History of Minimal polynomials of the Pisot numbers less than the golden ratio

   number-header: $m_r(x)$ Numbers:-  '1':-    number: x^3 - x - 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.-  '2':-    number: x^4 - x^3 - 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}[$\theta_{2}$].-  '3':-    number: x^5 - x^4 - x^3 + x^2 - 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}[$\theta_{3}$].-  '4':-    number: 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.-  '5':-    number: x^6 - x^5 - x^4 + 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}[$\theta_{5}$].-  '6':-    number: x^5 - x^3 - x^2 - x - 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}[$\theta_{6}$].-  '7':-    number: x^7 - x^6 - x^5 + x^2 - 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}[$\theta_{7}$].-  '8':-    number: x^6 - 2*x^5 + x^4 - x^2 + x - 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}[$\theta_{8}$].-  '9':-    number: x^5 - x^4 - x^2 - 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}[$\theta_{9}$].-  '10':-    number: x^8 - x^7 - x^6 + 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}[$\theta_{10}$].-  '11':-    number: x^7 - x^5 - x^4 - x^3 - x^2 - x - 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}[$\theta_{11}$].-  '12':-    number: x^9 - x^8 - x^7 + x^2 - 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}[$\theta_{12}$].-  '13':-    number: x^7 - x^6 - x^4 - 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}[$\theta_{13}$].-  '14':-    number: x^10 - x^9 - x^8 + 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}[$\theta_{14}$].-  '15':-    number: x^9 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 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}[$\theta_{15}$].-  '16':-    number: x^11 - x^10 - x^9 + x^2 - 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}[$\theta_{16}$].-  '17':-    number: x^9 - x^8 - x^6 - x^4 - 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}[$\theta_{17}$].-  '18':-    number: x^12 - x^11 - x^10 + 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}[$\theta_{18}$].-  '19':-    number: x^11 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 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}[$\theta_{19}$].-  '20':-    number: x^13 - x^12 - x^11 + x^2 - 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}[$\theta_{20}$].-  '21':-    number: x^11 - x^10 - x^8 - x^6 - x^4 - 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}[$\theta_{21}$].-  '22':-    number: x^14 - x^13 - x^12 + 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}[$\theta_{22}$].-  '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: $P_{12}(x)=(x-1)\,m_{23}(x)$, and the root of $m_{23}$ in $(1,\varphi)$-      is HREF{Pisot_numbers_less_than_the_golden_ratio#23}[$\theta_{23}$].-  '24':-    number: x^15 - x^14 - x^13 + x^2 - 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}[$\theta_{24}$].-  '25':-    number: x^13 - x^12 - x^10 - x^8 - x^6 - x^4 - 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}[$\theta_{25}$].-  '26':-    number: x^16 - x^15 - x^14 + 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}[$\theta_{26}$].-  '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: $P_{14}(x)=(x-1)\,m_{27}(x)$, and the root of $m_{27}$ in $(1,\varphi)$-      is HREF{Pisot_numbers_less_than_the_golden_ratio#27}[$\theta_{27}$].-  '28':-    number: x^17 - x^16 - x^15 + x^2 - 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}[$\theta_{28}$].-  '29':-    number: x^15 - x^14 - x^12 - x^10 - x^8 - x^6 - x^4 - 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}[$\theta_{29}$].-  '30':-    number: x^18 - x^17 - x^16 + 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}[$\theta_{30}$].-  '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: $P_{16}(x)=(x-1)\,m_{31}(x)$, and the root of $m_{31}$ in $(1,\varphi)$-      is HREF{Pisot_numbers_less_than_the_golden_ratio#31}[$\theta_{31}$].-  '32':-    number: x^19 - x^18 - x^17 + x^2 - 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}[$\theta_{32}$].-  '33':-    number: x^17 - x^16 - x^14 - x^12 - x^10 - x^8 - x^6 - x^4 - 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}[$\theta_{33}$].-  '34':-    number: x^20 - x^19 - x^18 + 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}[$\theta_{34}$].-  '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: $P_{18}(x)=(x-1)\,m_{35}(x)$, and the root of $m_{35}$ in $(1,\varphi)$-      is HREF{Pisot_numbers_less_than_the_golden_ratio#35}[$\theta_{35}$].-  '36':-    number: x^21 - x^20 - x^19 + x^2 - 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}[$\theta_{36}$].-  '37':-    number: x^19 - x^18 - x^16 - x^14 - x^12 - x^10 - x^8 - x^6 - x^4 - 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}[$\theta_{37}$].-  '38':-    number: x^22 - x^21 - x^20 + 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}[$\theta_{38}$].-  '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: $P_{20}(x)=(x-1)\,m_{39}(x)$, and the root of $m_{39}$ in $(1,\varphi)$-      is HREF{Pisot_numbers_less_than_the_golden_ratio#39}[$\theta_{39}$].-  '40':-    number: x^23 - x^22 - x^21 + x^2 - 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}[$\theta_{40}$].-  '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: $P_{21}(x)=(x^2-1)\,m_{41}(x)$, and the root of $m_{41}$ in $(1,\varphi)$-      is HREF{Pisot_numbers_less_than_the_golden_ratio#41}[$\theta_{41}$].-  '42':-    number: x^24 - x^23 - x^22 + 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}[$\theta_{42}$].-  '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: $P_{22}(x)=(x-1)\,m_{43}(x)$, and the root of $m_{43}$ in $(1,\varphi)$-      is HREF{Pisot_numbers_less_than_the_golden_ratio#43}[$\theta_{43}$].-  '44':-    number: x^25 - x^24 - x^23 + x^2 - 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}[$\theta_{44}$].-  '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: $P_{23}(x)=(x^2-1)\,m_{45}(x)$, and the root of $m_{45}$ in $(1,\varphi)$-      is HREF{Pisot_numbers_less_than_the_golden_ratio#45}[$\theta_{45}$].-  '46':-    number: x^26 - x^25 - x^24 + 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}[$\theta_{46}$].-  '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: $P_{24}(x)=(x-1)\,m_{47}(x)$, and the root of $m_{47}$ in $(1,\varphi)$-      is HREF{Pisot_numbers_less_than_the_golden_ratio#47}[$\theta_{47}$].-  '48':-    number: x^27 - x^26 - x^25 + x^2 - 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}[$\theta_{48}$].-  '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: $P_{25}(x)=(x^2-1)\,m_{49}(x)$, and the root of $m_{49}$ in $(1,\varphi)$-      is HREF{Pisot_numbers_less_than_the_golden_ratio#49}[$\theta_{49}$].-  '50':-    number: x^28 - x^27 - x^26 + 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}[$\theta_{50}$].+- 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}[$\theta_{79}$].