^42 - x^41 - x^40 + x^2 - 1- comment: $m_{78}(x)=Q_{40}(x)$, and the root of $m_{78}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#78}[$\theta_{78}$].-- params:- r: '79'- number: x^41 - x^39 - x^38 - x^37 - x^36 - x^35 - x^34 - x^33 - x^32 - x^31 - x^30- - x^29 - x^28 - x^27 - x^26 - x^25 - x^24 - x^23 - x^22 - x^21 - x^20 - x^19 -- x^18 - x^17 - x^16 - x^15 - x^14 - x^13 - x^12 - x^11 - x^10 - x^9 - x^8 - x^7- - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1- comment: $P_{40}(x)=(x-1)\,m_{79}(x)$, and the root of $m_{79}$ in $(1,\varphi)$- is HREF{Pisot_numbers_less_than_the_golden_ratio#79}[$\theta_{79}$].+ '1':+ comment: $P_{2}(x)=(x-1)\,m_{1}(x)$, and the root of $m_{1}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#1}[$\theta_{1}$], the plastic+ ratio and the smallest Pisot number.+ number: x^3 - x - 1+ '2':+ comment: $m_{2}(x)=Q_{2}(x)$, and the root of $m_{2}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#2}[$\theta_{2}$].+ number: x^4 - x^3 - 1+ '3':+ comment: $m_{3}(x)=Q_{3}(x)$, and the root of $m_{3}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#3}[$\theta_{3}$].+ number: x^5 - x^4 - x^3 + x^2 - 1+ '4':+ comment: $P_{3}(x)=(x^2-1)\,m_{4}(x)$, and the root of $m_{4}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#4}[$\theta_{4}$], the supergolden+ ratio.+ number: x^3 - x^2 - 1+ '5':+ comment: $m_{5}(x)=Q_{4}(x)$, and the root of $m_{5}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#5}[$\theta_{5}$].+ number: x^6 - x^5 - x^4 + x^2 - 1+ '6':+ comment: $P_{4}(x)=(x-1)\,m_{6}(x)$, and the root of $m_{6}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#6}[$\theta_{6}$].+ number: x^5 - x^3 - x^2 - x - 1+ '7':+ comment: $m_{7}(x)=Q_{5}(x)$, and the root of $m_{7}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#7}[$\theta_{7}$].+ number: x^7 - x^6 - x^5 + x^2 - 1+ '8':+ comment: $m_{8}(x)=E(x)$, and the root of $m_{8}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#8}[$\theta_{8}$].+ number: x^6 - 2*x^5 + x^4 - x^2 + x - 1+ '9':+ comment: $P_{5}(x)=(x^2-1)\,m_{9}(x)$, and the root of $m_{9}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#9}[$\theta_{9}$].+ number: x^5 - x^4 - x^2 - 1+ '10':+ comment: $m_{10}(x)=Q_{6}(x)$, and the root of $m_{10}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#10}[$\theta_{10}$].+ number: x^8 - x^7 - x^6 + x^2 - 1+ '11':+ comment: $P_{6}(x)=(x-1)\,m_{11}(x)$, and the root of $m_{11}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#11}[$\theta_{11}$].+ number: x^7 - x^5 - x^4 - x^3 - x^2 - x - 1+ '12':+ comment: $m_{12}(x)=Q_{7}(x)$, and the root of $m_{12}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#12}[$\theta_{12}$].+ number: x^9 - x^8 - x^7 + x^2 - 1+ '13':+ comment: $P_{7}(x)=(x^2-1)\,m_{13}(x)$, and the root of $m_{13}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#13}[$\theta_{13}$].+ number: x^7 - x^6 - x^4 - x^2 - 1+ '14':+ comment: $m_{14}(x)=Q_{8}(x)$, and the root of $m_{14}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#14}[$\theta_{14}$].+ number: x^10 - x^9 - x^8 + x^2 - 1+ '15':+ comment: $P_{8}(x)=(x-1)\,m_{15}(x)$, and the root of $m_{15}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#15}[$\theta_{15}$].+ number: x^9 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1+ '16':+ comment: $m_{16}(x)=Q_{9}(x)$, and the root of $m_{16}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#16}[$\theta_{16}$].+ number: x^11 - x^10 - x^9 + x^2 - 1+ '17':+ comment: $P_{9}(x)=(x^2-1)\,m_{17}(x)$, and the root of $m_{17}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#17}[$\theta_{17}$].+ number: x^9 - x^8 - x^6 - x^4 - x^2 - 1+ '18':+ comment: $m_{18}(x)=Q_{10}(x)$, and the root of $m_{18}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#18}[$\theta_{18}$].+ number: x^12 - x^11 - x^10 + x^2 - 1+ '19':+ comment: $P_{10}(x)=(x-1)\,m_{19}(x)$, and the root of $m_{19}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#19}[$\theta_{19}$].+ number: x^11 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1+ '20':+ comment: $m_{20}(x)=Q_{11}(x)$, and the root of $m_{20}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#20}[$\theta_{20}$].+ number: x^13 - x^12 - x^11 + x^2 - 1+ '21':+ comment: $P_{11}(x)=(x^2-1)\,m_{21}(x)$, and the root of $m_{21}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#21}[$\theta_{21}$].+ number: x^11 - x^10 - x^8 - x^6 - x^4 - x^2 - 1+ '22':+ comment: $m_{22}(x)=Q_{12}(x)$, and the root of $m_{22}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#22}[$\theta_{22}$].+ number: x^14 - x^13 - x^12 + x^2 - 1+ '23':+ comment: $P_{12}(x)=(x-1)\,m_{23}(x)$, and the root of $m_{23}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#23}[$\theta_{23}$].+ number: x^13 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x+ - 1+ '24':+ comment: $m_{24}(x)=Q_{13}(x)$, and the root of $m_{24}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#24}[$\theta_{24}$].+ number: x^15 - x^14 - x^13 + x^2 - 1+ '25':+ comment: $P_{13}(x)=(x^2-1)\,m_{25}(x)$, and the root of $m_{25}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#25}[$\theta_{25}$].+ number: x^13 - x^12 - x^10 - x^8 - x^6 - x^4 - x^2 - 1+ '26':+ comment: $m_{26}(x)=Q_{14}(x)$, and the root of $m_{26}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#26}[$\theta_{26}$].+ number: x^16 - x^15 - x^14 + x^2 - 1+ '27':+ comment: $P_{14}(x)=(x-1)\,m_{27}(x)$, and the root of $m_{27}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#27}[$\theta_{27}$].+ number: x^15 - x^13 - x^12 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4 -+ x^3 - x^2 - x - 1+ '28':+ comment: $m_{28}(x)=Q_{15}(x)$, and the root of $m_{28}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#28}[$\theta_{28}$].+ number: x^17 - x^16 - x^15 + x^2 - 1+ '29':+ comment: $P_{15}(x)=(x^2-1)\,m_{29}(x)$, and the root of $m_{29}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#29}[$\theta_{29}$].+ number: x^15 - x^14 - x^12 - x^10 - x^8 - x^6 - x^4 - x^2 - 1+ '30':+ comment: $m_{30}(x)=Q_{16}(x)$, and the root of $m_{30}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#30}[$\theta_{30}$].+ number: x^18 - x^17 - x^16 + x^2 - 1+ '31':+ comment: $P_{16}(x)=(x-1)\,m_{31}(x)$, and the root of $m_{31}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#31}[$\theta_{31}$].+ number: x^17 - x^15 - x^14 - x^13 - x^12 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6+ - x^5 - x^4 - x^3 - x^2 - x - 1+ '32':+ comment: $m_{32}(x)=Q_{17}(x)$, and the root of $m_{32}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#32}[$\theta_{32}$].+ number: x^19 - x^18 - x^17 + x^2 - 1+ '33':+ comment: $P_{17}(x)=(x^2-1)\,m_{33}(x)$, and the root of $m_{33}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#33}[$\theta_{33}$].+ number: x^17 - x^16 - x^14 - x^12 - x^10 - x^8 - x^6 - x^4 - x^2 - 1+ '34':+ comment: $m_{34}(x)=Q_{18}(x)$, and the root of $m_{34}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#34}[$\theta_{34}$].+ number: x^20 - x^19 - x^18 + x^2 - 1+ '35':+ comment: $P_{18}(x)=(x-1)\,m_{35}(x)$, and the root of $m_{35}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#35}[$\theta_{35}$].+ number: x^19 - x^17 - x^16 - x^15 - x^14 - x^13 - x^12 - x^11 - x^10 - x^9 - x^8+ - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1+ '36':+ comment: $m_{36}(x)=Q_{19}(x)$, and the root of $m_{36}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#36}[$\theta_{36}$].+ number: x^21 - x^20 - x^19 + x^2 - 1+ '37':+ comment: $P_{19}(x)=(x^2-1)\,m_{37}(x)$, and the root of $m_{37}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#37}[$\theta_{37}$].+ number: x^19 - x^18 - x^16 - x^14 - x^12 - x^10 - x^8 - x^6 - x^4 - x^2 - 1+ '38':+ comment: $m_{38}(x)=Q_{20}(x)$, and the root of $m_{38}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#38}[$\theta_{38}$].+ number: x^22 - x^21 - x^20 + x^2 - 1+ '39':+ comment: $P_{20}(x)=(x-1)\,m_{39}(x)$, and the root of $m_{39}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#39}[$\theta_{39}$].+ number: x^21 - x^19 - x^18 - x^17 - x^16 - x^15 - x^14 - x^13 - x^12 - x^11 -+ x^10 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1+ '40':+ comment: $m_{40}(x)=Q_{21}(x)$, and the root of $m_{40}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#40}[$\theta_{40}$].+ number: x^23 - x^22 - x^21 + x^2 - 1+ '41':+ comment: $P_{21}(x)=(x^2-1)\,m_{41}(x)$, and the root of $m_{41}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#41}[$\theta_{41}$].+ number: x^21 - x^20 - x^18 - x^16 - x^14 - x^12 - x^10 - x^8 - x^6 - x^4 - x^2+ - 1+ '42':+ comment: $m_{42}(x)=Q_{22}(x)$, and the root of $m_{42}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#42}[$\theta_{42}$].+ number: x^24 - x^23 - x^22 + x^2 - 1+ '43':+ comment: $P_{22}(x)=(x-1)\,m_{43}(x)$, and the root of $m_{43}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#43}[$\theta_{43}$].+ number: x^23 - x^21 - x^20 - x^19 - x^18 - x^17 - x^16 - x^15 - x^14 - x^13 -+ x^12 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1+ '44':+ comment: $m_{44}(x)=Q_{23}(x)$, and the root of $m_{44}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#44}[$\theta_{44}$].+ number: x^25 - x^24 - x^23 + x^2 - 1+ '45':+ comment: $P_{23}(x)=(x^2-1)\,m_{45}(x)$, and the root of $m_{45}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#45}[$\theta_{45}$].+ number: x^23 - x^22 - x^20 - x^18 - x^16 - x^14 - x^12 - x^10 - x^8 - x^6 - x^4+ - x^2 - 1+ '46':+ comment: $m_{46}(x)=Q_{24}(x)$, and the root of $m_{46}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#46}[$\theta_{46}$].+ number: x^26 - x^25 - x^24 + x^2 - 1+ '47':+ comment: $P_{24}(x)=(x-1)\,m_{47}(x)$, and the root of $m_{47}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#47}[$\theta_{47}$].+ number: x^25 - x^23 - x^22 - x^21 - x^20 - x^19 - x^18 - x^17 - x^16 - x^15 -+ x^14 - x^13 - x^12 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 -+ x^2 - x - 1+ '48':+ comment: $m_{48}(x)=Q_{25}(x)$, and the root of $m_{48}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#48}[$\theta_{48}$].+ number: x^27 - x^26 - x^25 + x^2 - 1+ '49':+ comment: $P_{25}(x)=(x^2-1)\,m_{49}(x)$, and the root of $m_{49}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#49}[$\theta_{49}$].+ number: x^25 - x^24 - x^22 - x^20 - x^18 - x^16 - x^14 - x^12 - x^10 - x^8 - x^6+ - x^4 - x^2 - 1+ '50':+ comment: $m_{50}(x)=Q_{26}(x)$, and the root of $m_{50}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#50}[$\theta_{50}$].+ number: x^28 - x^27 - x^26 + x^2 - 1+ '51':+ number: x^27 - x^25 - x^24 - x^23 - x^22 - x^21 - x^20 - x^19 - x^18 - x^17 -+ x^16 - x^15 - x^14 - x^13 - x^12 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6 - x^5+ - x^4 - x^3 - x^2 - x - 1+ comment: $P_{26}(x)=(x-1)\,m_{51}(x)$, and the root of $m_{51}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#51}[$\theta_{51}$].+ '52':+ number: x^29 - x^28 - x^27 + x^2 - 1+ comment: $m_{52}(x)=Q_{27}(x)$, and the root of $m_{52}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#52}[$\theta_{52}$].+ '53':+ number: x^27 - x^26 - x^24 - x^22 - x^20 - x^18 - x^16 - x^14 - x^12 - x^10 -+ x^8 - x^6 - x^4 - x^2 - 1+ comment: $P_{27}(x)=(x^2-1)\,m_{53}(x)$, and the root of $m_{53}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#53}[$\theta_{53}$].+ '54':+ number: x^30 - x^29 - x^28 + x^2 - 1+ comment: $m_{54}(x)=Q_{28}(x)$, and the root of $m_{54}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#54}[$\theta_{54}$].+ '55':+ number: x^29 - x^27 - x^26 - x^25 - x^24 - x^23 - x^22 - x^21 - x^20 - x^19 -+ x^18 - x^17 - x^16 - x^15 - x^14 - x^13 - x^12 - x^11 - x^10 - x^9 - x^8 - x^7+ - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1+ comment: $P_{28}(x)=(x-1)\,m_{55}(x)$, and the root of $m_{55}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#55}[$\theta_{55}$].+ '56':+ number: x^31 - x^30 - x^29 + x^2 - 1+ comment: $m_{56}(x)=Q_{29}(x)$, and the root of $m_{56}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#56}[$\theta_{56}$].+ '57':+ number: x^29 - x^28 - x^26 - x^24 - x^22 - x^20 - x^18 - x^16 - x^14 - x^12 -+ x^10 - x^8 - x^6 - x^4 - x^2 - 1+ comment: $P_{29}(x)=(x^2-1)\,m_{57}(x)$, and the root of $m_{57}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#57}[$\theta_{57}$].+ '58':+ number: x^32 - x^31 - x^30 + x^2 - 1+ comment: $m_{58}(x)=Q_{30}(x)$, and the root of $m_{58}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#58}[$\theta_{58}$].+ '59':+ number: x^31 - x^29 - x^28 - x^27 - x^26 - x^25 - x^24 - x^23 - x^22 - x^21 -+ x^20 - x^19 - x^18 - x^17 - x^16 - x^15 - x^14 - x^13 - x^12 - x^11 - x^10 -+ x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1+ comment: $P_{30}(x)=(x-1)\,m_{59}(x)$, and the root of $m_{59}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#59}[$\theta_{59}$].+ '60':+ number: x^33 - x^32 - x^31 + x^2 - 1+ comment: $m_{60}(x)=Q_{31}(x)$, and the root of $m_{60}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#60}[$\theta_{60}$].+ '61':+ number: x^31 - x^30 - x^28 - x^26 - x^24 - x^22 - x^20 - x^18 - x^16 - x^14 -+ x^12 - x^10 - x^8 - x^6 - x^4 - x^2 - 1+ comment: $P_{31}(x)=(x^2-1)\,m_{61}(x)$, and the root of $m_{61}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#61}[$\theta_{61}$].+ '62':+ number: x^34 - x^33 - x^32 + x^2 - 1+ comment: $m_{62}(x)=Q_{32}(x)$, and the root of $m_{62}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#62}[$\theta_{62}$].+ '63':+ number: x^33 - x^31 - x^30 - x^29 - x^28 - x^27 - x^26 - x^25 - x^24 - x^23 -+ x^22 - x^21 - x^20 - x^19 - x^18 - x^17 - x^16 - x^15 - x^14 - x^13 - x^12 -+ x^11 - x^10 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1+ comment: $P_{32}(x)=(x-1)\,m_{63}(x)$, and the root of $m_{63}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#63}[$\theta_{63}$].+ '64':+ number: x^35 - x^34 - x^33 + x^2 - 1+ comment: $m_{64}(x)=Q_{33}(x)$, and the root of $m_{64}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#64}[$\theta_{64}$].+ '65':+ number: x^33 - x^32 - x^30 - x^28 - x^26 - x^24 - x^22 - x^20 - x^18 - x^16 -+ x^14 - x^12 - x^10 - x^8 - x^6 - x^4 - x^2 - 1+ comment: $P_{33}(x)=(x^2-1)\,m_{65}(x)$, and the root of $m_{65}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#65}[$\theta_{65}$].+ '66':+ number: x^36 - x^35 - x^34 + x^2 - 1+ comment: $m_{66}(x)=Q_{34}(x)$, and the root of $m_{66}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#66}[$\theta_{66}$].+ '67':+ number: x^35 - x^33 - x^32 - x^31 - x^30 - x^29 - x^28 - x^27 - x^26 - x^25 -+ x^24 - x^23 - x^22 - x^21 - x^20 - x^19 - x^18 - x^17 - x^16 - x^15 - x^14 -+ x^13 - x^12 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 -+ x - 1+ comment: $P_{34}(x)=(x-1)\,m_{67}(x)$, and the root of $m_{67}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#67}[$\theta_{67}$].+ '68':+ number: x^37 - x^36 - x^35 + x^2 - 1+ comment: $m_{68}(x)=Q_{35}(x)$, and the root of $m_{68}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#68}[$\theta_{68}$].+ '69':+ number: x^35 - x^34 - x^32 - x^30 - x^28 - x^26 - x^24 - x^22 - x^20 - x^18 -+ x^16 - x^14 - x^12 - x^10 - x^8 - x^6 - x^4 - x^2 - 1+ comment: $P_{35}(x)=(x^2-1)\,m_{69}(x)$, and the root of $m_{69}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#69}[$\theta_{69}$].+ '70':+ number: x^38 - x^37 - x^36 + x^2 - 1+ comment: $m_{70}(x)=Q_{36}(x)$, and the root of $m_{70}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#70}[$\theta_{70}$].+ '71':+ number: x^37 - x^35 - x^34 - x^33 - x^32 - x^31 - x^30 - x^29 - x^28 - x^27 -+ x^26 - x^25 - x^24 - x^23 - x^22 - x^21 - x^20 - x^19 - x^18 - x^17 - x^16 -+ x^15 - x^14 - x^13 - x^12 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4+ - x^3 - x^2 - x - 1+ comment: $P_{36}(x)=(x-1)\,m_{71}(x)$, and the root of $m_{71}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#71}[$\theta_{71}$].+ '72':+ number: x^39 - x^38 - x^37 + x^2 - 1+ comment: $m_{72}(x)=Q_{37}(x)$, and the root of $m_{72}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#72}[$\theta_{72}$].+ '73':+ number: x^37 - x^36 - x^34 - x^32 - x^30 - x^28 - x^26 - x^24 - x^22 - x^20 -+ x^18 - x^16 - x^14 - x^12 - x^10 - x^8 - x^6 - x^4 - x^2 - 1+ comment: $P_{37}(x)=(x^2-1)\,m_{73}(x)$, and the root of $m_{73}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#73}[$\theta_{73}$].+ '74':+ number: x^40 - x^39 - x^38 + x^2 - 1+ comment: $m_{74}(x)=Q_{38}(x)$, and the root of $m_{74}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#74}[$\theta_{74}$].+ '75':+ number: x^39 - x^37 - x^36 - x^35 - x^34 - x^33 - x^32 - x^31 - x^30 - x^29 -+ x^28 - x^27 - x^26 - x^25 - x^24 - x^23 - x^22 - x^21 - x^20 - x^19 - x^18 -+ x^17 - x^16 - x^15 - x^14 - x^13 - x^12 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6+ - x^5 - x^4 - x^3 - x^2 - x - 1+ comment: $P_{38}(x)=(x-1)\,m_{75}(x)$, and the root of $m_{75}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#75}[$\theta_{75}$].+ '76':+ number: x^41 - x^40 - x^39 + x^2 - 1+ comment: $m_{76}(x)=Q_{39}(x)$, and the root of $m_{76}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#76}[$\theta_{76}$].+ '77':+ number: x^39 - x^38 - x^36 - x^34 - x^32 - x^30 - x^28 - x^26 - x^24 - x^22 -+ x^20 - x^18 - x^16 - x^14 - x^12 - x^10 - x^8 - x^6 - x^4 - x^2 - 1+ comment: $P_{39}(x)=(x^2-1)\,m_{77}(x)$, and the root of $m_{77}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#77}[$\theta_{77}$].+ '78':+ number: x^42 - x^41 - x^40 + x^2 - 1+ comment: $m_{78}(x)=Q_{40}(x)$, and the root of $m_{78}$ in $(1,\varphi)$ is HREF{Pisot_numbers_less_than_the_golden_ratio#78}[$\theta_{78}$].+ '79':+ number: x^41 - x^39 - x^38 - x^37 - x^36 - x^35 - x^34 - x^33 - x^32 - x^31 -+ x^30 - x^29 - x^28 - x^27 - x^26 - x^25 - x^24 - x^23 - x^22 - x^21 - x^20 -+ x^19 - x^18 - x^17 - x^16 - x^15 - x^14 - x^13 - x^12 - x^11 - x^10 - x^9 -+ x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1+ comment: $P_{40}(x)=(x-1)\,m_{79}(x)$, and the root of $m_{79}$ in $(1,\varphi)$+ is HREF{Pisot_numbers_less_than_the_golden_ratio#79}[$\theta_{79}$].
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