1.はじめに


A5+B5+C5=D5+E5+F5のパラメータ解を求めてみたいと思います。

解を求める方法は、オイラーがA4+B4=C4+D4の解を求めた時の方法を拡張するだけです。
但し、パラメータの項目数がやたらと多いですが。
数値例も最後に付け加えておきましたが、計算して確認する場合はUBASICを使用すると良いでしょう。
UBASICは2800桁位まで多倍長計算できる優秀かつ有名なフリーウエアです。


UBASICのホームページ:http://www.rkmath.rikkyo.ac.jp/~kida/ubasic.htm

2.定理
      
     
      
   A5+B5+C5=D5+E5+F5のパラメータ解が存在する。

    a1,a2,a3 は整数とする。


A=-a1^18*a2^6-a1^18*a2^5*a3-2*a1^18*a2^4*a3^2-a1^18*a2^3*a3^3-2*a1^18*a2^2*a3^4
-a1^18*a2*a3^5-a1^18*a3^6-a1^17*a2^7-4*a1^17*a2^6*a3-5*a1^17*a2^5*a3^2-5*a1^17*a2^4*a3^3
-5*a1^17*a2^3*a3^4-5*a1^17*a2^2*a3^5-4*a1^17*a2*a3^6-a1^17*a3^7-2*a1^16*a2^8
-3*a1^16*a2^7*a3-11*a1^16*a2^6*a3^2-6*a1^16*a2^5*a3^3-9*a1^16*a2^4*a3^4-6*a1^16*a2^3*a3^5
-11*a1^16*a2^2*a3^6-3*a1^16*a2*a3^7-2*a1^16*a3^8-2*a1^15*a2^9-4*a1^15*a2^8*a3
-8*a1^15*a2^7*a3^2-14*a1^15*a2^6*a3^3-8*a1^15*a2^5*a3^4-8*a1^15*a2^4*a3^5
-14*a1^15*a2^3*a3^6-8*a1^15*a2^2*a3^7-4*a1^15*a2*a3^8-2*a1^15*a3^9-a1^14*a2^10
+3*a1^14*a2^9*a3+3*a1^14*a2^7*a3^3-5*a1^14*a2^6*a3^4+4*a1^14*a2^5*a3^5-15*a1^14*a2^4*a3^6
-11*a1^14*a2^3*a3^7-12*a1^14*a2^2*a3^8-3*a1^14*a2*a3^9-3*a1^14*a3^10-2*a1^13*a2^11
+7*a1^13*a2^10*a3+15*a1^13*a2^9*a3^2+17*a1^13*a2^8*a3^3+23*a1^13*a2^7*a3^4
+16*a1^13*a2^6*a3^5+4*a1^13*a2^5*a3^6-7*a1^13*a2^4*a3^7-13*a1^13*a2^3*a3^8
-3*a1^13*a2^2*a3^9+a1^13*a2*a3^10-2*a1^13*a3^11-a1^12*a2^12+7*a1^12*a2^11*a3
+22*a1^12*a2^10*a3^2+34*a1^12*a2^9*a3^3+36*a1^12*a2^8*a3^4+37*a1^12*a2^7*a3^5
+20*a1^12*a2^6*a3^6-a1^12*a2^5*a3^7-16*a1^12*a2^4*a3^8-10*a1^12*a2^3*a3^9
+2*a1^12*a2^2*a3^10+a1^12*a2*a3^11-3*a1^12*a3^12-2*a1^11*a2^13+7*a1^11*a2^12*a3
+15*a1^11*a2^11*a3^2+32*a1^11*a2^10*a3^3+42*a1^11*a2^9*a3^4+35*a1^11*a2^8*a3^5
+43*a1^11*a2^7*a3^6+29*a1^11*a2^6*a3^7+5*a1^11*a2^5*a3^8+6*a1^11*a2^4*a3^9
+12*a1^11*a2^3*a3^10+9*a1^11*a2^2*a3^11+a1^11*a2*a3^12-2*a1^11*a3^13-a1^10*a2^14
+3*a1^10*a2^13*a3+10*a1^10*a2^12*a3^2+16*a1^10*a2^11*a3^3+24*a1^10*a2^10*a3^4
+16*a1^10*a2^9*a3^5+21*a1^10*a2^8*a3^6+18*a1^10*a2^7*a3^7+27*a1^10*a2^6*a3^8
-4*a1^10*a2^5*a3^9+24*a1^10*a2^4*a3^10+8*a1^10*a2^3*a3^11+2*a1^10*a2^2*a3^12
-3*a1^10*a2*a3^13-3*a1^10*a3^14-2*a1^9*a2^15-a1^9*a2^14*a3-9*a1^9*a2^13*a3^2
-6*a1^9*a2^12*a3^3-14*a1^9*a2^11*a3^4-24*a1^9*a2^10*a3^5-14*a1^9*a2^9*a3^6
-10*a1^9*a2^8*a3^7+22*a1^9*a2^7*a3^8+26*a1^9*a2^6*a3^9+28*a1^9*a2^5*a3^10
+34*a1^9*a2^4*a3^11+10*a1^9*a2^3*a3^12-3*a1^9*a2^2*a3^13-7*a1^9*a2*a3^14
-2*a1^9*a3^15-4*a1^8*a2^14*a3^2-11*a1^8*a2^13*a3^3-20*a1^8*a2^12*a3^4
-37*a1^8*a2^11*a3^5-51*a1^8*a2^10*a3^6-46*a1^8*a2^9*a3^7-8*a1^8*a2^8*a3^8
-4*a1^8*a2^7*a3^9+47*a1^8*a2^6*a3^10+27*a1^8*a2^5*a3^11+40*a1^8*a2^4*a3^12
+5*a1^8*a2^3*a3^13-12*a1^8*a2^2*a3^14-6*a1^8*a2*a3^15-2*a1^8*a3^16-a1^7*a2^17
-a1^7*a2^16*a3-8*a1^7*a2^15*a3^2-9*a1^7*a2^14*a3^3-29*a1^7*a2^13*a3^4-39*a1^7*a2^12*a3^5
-61*a1^7*a2^11*a3^6-78*a1^7*a2^10*a3^7-38*a1^7*a2^9*a3^8-28*a1^7*a2^8*a3^9
+12*a1^7*a2^7*a3^10+27*a1^7*a2^6*a3^11+25*a1^7*a2^5*a3^12+19*a1^7*a2^4*a3^13
-17*a1^7*a2^3*a3^14-14*a1^7*a2^2*a3^15-7*a1^7*a2*a3^16-a1^7*a3^17+a1^6*a2^18
+2*a1^6*a2^17*a3+5*a1^6*a2^16*a3^2+6*a1^6*a2^15*a3^3-a1^6*a2^14*a3^4-20*a1^6*a2^13*a3^5
-56*a1^6*a2^12*a3^6-75*a1^6*a2^11*a3^7-77*a1^6*a2^10*a3^8-38*a1^6*a2^9*a3^9
+15*a1^6*a2^8*a3^10+27*a1^6*a2^7*a3^11+54*a1^6*a2^6*a3^12+32*a1^6*a2^5*a3^13
-a1^6*a2^4*a3^14-14*a1^6*a2^3*a3^15-15*a1^6*a2^2*a3^16-4*a1^6*a2*a3^17-a1^6*a3^18
+3*a1^5*a2^18*a3+7*a1^5*a2^17*a3^2+18*a1^5*a2^16*a3^3+16*a1^5*a2^15*a3^4+12*a1^5*a2^14*a3^5
-28*a1^5*a2^13*a3^6-57*a1^5*a2^12*a3^7-67*a1^5*a2^11*a3^8-60*a1^5*a2^10*a3^9
+12*a1^5*a2^9*a3^10+27*a1^5*a2^8*a3^11+45*a1^5*a2^7*a3^12+36*a1^5*a2^6*a3^13
-8*a1^5*a2^5*a3^14-20*a1^5*a2^4*a3^15-26*a1^5*a2^3*a3^16-11*a1^5*a2^2*a3^17
-3*a1^5*a2*a3^18+6*a1^4*a2^18*a3^2+15*a1^4*a2^17*a3^3+27*a1^4*a2^16*a3^4
+28*a1^4*a2^15*a3^5+5*a1^4*a2^14*a3^6-27*a1^4*a2^13*a3^7-64*a1^4*a2^12*a3^8
-58*a1^4*a2^11*a3^9-24*a1^4*a2^10*a3^10+26*a1^4*a2^9*a3^11+48*a1^4*a2^8*a3^12
+51*a1^4*a2^7*a3^13+15*a1^4*a2^6*a3^14-8*a1^4*a2^5*a3^15-27*a1^4*a2^4*a3^16
-15*a1^4*a2^3*a3^17-6*a1^4*a2^2*a3^18+7*a1^3*a2^18*a3^3+15*a1^3*a2^17*a3^4
+26*a1^3*a2^16*a3^5+22*a1^3*a2^15*a3^6+13*a1^3*a2^14*a3^7-13*a1^3*a2^13*a3^8-26*a1^3*a2^12*a3^9
-8*a1^3*a2^11*a3^10+4*a1^3*a2^10*a3^11+34*a1^3*a2^9*a3^12+33*a1^3*a2^8*a3^13
+19*a1^3*a2^7*a3^14+2*a1^3*a2^6*a3^15-18*a1^3*a2^5*a3^16-15*a1^3*a2^4*a3^17
-7*a1^3*a2^3*a3^18+6*a1^2*a2^18*a3^4+11*a1^2*a2^17*a3^5+15*a1^2*a2^16*a3^6
+16*a1^2*a2^15*a3^7+4*a1^2*a2^14*a3^8-7*a1^2*a2^13*a3^9-10*a1^2*a2^12*a3^10
-3*a1^2*a2^11*a3^11+2*a1^2*a2^10*a3^12+17*a1^2*a2^9*a3^13+8*a1^2*a2^8*a3^14
+10*a1^2*a2^7*a3^15-5*a1^2*a2^6*a3^16-7*a1^2*a2^5*a3^17-6*a1^2*a2^4*a3^18
+3*a1*a2^18*a3^5+4*a1*a2^17*a3^6+7*a1*a2^16*a3^7+8*a1*a2^15*a3^8+5*a1*a2^14*a3^9
+5*a1*a2^13*a3^10+5*a1*a2^12*a3^11+5*a1*a2^11*a3^12+5*a1*a2^10*a3^13+5*a1*a2^9*a3^14
+2*a1*a2^8*a3^15+a1*a2^7*a3^16-2*a1*a2^6*a3^17-3*a1*a2^5*a3^18+a2^18*a3^6+a2^17*a3^7
+2*a2^16*a3^8+2*a2^15*a3^9+a2^14*a3^10+2*a2^13*a3^11+a2^12*a3^12+2*a2^11*a3^13
+a2^10*a3^14+2*a2^9*a3^15+a2^7*a3^17-a2^6*a3^18

      
B=-a1^18*a2^6-3*a1^18*a2^5*a3-6*a1^18*a2^4*a3^2-7*a1^18*a2^3*a3^3-6*a1^18*a2^2*a3^4
-3*a1^18*a2*a3^5-a1^18*a3^6-a1^17*a2^7-4*a1^17*a2^6*a3-11*a1^17*a2^5*a3^2-15*a1^17*a2^4*a3^3
-15*a1^17*a2^3*a3^4-7*a1^17*a2^2*a3^5-2*a1^17*a2*a3^6+a1^17*a3^7-2*a1^16*a2^8-7*a1^16*a2^7*a3
-15*a1^16*a2^6*a3^2-26*a1^16*a2^5*a3^3-27*a1^16*a2^4*a3^4-18*a1^16*a2^3*a3^5-5*a1^16*a2^2*a3^6
+a1^16*a2*a3^7-2*a1^15*a2^9-6*a1^15*a2^8*a3-14*a1^15*a2^7*a3^2-14*a1^15*a2^6*a3^3
-20*a1^15*a2^5*a3^4-8*a1^15*a2^4*a3^5+2*a1^15*a2^3*a3^6+10*a1^15*a2^2*a3^7+2*a1^15*a2*a3^8
+2*a1^15*a3^9-3*a1^14*a2^10-7*a1^14*a2^9*a3-12*a1^14*a2^8*a3^2-17*a1^14*a2^7*a3^3
-a1^14*a2^6*a3^4-8*a1^14*a2^5*a3^5+15*a1^14*a2^4*a3^6+19*a1^14*a2^3*a3^7+8*a1^14*a2^2*a3^8
+5*a1^14*a2*a3^9+a1^14*a3^10-2*a1^13*a2^11-3*a1^13*a2^10*a3-3*a1^13*a2^9*a3^2+5*a1^13*a2^8*a3^3
+19*a1^13*a2^7*a3^4+32*a1^13*a2^6*a3^5+36*a1^13*a2^5*a3^6+51*a1^13*a2^4*a3^7+33*a1^13*a2^3*a3^8
+17*a1^13*a2^2*a3^9+5*a1^13*a2*a3^10+2*a1^13*a3^11-3*a1^12*a2^12+a1^12*a2^11*a3
+2*a1^12*a2^10*a3^2+10*a1^12*a2^9*a3^3+40*a1^12*a2^8*a3^4+25*a1^12*a2^7*a3^5
+54*a1^12*a2^6*a3^6+45*a1^12*a2^5*a3^7+48*a1^12*a2^4*a3^8+34*a1^12*a2^3*a3^9+2*a1^12*a2^2*a3^10
+5*a1^12*a2*a3^11+a1^12*a3^12-2*a1^11*a2^13+a1^11*a2^12*a3+9*a1^11*a2^11*a3^2+8*a1^11*a2^10*a3^3
+34*a1^11*a2^9*a3^4+27*a1^11*a2^8*a3^5+27*a1^11*a2^7*a3^6+27*a1^11*a2^6*a3^7+27*a1^11*a2^5*a3^8
+26*a1^11*a2^4*a3^9+4*a1^11*a2^3*a3^10-3*a1^11*a2^2*a3^11+5*a1^11*a2*a3^12+2*a1^11*a3^13
-3*a1^10*a2^14+a1^10*a2^13*a3+2*a1^10*a2^12*a3^2+12*a1^10*a2^11*a3^3+24*a1^10*a2^10*a3^4
+28*a1^10*a2^9*a3^5+47*a1^10*a2^8*a3^6+12*a1^10*a2^7*a3^7+15*a1^10*a2^6*a3^8+12*a1^10*a2^5*a3^9
-24*a1^10*a2^4*a3^10-8*a1^10*a2^3*a3^11-10*a1^10*a2^2*a3^12+5*a1^10*a2*a3^13+a1^10*a3^14
-2*a1^9*a2^15-3*a1^9*a2^14*a3-3*a1^9*a2^13*a3^2-10*a1^9*a2^12*a3^3+6*a1^9*a2^11*a3^4
-4*a1^9*a2^10*a3^5+26*a1^9*a2^9*a3^6-4*a1^9*a2^8*a3^7-28*a1^9*a2^7*a3^8-38*a1^9*a2^6*a3^9
-60*a1^9*a2^5*a3^10-58*a1^9*a2^4*a3^11-26*a1^9*a2^3*a3^12-7*a1^9*a2^2*a3^13+5*a1^9*a2*a3^14
+2*a1^9*a3^15-2*a1^8*a2^16-4*a1^8*a2^15*a3-12*a1^8*a2^14*a3^2-13*a1^8*a2^13*a3^3
-16*a1^8*a2^12*a3^4+5*a1^8*a2^11*a3^5+27*a1^8*a2^10*a3^6+22*a1^8*a2^9*a3^7-8*a1^8*a2^8*a3^8
-38*a1^8*a2^7*a3^9-77*a1^8*a2^6*a3^10-67*a1^8*a2^5*a3^11-64*a1^8*a2^4*a3^12-13*a1^8*a2^3*a3^13
+4*a1^8*a2^2*a3^14+8*a1^8*a2*a3^15+2*a1^8*a3^16-a1^7*a2^17-3*a1^7*a2^16*a3-8*a1^7*a2^15*a3^2
-11*a1^7*a2^14*a3^3-7*a1^7*a2^13*a3^4-a1^7*a2^12*a3^5+29*a1^7*a2^11*a3^6+18*a1^7*a2^10*a3^7
-10*a1^7*a2^9*a3^8-46*a1^7*a2^8*a3^9-78*a1^7*a2^7*a3^10-75*a1^7*a2^6*a3^11-57*a1^7*a2^5*a3^12
-27*a1^7*a2^4*a3^13+13*a1^7*a2^3*a3^14+16*a1^7*a2^2*a3^15+7*a1^7*a2*a3^16+a1^7*a3^17-a1^6*a2^18
-4*a1^6*a2^17*a3-11*a1^6*a2^16*a3^2-14*a1^6*a2^15*a3^3-15*a1^6*a2^14*a3^4+4*a1^6*a2^13*a3^5
+20*a1^6*a2^12*a3^6+43*a1^6*a2^11*a3^7+21*a1^6*a2^10*a3^8-14*a1^6*a2^9*a3^9-51*a1^6*a2^8*a3^10
-61*a1^6*a2^7*a3^11-56*a1^6*a2^6*a3^12-28*a1^6*a2^5*a3^13+5*a1^6*a2^4*a3^14+22*a1^6*a2^3*a3^15
+15*a1^6*a2^2*a3^16+4*a1^6*a2*a3^17+a1^6*a3^18-a1^5*a2^18*a3-5*a1^5*a2^17*a3^2-6*a1^5*a2^16*a3^3
-8*a1^5*a2^15*a3^4+4*a1^5*a2^14*a3^5+16*a1^5*a2^13*a3^6+37*a1^5*a2^12*a3^7+35*a1^5*a2^11*a3^8
+16*a1^5*a2^10*a3^9-24*a1^5*a2^9*a3^10-37*a1^5*a2^8*a3^11-39*a1^5*a2^7*a3^12-20*a1^5*a2^6*a3^13
+12*a1^5*a2^5*a3^14+28*a1^5*a2^4*a3^15+26*a1^5*a2^3*a3^16+11*a1^5*a2^2*a3^17+3*a1^5*a2*a3^18
-2*a1^4*a2^18*a3^2-5*a1^4*a2^17*a3^3-9*a1^4*a2^16*a3^4-8*a1^4*a2^15*a3^5-5*a1^4*a2^14*a3^6
+23*a1^4*a2^13*a3^7+36*a1^4*a2^12*a3^8+42*a1^4*a2^11*a3^9+24*a1^4*a2^10*a3^10-14*a1^4*a2^9*a3^11
-20*a1^4*a2^8*a3^12-29*a1^4*a2^7*a3^13-a1^4*a2^6*a3^14+16*a1^4*a2^5*a3^15+27*a1^4*a2^4*a3^16
+15*a1^4*a2^3*a3^17+6*a1^4*a2^2*a3^18-a1^3*a2^18*a3^3-5*a1^3*a2^17*a3^4-6*a1^3*a2^16*a3^5
-14*a1^3*a2^15*a3^6+3*a1^3*a2^14*a3^7+17*a1^3*a2^13*a3^8+34*a1^3*a2^12*a3^9+32*a1^3*a2^11*a3^10
+16*a1^3*a2^10*a3^11-6*a1^3*a2^9*a3^12-11*a1^3*a2^8*a3^13-9*a1^3*a2^7*a3^14+6*a1^3*a2^6*a3^15
+18*a1^3*a2^5*a3^16+15*a1^3*a2^4*a3^17+7*a1^3*a2^3*a3^18-2*a1^2*a2^18*a3^4-5*a1^2*a2^17*a3^5
-11*a1^2*a2^16*a3^6-8*a1^2*a2^15*a3^7+15*a1^2*a2^13*a3^9+22*a1^2*a2^12*a3^10+15*a1^2*a2^11*a3^11
+10*a1^2*a2^10*a3^12-9*a1^2*a2^9*a3^13-4*a1^2*a2^8*a3^14-8*a1^2*a2^7*a3^15+5*a1^2*a2^6*a3^16
+7*a1^2*a2^5*a3^17+6*a1^2*a2^4*a3^18-a1*a2^18*a3^5-4*a1*a2^17*a3^6-3*a1*a2^16*a3^7
-4*a1*a2^15*a3^8+3*a1*a2^14*a3^9+7*a1*a2^13*a3^10+7*a1*a2^12*a3^11+7*a1*a2^11*a3^12+3*a1*a2^10*a3^13-a1*a2^9*a3^14-a1*a2^7*a3^16+2*a1*a2^6*a3^17+3*a1*a2^5*a3^18-a2^18*a3^6-a2^17*a3^7
-2*a2^16*a3^8-2*a2^15*a3^9-a2^14*a3^10-2*a2^13*a3^11-a2^12*a3^12-2*a2^11*a3^13-a2^10*a3^14
-2*a2^9*a3^15-a2^7*a3^17+a2^6*a3^18


C=a1^18*a2^6+3*a1^18*a2^5*a3+6*a1^18*a2^4*a3^2+7*a1^18*a2^3*a3^3+6*a1^18*a2^2*a3^4
+3*a1^18*a2*a3^5+a1^18*a3^6+a1^17*a2^7+4*a1^17*a2^6*a3+11*a1^17*a2^5*a3^2+15*a1^17*a2^4*a3^3
+15*a1^17*a2^3*a3^4+7*a1^17*a2^2*a3^5+2*a1^17*a2*a3^6-a1^17*a3^7+2*a1^16*a2^8+7*a1^16*a2^7*a3
+15*a1^16*a2^6*a3^2+26*a1^16*a2^5*a3^3+27*a1^16*a2^4*a3^4+18*a1^16*a2^3*a3^5+5*a1^16*a2^2*a3^6
-a1^16*a2*a3^7+2*a1^15*a2^9+8*a1^15*a2^8*a3+16*a1^15*a2^7*a3^2+22*a1^15*a2^6*a3^3
+28*a1^15*a2^5*a3^4+16*a1^15*a2^4*a3^5+6*a1^15*a2^3*a3^6-8*a1^15*a2^2*a3^7-2*a1^15*a3^9
+a1^14*a2^10+5*a1^14*a2^9*a3+4*a1^14*a2^8*a3^2+13*a1^14*a2^7*a3^3+5*a1^14*a2^6*a3^4
+12*a1^14*a2^5*a3^5-a1^14*a2^4*a3^6-9*a1^14*a2^3*a3^7-4*a1^14*a2^2*a3^8-a1^14*a2*a3^9
-a1^14*a3^10+2*a1^13*a2^11+5*a1^13*a2^10*a3-7*a1^13*a2^9*a3^2-13*a1^13*a2^8*a3^3
-27*a1^13*a2^7*a3^4-28*a1^13*a2^6*a3^5-20*a1^13*a2^5*a3^6-29*a1^13*a2^4*a3^7
-11*a1^13*a2^3*a3^8-9*a1^13*a2^2*a3^9+3*a1^13*a2*a3^10-2*a1^13*a3^11+a1^12*a2^12
+5*a1^12*a2^11*a3-10*a1^12*a2^10*a3^2-26*a1^12*a2^9*a3^3-64*a1^12*a2^8*a3^4
-57*a1^12*a2^7*a3^5-56*a1^12*a2^6*a3^6-39*a1^12*a2^5*a3^7-20*a1^12*a2^4*a3^8
-6*a1^12*a2^3*a3^9+10*a1^12*a2^2*a3^10+7*a1^12*a2*a3^11-a1^12*a3^12+2*a1^11*a2^13
+5*a1^11*a2^12*a3-3*a1^11*a2^11*a3^2-8*a1^11*a2^10*a3^3-58*a1^11*a2^9*a3^4
-67*a1^11*a2^8*a3^5-75*a1^11*a2^7*a3^6-61*a1^11*a2^6*a3^7-37*a1^11*a2^5*a3^8
-14*a1^11*a2^4*a3^9+16*a1^11*a2^3*a3^10+15*a1^11*a2^2*a3^11+7*a1^11*a2*a3^12-2*a1^11*a3^13
+a1^10*a2^14+5*a1^10*a2^13*a3+2*a1^10*a2^12*a3^2+4*a1^10*a2^11*a3^3-24*a1^10*a2^10*a3^4
-60*a1^10*a2^9*a3^5-77*a1^10*a2^8*a3^6-78*a1^10*a2^7*a3^7-51*a1^10*a2^6*a3^8
-24*a1^10*a2^5*a3^9+24*a1^10*a2^4*a3^10+32*a1^10*a2^3*a3^11+22*a1^10*a2^2*a3^12
+7*a1^10*a2*a3^13-a1^10*a3^14+2*a1^9*a2^15+5*a1^9*a2^14*a3+17*a1^9*a2^13*a3^2
+34*a1^9*a2^12*a3^3+26*a1^9*a2^11*a3^4+12*a1^9*a2^10*a3^5-38*a1^9*a2^9*a3^6
-38*a1^9*a2^8*a3^7-46*a1^9*a2^7*a3^8-14*a1^9*a2^6*a3^9+16*a1^9*a2^5*a3^10+42*a1^9*a2^4*a3^11
+34*a1^9*a2^3*a3^12+15*a1^9*a2^2*a3^13+3*a1^9*a2*a3^14-2*a1^9*a3^1+2*a1^8*a2^15*a3
+8*a1^8*a2^14*a3^2+33*a1^8*a2^13*a3^3+48*a1^8*a2^12*a3^4+27*a1^8*a2^11*a3^5
+15*a1^8*a2^10*a3^6-28*a1^8*a2^9*a3^7-8*a1^8*a2^8*a3^8-10*a1^8*a2^7*a3^9
+21*a1^8*a2^6*a3^10+35*a1^8*a2^5*a3^11+36*a1^8*a2^4*a3^12+17*a1^8*a2^3*a3^13
-4*a1^8*a2*a3^15-2*a1^8*a3^16+a1^7*a2^17+a1^7*a2^16*a3+10*a1^7*a2^15*a3^2
+19*a1^7*a2^14*a3^3+51*a1^7*a2^13*a3^4+45*a1^7*a2^12*a3^5+27*a1^7*a2^11*a3^6
+12*a1^7*a2^10*a3^7-4*a1^7*a2^9*a3^8+22*a1^7*a2^8*a3^9+18*a1^7*a2^7*a3^10
+43*a1^7*a2^6*a3^11+37*a1^7*a2^5*a3^12+23*a1^7*a2^4*a3^13+3*a1^7*a2^3*a3^14
-8*a1^7*a2^2*a3^15-3*a1^7*a2*a3^16-a1^7*a3^17-a1^6*a2^18-2*a1^6*a2^17*a3-5*a1^6*a2^16*a3^2
+2*a1^6*a2^15*a3^3+15*a1^6*a2^14*a3^4+36*a1^6*a2^13*a3^5+54*a1^6*a2^12*a3^6
+27*a1^6*a2^11*a3^7+47*a1^6*a2^10*a3^8+26*a1^6*a2^9*a3^9+27*a1^6*a2^8*a3^10
+29*a1^6*a2^7*a3^11+20*a1^6*a2^6*a3^12+16*a1^6*a2^5*a3^13-5*a1^6*a2^4*a3^14
-14*a1^6*a2^3*a3^15-11*a1^6*a2^2*a3^16-4*a1^6*a2*a3^17-a1^6*a3^18-3*a1^5*a2^18*a3
-7*a1^5*a2^17*a3^2-18*a1^5*a2^16*a3^3-8*a1^5*a2^15*a3^4-8*a1^5*a2^14*a3^5
+32*a1^5*a2^13*a3^6+25*a1^5*a2^12*a3^7+27*a1^5*a2^11*a3^8+28*a1^5*a2^10*a3^9
-4*a1^5*a2^9*a3^10+5*a1^5*a2^8*a3^11-a1^5*a2^7*a3^12+4*a1^5*a2^6*a3^13+4*a1^5*a2^5*a3^14
-8*a1^5*a2^4*a3^15-6*a1^5*a2^3*a3^16-5*a1^5*a2^2*a3^17-a1^5*a2*a3^18-6*a1^4*a2^18*a3^2
-15*a1^4*a2^17*a3^3-27*a1^4*a2^16*a3^4-20*a1^4*a2^15*a3^5-a1^4*a2^14*a3^6
+19*a1^4*a2^13*a3^7+40*a1^4*a2^12*a3^8+34*a1^4*a2^11*a3^9+24*a1^4*a2^10*a3^10
+6*a1^4*a2^9*a3^11-16*a1^4*a2^8*a3^12-7*a1^4*a2^7*a3^13-15*a1^4*a2^6*a3^14
-8*a1^4*a2^5*a3^15-9*a1^4*a2^4*a3^16-5*a1^4*a2^3*a3^17-2*a1^4*a2^2*a3^18-7*a1^3*a2^18*a3^3
-15*a1^3*a2^17*a3^4-26*a1^3*a2^16*a3^5-14*a1^3*a2^15*a3^6-17*a1^3*a2^14*a3^7+5*a1^3*a2^13*a3^8
+10*a1^3*a2^12*a3^9+8*a1^3*a2^11*a3^10+12*a1^3*a2^10*a3^11-10*a1^3*a2^9*a3^12
-13*a1^3*a2^8*a3^13-11*a1^3*a2^7*a3^14-14*a1^3*a2^6*a3^15-6*a1^3*a2^5*a3^16
-5*a1^3*a2^4*a3^17-a1^3*a2^3*a3^18-6*a1^2*a2^18*a3^4-11*a1^2*a2^17*a3^5-15*a1^2*a2^16*a3^6
-14*a1^2*a2^15*a3^7-12*a1^2*a2^14*a3^8-3*a1^2*a2^13*a3^9+2*a1^2*a2^12*a3^10
+9*a1^2*a2^11*a3^11+2*a1^2*a2^10*a3^12-3*a1^2*a2^9*a3^13-12*a1^2*a2^8*a3^14
-8*a1^2*a2^7*a3^15-11*a1^2*a2^6*a3^16-5*a1^2*a2^5*a3^17-2*a1^2*a2^4*a3^18
-3*a1*a2^18*a3^5-4*a1*a2^17*a3^6-7*a1*a2^16*a3^7-6*a1*a2^15*a3^8-7*a1*a2^14*a3^9
-3*a1*a2^13*a3^10+a1*a2^12*a3^11+a1*a2^11*a3^12+a1*a2^10*a3^13-3*a1*a2^9*a3^14
-4*a1*a2^8*a3^15-3*a1*a2^7*a3^16-4*a1*a2^6*a3^17-a1*a2^5*a3^18-a2^18*a3^6
-a2^17*a3^7-2*a2^16*a3^8-2*a2^15*a3^9-3*a2^14*a3^10-2*a2^13*a3^11-3*a2^12*a3^12
-2*a2^11*a3^13-3*a2^10*a3^14-2*a2^9*a3^15-2*a2^8*a3^16-a2^7*a3^17-a2^6*a3^18



D=-a1^18*a2^6-a1^18*a2^5*a3-2*a1^18*a2^4*a3^2-a1^18*a2^3*a3^3-2*a1^18*a2^2*a3^4
-a1^18*a2*a3^5-a1^18*a3^6-a1^17*a2^7-4*a1^17*a2^6*a3-5*a1^17*a2^5*a3^2-5*a1^17*a2^4*a3^3
-5*a1^17*a2^3*a3^4-5*a1^17*a2^2*a3^5-4*a1^17*a2*a3^6-a1^17*a3^7-2*a1^16*a2^8-3*a1^16*a2^7*a3
-11*a1^16*a2^6*a3^2-6*a1^16*a2^5*a3^3-9*a1^16*a2^4*a3^4-6*a1^16*a2^3*a3^5-11*a1^16*a2^2*a3^6
-3*a1^16*a2*a3^7-2*a1^16*a3^8-2*a1^15*a2^9-4*a1^15*a2^8*a3-8*a1^15*a2^7*a3^2-14*a1^15*a2^6*a3^3
-8*a1^15*a2^5*a3^4-8*a1^15*a2^4*a3^5-14*a1^15*a2^3*a3^6-8*a1^15*a2^2*a3^7-4*a1^15*a2*a3^8
-2*a1^15*a3^9-3*a1^14*a2^10-3*a1^14*a2^9*a3-12*a1^14*a2^8*a3^2-11*a1^14*a2^7*a3^3
-15*a1^14*a2^6*a3^4+4*a1^14*a2^5*a3^5-5*a1^14*a2^4*a3^6+3*a1^14*a2^3*a3^7+3*a1^14*a2*a3^9
-a1^14*a3^10-2*a1^13*a2^11+a1^13*a2^10*a3-3*a1^13*a2^9*a3^2-13*a1^13*a2^8*a3^3-7*a1^13*a2^7*a3^4
+4*a1^13*a2^6*a3^5+16*a1^13*a2^5*a3^6+23*a1^13*a2^4*a3^7+17*a1^13*a2^3*a3^8+15*a1^13*a2^2*a3^9
+7*a1^13*a2*a3^10-2*a1^13*a3^11-3*a1^12*a2^12+a1^12*a2^11*a3+2*a1^12*a2^10*a3^2
-10*a1^12*a2^9*a3^3-16*a1^12*a2^8*a3^4-a1^12*a2^7*a3^5+20*a1^12*a2^6*a3^6+37*a1^12*a2^5*a3^7
+36*a1^12*a2^4*a3^8+34*a1^12*a2^3*a3^9+22*a1^12*a2^2*a3^10+7*a1^12*a2*a3^11-a1^12*a3^12
-2*a1^11*a2^13+a1^11*a2^12*a3+9*a1^11*a2^11*a3^2+12*a1^11*a2^10*a3^3+6*a1^11*a2^9*a3^4
+5*a1^11*a2^8*a3^5+29*a1^11*a2^7*a3^6+43*a1^11*a2^6*a3^7+35*a1^11*a2^5*a3^8+42*a1^11*a2^4*a3^9
+32*a1^11*a2^3*a3^10+15*a1^11*a2^2*a3^11+7*a1^11*a2*a3^12-2*a1^11*a3^13-3*a1^10*a2^14
-3*a1^10*a2^13*a3+2*a1^10*a2^12*a3^2+8*a1^10*a2^11*a3^3+24*a1^10*a2^10*a3^4-4*a1^10*a2^9*a3^5
+27*a1^10*a2^8*a3^6+18*a1^10*a2^7*a3^7+21*a1^10*a2^6*a3^8+16*a1^10*a2^5*a3^9+24*a1^10*a2^4*a3^10+16*a1^10*a2^3*a3^11+10*a1^10*a2^2*a3^12+3*a1^10*a2*a3^13-a1^10*a3^14-2*a1^9*a2^15-7*a1^9*a2^14*a3-3*a1^9*a2^13*a3^2+10*a1^9*a2^12*a3^3
+34*a1^9*a2^11*a3^4+28*a1^9*a2^10*a3^5+26*a1^9*a2^9*a3^6+22*a1^9*a2^8*a3^7-10*a1^9*a2^7*a3^8
-14*a1^9*a2^6*a3^9-24*a1^9*a2^5*a3^10-14*a1^9*a2^4*a3^11-6*a1^9*a2^3*a3^12-9*a1^9*a2^2*a3^13
-a1^9*a2*a3^14-2*a1^9*a3^15-2*a1^8*a2^16-6*a1^8*a2^15*a3-12*a1^8*a2^14*a3^2+5*a1^8*a2^13*a3^3
+40*a1^8*a2^12*a3^4+27*a1^8*a2^11*a3^5+47*a1^8*a2^10*a3^6-4*a1^8*a2^9*a3^7-8*a1^8*a2^8*a3^8
-46*a1^8*a2^7*a3^9-51*a1^8*a2^6*a3^10-37*a1^8*a2^5*a3^11-20*a1^8*a2^4*a3^12-11*a1^8*a2^3*a3^13
-4*a1^8*a2^2*a3^14-a1^7*a2^17-7*a1^7*a2^16*a3-14*a1^7*a2^15*a3^2-17*a1^7*a2^14*a3^3
+19*a1^7*a2^13*a3^4+25*a1^7*a2^12*a3^5+27*a1^7*a2^11*a3^6+12*a1^7*a2^10*a3^7-28*a1^7*a2^9*a3^8
-38*a1^7*a2^8*a3^9-78*a1^7*a2^7*a3^10-61*a1^7*a2^6*a3^11-39*a1^7*a2^5*a3^12-29*a1^7*a2^4*a3^13
-9*a1^7*a2^3*a3^14-8*a1^7*a2^2*a3^15-a1^7*a2*a3^16-a1^7*a3^17-a1^6*a2^18-4*a1^6*a2^17*a3
-15*a1^6*a2^16*a3^2-14*a1^6*a2^15*a3^3-a1^6*a2^14*a3^4+32*a1^6*a2^13*a3^5+54*a1^6*a2^12*a3^6
+27*a1^6*a2^11*a3^7+15*a1^6*a2^10*a3^8-38*a1^6*a2^9*a3^9-77*a1^6*a2^8*a3^10-75*a1^6*a2^7*a3^11
-56*a1^6*a2^6*a3^12-20*a1^6*a2^5*a3^13-a1^6*a2^4*a3^14+6*a1^6*a2^3*a3^15+5*a1^6*a2^2*a3^16
+2*a1^6*a2*a3^17+a1^6*a3^18-3*a1^5*a2^18*a3-11*a1^5*a2^17*a3^2-26*a1^5*a2^16*a3^3
-20*a1^5*a2^15*a3^4-8*a1^5*a2^14*a3^5+36*a1^5*a2^13*a3^6+45*a1^5*a2^12*a3^7+27*a1^5*a2^11*a3^8
+12*a1^5*a2^10*a3^9-60*a1^5*a2^9*a3^10-67*a1^5*a2^8*a3^11-57*a1^5*a2^7*a3^12-28*a1^5*a2^6*a3^13
+12*a1^5*a2^5*a3^14+16*a1^5*a2^4*a3^15+18*a1^5*a2^3*a3^16+7*a1^5*a2^2*a3^17+3*a1^5*a2*a3^18
-6*a1^4*a2^18*a3^2-15*a1^4*a2^17*a3^3-27*a1^4*a2^16*a3^4-8*a1^4*a2^15*a3^5+15*a1^4*a2^14*a3^6
+51*a1^4*a2^13*a3^7+48*a1^4*a2^12*a3^8+26*a1^4*a2^11*a3^9-24*a1^4*a2^10*a3^10-58*a1^4*a2^9*a3^11
-64*a1^4*a2^8*a3^12-27*a1^4*a2^7*a3^13+5*a1^4*a2^6*a3^14+28*a1^4*a2^5*a3^15+27*a1^4*a2^4*a3^16
+15*a1^4*a2^3*a3^17+6*a1^4*a2^2*a3^18-7*a1^3*a2^18*a3^3-15*a1^3*a2^17*a3^4-18*a1^3*a2^16*a3^5
+2*a1^3*a2^15*a3^6+19*a1^3*a2^14*a3^7+33*a1^3*a2^13*a3^8+34*a1^3*a2^12*a3^9+4*a1^3*a2^11*a3^10
-8*a1^3*a2^10*a3^11-26*a1^3*a2^9*a3^12-13*a1^3*a2^8*a3^13+13*a1^3*a2^7*a3^14+22*a1^3*a2^6*a3^15
+26*a1^3*a2^5*a3^16+15*a1^3*a2^4*a3^17+7*a1^3*a2^3*a3^18-6*a1^2*a2^18*a3^4-7*a1^2*a2^17*a3^5
-5*a1^2*a2^16*a3^6+10*a1^2*a2^15*a3^7+8*a1^2*a2^14*a3^8+17*a1^2*a2^13*a3^9+2*a1^2*a2^12*a3^10
-3*a1^2*a2^11*a3^11-10*a1^2*a2^10*a3^12-7*a1^2*a2^9*a3^13+4*a1^2*a2^8*a3^14+16*a1^2*a2^7*a3^15
+15*a1^2*a2^6*a3^16+11*a1^2*a2^5*a3^17+6*a1^2*a2^4*a3^18-3*a1*a2^18*a3^5-2*a1*a2^17*a3^6
+a1*a2^16*a3^7+2*a1*a2^15*a3^8+5*a1*a2^14*a3^9+5*a1*a2^13*a3^10+5*a1*a2^12*a3^11
+5*a1*a2^11*a3^12+5*a1*a2^10*a3^13+5*a1*a2^9*a3^14+8*a1*a2^8*a3^15+7*a1*a2^7*a3^16
+4*a1*a2^6*a3^17+3*a1*a2^5*a3^18-a2^18*a3^6+a2^17*a3^7+2*a2^15*a3^9+a2^14*a3^10+2*a2^13*a3^11
+a2^12*a3^12+2*a2^11*a3^13+a2^10*a3^14+2*a2^9*a3^15+2*a2^8*a3^16+a2^7*a3^17+a2^6*a3^18


E=a1^18*a2^6+3*a1^18*a2^5*a3+6*a1^18*a2^4*a3^2+7*a1^18*a2^3*a3^3+6*a1^18*a2^2*a3^4
+3*a1^18*a2*a3^5+a1^18*a3^6-a1^17*a2^7+2*a1^17*a2^6*a3+7*a1^17*a2^5*a3^2+15*a1^17*a2^4*a3^3
+15*a1^17*a2^3*a3^4+11*a1^17*a2^2*a3^5+4*a1^17*a2*a3^6+a1^17*a3^7-a1^16*a2^7*a3
+5*a1^16*a2^6*a3^2+18*a1^16*a2^5*a3^3+27*a1^16*a2^4*a3^4+26*a1^16*a2^3*a3^5+15*a1^16*a2^2*a3^6
+7*a1^16*a2*a3^7+2*a1^16*a3^8-2*a1^15*a2^9-8*a1^15*a2^7*a3^2+6*a1^15*a2^6*a3^3
+16*a1^15*a2^5*a3^4+28*a1^15*a2^4*a3^5+22*a1^15*a2^3*a3^6+16*a1^15*a2^2*a3^7+8*a1^15*a2*a3^8
+2*a1^15*a3^9-a1^14*a2^10-a1^14*a2^9*a3-4*a1^14*a2^8*a3^2-9*a1^14*a2^7*a3^3-a1^14*a2^6*a3^4
+12*a1^14*a2^5*a3^5+5*a1^14*a2^4*a3^6+13*a1^14*a2^3*a3^7+4*a1^14*a2^2*a3^8+5*a1^14*a2*a3^9
+a1^14*a3^10-2*a1^13*a2^11+3*a1^13*a2^10*a3-9*a1^13*a2^9*a3^2-11*a1^13*a2^8*a3^3
-29*a1^13*a2^7*a3^4-20*a1^13*a2^6*a3^5-28*a1^13*a2^5*a3^6-27*a1^13*a2^4*a3^7-13*a1^13*a2^3*a3^8
-7*a1^13*a2^2*a3^9+5*a1^13*a2*a3^10+2*a1^13*a3^11-a1^12*a2^12+7*a1^12*a2^11*a3
+10*a1^12*a2^10*a3^2-6*a1^12*a2^9*a3^3-20*a1^12*a2^8*a3^4-39*a1^12*a2^7*a3^5-56*a1^12*a2^6*a3^6
-57*a1^12*a2^5*a3^7-64*a1^12*a2^4*a3^8-26*a1^12*a2^3*a3^9-10*a1^12*a2^2*a3^10+5*a1^12*a2*a3^11
+a1^12*a3^12-2*a1^11*a2^13+7*a1^11*a2^12*a3+15*a1^11*a2^11*a3^2+16*a1^11*a2^10*a3^3
-14*a1^11*a2^9*a3^4-37*a1^11*a2^8*a3^5-61*a1^11*a2^7*a3^6-75*a1^11*a2^6*a3^7-67*a1^11*a2^5*a3^8
-58*a1^11*a2^4*a3^9-8*a1^11*a2^3*a3^10-3*a1^11*a2^2*a3^11+5*a1^11*a2*a3^12+2*a1^11*a3^13
-a1^10*a2^14+7*a1^10*a2^13*a3+22*a1^10*a2^12*a3^2+32*a1^10*a2^11*a3^3+24*a1^10*a2^10*a3^4
-24*a1^10*a2^9*a3^5-51*a1^10*a2^8*a3^6-78*a1^10*a2^7*a3^7-77*a1^10*a2^6*a3^8-60*a1^10*a2^5*a3^9
-24*a1^10*a2^4*a3^10+4*a1^10*a2^3*a3^11+2*a1^10*a2^2*a3^12+5*a1^10*a2*a3^13+a1^10*a3^14
-2*a1^9*a2^15+3*a1^9*a2^14*a3+15*a1^9*a2^13*a3^2+34*a1^9*a2^12*a3^3+42*a1^9*a2^11*a3^4
+16*a1^9*a2^10*a3^5-14*a1^9*a2^9*a3^6-46*a1^9*a2^8*a3^7-38*a1^9*a2^7*a3^8-38*a1^9*a2^6*a3^9
+12*a1^9*a2^5*a3^10+26*a1^9*a2^4*a3^11+34*a1^9*a2^3*a3^12+17*a1^9*a2^2*a3^13+5*a1^9*a2*a3^14
+2*a1^9*a3^15-2*a1^8*a2^16-4*a1^8*a2^15*a3+17*a1^8*a2^13*a3^3+36*a1^8*a2^12*a3^4
+35*a1^8*a2^11*a3^5+21*a1^8*a2^10*a3^6-10*a1^8*a2^9*a3^7-8*a1^8*a2^8*a3^8-28*a1^8*a2^7*a3^9+15*a1^8*a2^6*a3^10+27*a1^8*a2^5*a3^11+48*a1^8*a2^4*a3^12+33*a1^8*a2^3*a3^13+8*a1^8*a2^2*a3^14
+2*a1^8*a2*a3^15-a1^7*a2^17-3*a1^7*a2^16*a3-8*a1^7*a2^15*a3^2+3*a1^7*a2^14*a3^3
+23*a1^7*a2^13*a3^4+37*a1^7*a2^12*a3^5+43*a1^7*a2^11*a3^6+18*a1^7*a2^10*a3^7+22*a1^7*a2^9*a3^8
-4*a1^7*a2^8*a3^9+12*a1^7*a2^7*a3^10+27*a1^7*a2^6*a3^11+45*a1^7*a2^5*a3^12+51*a1^7*a2^4*a3^13
+19*a1^7*a2^3*a3^14+10*a1^7*a2^2*a3^15+a1^7*a2*a3^16+a1^7*a3^17-a1^6*a2^18-4*a1^6*a2^17*a3-11*a1^6*a2^16*a3^2-14*a1^6*a2^15*a3^3-5*a1^6*a2^14*a3^4+16*a1^6*a2^13*a3^5+20*a1^6*a2^12*a3^6
+29*a1^6*a2^11*a3^7+27*a1^6*a2^10*a3^8+26*a1^6*a2^9*a3^9+47*a1^6*a2^8*a3^10+27*a1^6*a2^7*a3^11
+54*a1^6*a2^6*a3^12+36*a1^6*a2^5*a3^13+15*a1^6*a2^4*a3^14+2*a1^6*a2^3*a3^15-5*a1^6*a2^2*a3^16
-2*a1^6*a2*a3^17-a1^6*a3^18-a1^5*a2^18*a3-5*a1^5*a2^17*a3^2-6*a1^5*a2^16*a3^3-8*a1^5*a2^15*a3^4
+4*a1^5*a2^14*a3^5+4*a1^5*a2^13*a3^6-a1^5*a2^12*a3^7+5*a1^5*a2^11*a3^8-4*a1^5*a2^10*a3^9
+28*a1^5*a2^9*a3^10+27*a1^5*a2^8*a3^11+25*a1^5*a2^7*a3^12+32*a1^5*a2^6*a3^13-8*a1^5*a2^5*a3^14
-8*a1^5*a2^4*a3^15-18*a1^5*a2^3*a3^16-7*a1^5*a2^2*a3^17-3*a1^5*a2*a3^18-2*a1^4*a2^18*a3^2
-5*a1^4*a2^17*a3^3-9*a1^4*a2^16*a3^4-8*a1^4*a2^15*a3^5-15*a1^4*a2^14*a3^6-7*a1^4*a2^13*a3^7
-16*a1^4*a2^12*a3^8+6*a1^4*a2^11*a3^9+24*a1^4*a2^10*a3^10+34*a1^4*a2^9*a3^11+40*a1^4*a2^8*a3^12
+19*a1^4*a2^7*a3^13-a1^4*a2^6*a3^14-20*a1^4*a2^5*a3^15-27*a1^4*a2^4*a3^16-15*a1^4*a2^3*a3^17
-6*a1^4*a2^2*a3^18-a1^3*a2^18*a3^3-5*a1^3*a2^17*a3^4-6*a1^3*a2^16*a3^5-14*a1^3*a2^15*a3^6
-11*a1^3*a2^14*a3^7-13*a1^3*a2^13*a3^8-10*a1^3*a2^12*a3^9+12*a1^3*a2^11*a3^10+8*a1^3*a2^10*a3^11
+10*a1^3*a2^9*a3^12+5*a1^3*a2^8*a3^13-17*a1^3*a2^7*a3^14-14*a1^3*a2^6*a3^15-26*a1^3*a2^5*a3^16
-15*a1^3*a2^4*a3^17-7*a1^3*a2^3*a3^18-2*a1^2*a2^18*a3^4-5*a1^2*a2^17*a3^5-11*a1^2*a2^16*a3^6
-8*a1^2*a2^15*a3^7-12*a1^2*a2^14*a3^8-3*a1^2*a2^13*a3^9+2*a1^2*a2^12*a3^10+9*a1^2*a2^11*a3^11
+2*a1^2*a2^10*a3^12-3*a1^2*a2^9*a3^13-12*a1^2*a2^8*a3^14-14*a1^2*a2^7*a3^15-15*a1^2*a2^6*a3^16
-11*a1^2*a2^5*a3^17-6*a1^2*a2^4*a3^18-a1*a2^18*a3^5-4*a1*a2^17*a3^6-3*a1*a2^16*a3^7
-4*a1*a2^15*a3^8-3*a1*a2^14*a3^9+a1*a2^13*a3^10+a1*a2^12*a3^11+a1*a2^11*a3^12-3*a1*a2^10*a3^13
-7*a1*a2^9*a3^14-6*a1*a2^8*a3^15-7*a1*a2^7*a3^16-4*a1*a2^6*a3^17-3*a1*a2^5*a3^18-a2^18*a3^6
-a2^17*a3^7-2*a2^16*a3^8-2*a2^15*a3^9-3*a2^14*a3^10-2*a2^13*a3^11-3*a2^12*a3^12-2*a2^11*a3^13
-3*a2^10*a3^14-2*a2^9*a3^15-2*a2^8*a3^16-a2^7*a3^17-a2^6*a3^18


F=-a1^18*a2^6-3*a1^18*a2^5*a3-6*a1^18*a2^4*a3^2-7*a1^18*a2^3*a3^3-6*a1^18*a2^2*a3^4
-3*a1^18*a2*a3^5-a1^18*a3^6+a1^17*a2^7-2*a1^17*a2^6*a3-7*a1^17*a2^5*a3^2-15*a1^17*a2^4*a3^3
-15*a1^17*a2^3*a3^4-11*a1^17*a2^2*a3^5-4*a1^17*a2*a3^6-a1^17*a3^7+a1^16*a2^7*a3
-5*a1^16*a2^6*a3^2-18*a1^16*a2^5*a3^3-27*a1^16*a2^4*a3^4-26*a1^16*a2^3*a3^5-15*a1^16*a2^2*a3^6
-7*a1^16*a2*a3^7-2*a1^16*a3^8+2*a1^15*a2^9+2*a1^15*a2^8*a3+10*a1^15*a2^7*a3^2+2*a1^15*a2^6*a3^3
-8*a1^15*a2^5*a3^4-20*a1^15*a2^4*a3^5-14*a1^15*a2^3*a3^6-14*a1^15*a2^2*a3^7-6*a1^15*a2*a3^8
-2*a1^15*a3^9+a1^14*a2^10+5*a1^14*a2^9*a3+8*a1^14*a2^8*a3^2+19*a1^14*a2^7*a3^3
+15*a1^14*a2^6*a3^4-8*a1^14*a2^5*a3^5-a1^14*a2^4*a3^6-17*a1^14*a2^3*a3^7-12*a1^14*a2^2*a3^8
-7*a1^14*a2*a3^9-3*a1^14*a3^10+2*a1^13*a2^11+5*a1^13*a2^10*a3+17*a1^13*a2^9*a3^2
+33*a1^13*a2^8*a3^3+51*a1^13*a2^7*a3^4+36*a1^13*a2^6*a3^5+32*a1^13*a2^5*a3^6+19*a1^13*a2^4*a3^7
+5*a1^13*a2^3*a3^8-3*a1^13*a2^2*a3^9-3*a1^13*a2*a3^10-2*a1^13*a3^11+a1^12*a2^12
+5*a1^12*a2^11*a3+2*a1^12*a2^10*a3^2+34*a1^12*a2^9*a3^3+48*a1^12*a2^8*a3^4+45*a1^12*a2^7*a3^5
+54*a1^12*a2^6*a3^6+25*a1^12*a2^5*a3^7+40*a1^12*a2^4*a3^8+10*a1^12*a2^3*a3^9+2*a1^12*a2^2*a3^10
+a1^12*a2*a3^11-3*a1^12*a3^12+2*a1^11*a2^13+5*a1^11*a2^12*a3-3*a1^11*a2^11*a3^2
+4*a1^11*a2^10*a3^3+26*a1^11*a2^9*a3^4+27*a1^11*a2^8*a3^5+27*a1^11*a2^7*a3^6+27*a1^11*a2^6*a3^7
+27*a1^11*a2^5*a3^8+34*a1^11*a2^4*a3^9+8*a1^11*a2^3*a3^10+9*a1^11*a2^2*a3^11+a1^11*a2*a3^12
-2*a1^11*a3^13+a1^10*a2^14+5*a1^10*a2^13*a3-10*a1^10*a2^12*a3^2-8*a1^10*a2^11*a3^3
-24*a1^10*a2^10*a3^4+12*a1^10*a2^9*a3^5+15*a1^10*a2^8*a3^6+12*a1^10*a2^7*a3^7+47*a1^10*a2^6*a3^8
+28*a1^10*a2^5*a3^9+24*a1^10*a2^4*a3^10+12*a1^10*a2^3*a3^11+2*a1^10*a2^2*a3^12+a1^10*a2*a3^13
-3*a1^10*a3^14+2*a1^9*a2^15+5*a1^9*a2^14*a3-7*a1^9*a2^13*a3^2-26*a1^9*a2^12*a3^3
-58*a1^9*a2^11*a3^4-60*a1^9*a2^10*a3^5-38*a1^9*a2^9*a3^6-28*a1^9*a2^8*a3^7-4*a1^9*a2^7*a3^8
+26*a1^9*a2^6*a3^9-4*a1^9*a2^5*a3^10+6*a1^9*a2^4*a3^11-10*a1^9*a2^3*a3^12-3*a1^9*a2^2*a3^13
-3*a1^9*a2*a3^14-2*a1^9*a3^15+2*a1^8*a2^16+8*a1^8*a2^15*a3+4*a1^8*a2^14*a3^2-13*a1^8*a2^13*a3^3-64*a1^8*a2^12*a3^4-67*a1^8*a2^11*a3^5-77*a1^8*a2^10*a3^6-38*a1^8*a2^9*a3^7-8*a1^8*a2^8*a3^8
+22*a1^8*a2^7*a3^9+27*a1^8*a2^6*a3^10+5*a1^8*a2^5*a3^11-16*a1^8*a2^4*a3^12-13*a1^8*a2^3*a3^13
-12*a1^8*a2^2*a3^14-4*a1^8*a2*a3^15-2*a1^8*a3^16+a1^7*a2^17+7*a1^7*a2^16*a3+16*a1^7*a2^15*a3^2
+13*a1^7*a2^14*a3^3-27*a1^7*a2^13*a3^4-57*a1^7*a2^12*a3^5-75*a1^7*a2^11*a3^6-78*a1^7*a2^10*a3^7
-46*a1^7*a2^9*a3^8-10*a1^7*a2^8*a3^9+18*a1^7*a2^7*a3^10+29*a1^7*a2^6*a3^11-a1^7*a2^5*a3^12
-7*a1^7*a2^4*a3^13-11*a1^7*a2^3*a3^14-8*a1^7*a2^2*a3^15-3*a1^7*a2*a3^16-a1^7*a3^17+a1^6*a2^18
+4*a1^6*a2^17*a3+15*a1^6*a2^16*a3^2+22*a1^6*a2^15*a3^3+5*a1^6*a2^14*a3^4-28*a1^6*a2^13*a3^5
-56*a1^6*a2^12*a3^6-61*a1^6*a2^11*a3^7-51*a1^6*a2^10*a3^8-14*a1^6*a2^9*a3^9+21*a1^6*a2^8*a3^10
+43*a1^6*a2^7*a3^11+20*a1^6*a2^6*a3^12+4*a1^6*a2^5*a3^13-15*a1^6*a2^4*a3^14-14*a1^6*a2^3*a3^15
-11*a1^6*a2^2*a3^16-4*a1^6*a2*a3^17-a1^6*a3^18+3*a1^5*a2^18*a3+11*a1^5*a2^17*a3^2
+26*a1^5*a2^16*a3^3+28*a1^5*a2^15*a3^4+12*a1^5*a2^14*a3^5-20*a1^5*a2^13*a3^6-39*a1^5*a2^12*a3^7
-37*a1^5*a2^11*a3^8-24*a1^5*a2^10*a3^9+16*a1^5*a2^9*a3^10+35*a1^5*a2^8*a3^11+37*a1^5*a2^7*a3^12
+16*a1^5*a2^6*a3^13+4*a1^5*a2^5*a3^14-8*a1^5*a2^4*a3^15-6*a1^5*a2^3*a3^16-5*a1^5*a2^2*a3^17
-a1^5*a2*a3^18+6*a1^4*a2^18*a3^2+15*a1^4*a2^17*a3^3+27*a1^4*a2^16*a3^4+16*a1^4*a2^15*a3^5
-a1^4*a2^14*a3^6-29*a1^4*a2^13*a3^7-20*a1^4*a2^12*a3^8-14*a1^4*a2^11*a3^9+24*a1^4*a2^10*a3^10
+42*a1^4*a2^9*a3^11+36*a1^4*a2^8*a3^12+23*a1^4*a2^7*a3^13-5*a1^4*a2^6*a3^14-8*a1^4*a2^5*a3^15
-9*a1^4*a2^4*a3^16-5*a1^4*a2^3*a3^17-2*a1^4*a2^2*a3^18+7*a1^3*a2^18*a3^3+15*a1^3*a2^17*a3^4
+18*a1^3*a2^16*a3^5+6*a1^3*a2^15*a3^6-9*a1^3*a2^14*a3^7-11*a1^3*a2^13*a3^8-6*a1^3*a2^12*a3^9
+16*a1^3*a2^11*a3^10+32*a1^3*a2^10*a3^11+34*a1^3*a2^9*a3^12+17*a1^3*a2^8*a3^13+3*a1^3*a2^7*a3^14
-14*a1^3*a2^6*a3^15-6*a1^3*a2^5*a3^16-5*a1^3*a2^4*a3^17-a1^3*a2^3*a3^18+6*a1^2*a2^18*a3^4
+7*a1^2*a2^17*a3^5+5*a1^2*a2^16*a3^6-8*a1^2*a2^15*a3^7-4*a1^2*a2^14*a3^8-9*a1^2*a2^13*a3^9
+10*a1^2*a2^12*a3^10+15*a1^2*a2^11*a3^11+22*a1^2*a2^10*a3^12+15*a1^2*a2^9*a3^13
-8*a1^2*a2^7*a3^15-11*a1^2*a2^6*a3^16-5*a1^2*a2^5*a3^17-2*a1^2*a2^4*a3^18+3*a1*a2^18*a3^5
+2*a1*a2^17*a3^6-a1*a2^16*a3^7-a1*a2^14*a3^9+3*a1*a2^13*a3^10+7*a1*a2^12*a3^11+7*a1*a2^11*a3^12
+7*a1*a2^10*a3^13+3*a1*a2^9*a3^14-4*a1*a2^8*a3^15-3*a1*a2^7*a3^16-4*a1*a2^6*a3^17-a1*a2^5*a3^18
+a2^18*a3^6-a2^17*a3^7-2*a2^15*a3^9-a2^14*a3^10-2*a2^13*a3^11-a2^12*a3^12-2*a2^11*a3^13
-a2^10*a3^14-2*a2^9*a3^15-2*a2^8*a3^16-a2^7*a3^17-a2^6*a3^18


     

証明

 a1,a2,a3,c1,c2 は整数とする。

(a1*x+c1)^5+(a2*x+c2)^5+(a3*x+1)^5=(a1*x+1)^5+(a2*x+c1)^5+(a3*x+c2)^5
として、上記式を展開すると

(-5*a1^4+5*a2^4*c2+5*a3^4+5*a1^4*c1-5*a3^4*c2-5*a2^4*c1)*x^4
+(-10*a1^3-10*a2^3*c1^2+10*a1^3*c1^2-10*a3^3*c2^2+10*a2^3*c2^2+10*a3^3)*x^3
+(10*a1^2*c1^3-10*a1^2-10*a3^2*c2^3-10*a2^2*c1^3+10*a3^2+10*a2^2*c2^3)*x^2
+(5*a1*c1^4-5*a3*c2^4+5*a2*c2^4+5*a3-5*a1-5*a2*c1^4)*x
となる。

x^4,x^3の係数を0にするようにc1,c2を求める。

c1=(a1^6*a3^2+2*a1^2*a2^5*a3+3*a1^3*a2^2*a3^3-a1^2*a3^6+2*a1*a2^5*a3^2+a1^2*a2^6+a2^3*a3^5
+a1^5*a3^3+2*a1^4*a2^3*a3+2*a1^4*a2*a3^3+a1^6*a2*a3+2*a1^5*a2^2*a3+2*a1^5*a2*a3^2+2*a1*a2^4*a3^3
+2*a1*a2^3*a3^4-a1*a2*a3^6+a1*a3*a2^6+2*a1^3*a2^4*a3+3*a1^3*a2^3*a3^2+2*a1^3*a2*a3^4
+3*a1^2*a2^4*a3^2+3*a1^2*a2^3*a3^3+a1^2*a2^2*a3^4+3*a1^4*a2^2*a3^2+a1^4*a3^4+a1^4*a2^4
+a1^3*a3^5+a2^5*a3^3+a1^3*a2^5-a2^2*a3^6+a1^5*a2^3+a2^4*a3^4+a1^6*a2^2+a3^2*a2^6)
/(a1^6*a3^2+3*a1^3*a2^2*a3^3+a1^2*a3^6-a1^2*a2^6+a2^3*a3^5+a1^5*a3^3+2*a1^4*a2^3*a3
+2*a1^4*a2*a3^3+a1^6*a2*a3+2*a1^5*a2^2*a3+2*a1^5*a2*a3^2+2*a1*a2^4*a3^3+2*a1*a2^3*a3^4
+2*a1*a2^2*a3^5+a1*a2*a3^6-a1*a3*a2^6+2*a1^3*a2^4*a3+3*a1^3*a2^3*a3^2+2*a1^3*a2*a3^4
+a1^2*a2^4*a3^2+3*a1^2*a2^3*a3^3+3*a1^2*a2^2*a3^4+2*a1^2*a2*a3^5+3*a1^4*a2^2*a3^2+a1^4*a3^4
+a1^4*a2^4+a1^3*a3^5+a2^5*a3^3+a1^3*a2^5+a2^2*a3^6+a1^5*a2^3+a2^4*a3^4+a1^6*a2^2-a3^2*a2^6)

     
c2=-(a1^6*a3^2-2*a1^2*a2^5*a3-3*a1^3*a2^2*a3^3-a1^2*a3^6-2*a1*a2^5*a3^2-a1^2*a2^6-a2^3*a3^5
-a1^5*a3^3-2*a1^4*a2^3*a3-2*a1^4*a2*a3^3+a1^6*a2*a3-2*a1*a2^4*a3^3-2*a1*a2^3*a3^4-2*a1*a2^2*a3^5
-a1*a2*a3^6-a1*a3*a2^6-2*a1^3*a2^4*a3-3*a1^3*a2^3*a3^2-2*a1^3*a2*a3^4-3*a1^2*a2^4*a3^2
-3*a1^2*a2^3*a3^3-3*a1^2*a2^2*a3^4-2*a1^2*a2*a3^5-a1^4*a2^2*a3^2-a1^4*a3^4-a1^4*a2^4
-a1^3*a3^5-a2^5*a3^3-a1^3*a2^5-a2^2*a3^6-a1^5*a2^3-a2^4*a3^4+a1^6*a2^2-a3^2*a2^6)
/(a1^6*a3^2+3*a1^3*a2^2*a3^3+a1^2*a3^6-a1^2*a2^6+a2^3*a3^5+a1^5*a3^3+2*a1^4*a2^3*a3
+2*a1^4*a2*a3^3+a1^6*a2*a3+2*a1^5*a2^2*a3+2*a1^5*a2*a3^2+2*a1*a2^4*a3^3+2*a1*a2^3*a3^4
+2*a1*a2^2*a3^5+a1*a2*a3^6-a1*a3*a2^6+2*a1^3*a2^4*a3+3*a1^3*a2^3*a3^2+2*a1^3*a2*a3^4
+a1^2*a2^4*a3^2+3*a1^2*a2^3*a3^3+3*a1^2*a2^2*a3^4+2*a1^2*a2*a3^5+3*a1^4*a2^2*a3^2+a1^4*a3^4
+a1^4*a2^4+a1^3*a3^5+a2^5*a3^3+a1^3*a2^5+a2^2*a3^6+a1^5*a2^3+a2^4*a3^4+a1^6*a2^2-a3^2*a2^6)
  
また、x^2,x の係数より
x=-1/2*(a2*c2^4-a3*c2^4+a1*c1^4-a2*c1^4-a1+a3)
/(-a1^2+a2^2*c2^3-a3^2*c2^3-a2^2*c1^3+a3^2+a1^2*c1^3)
           
  
xにc1,c2を代入して整理すると24次のパラメータ解が得られる。


     
3.数値例

G=gcd(A,B,C,D,E,F) > 1 の場合は、Gで割って既約な値にしています。

36231625^5 + 23603111^5 + 6955753^5 = 13636793^5 + 36873121^5 + 16280575^5

2155548251^5 + 1480455289^5 + 370704359^5 = 542132891^5 + 2188706129^5 + 1275868879^5

16203947943^5 + 10362375972^5 + 7424530508^5 = 9547600853^5 + 16279971813^5 + 8163281757^5

18149116727^5 + 12777715345^5 + 7235131721^5 = 6035024351^5 + 17544075745^5 + 14582863697^5



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