1.Introduction
I show that there are many parameter solutions of 6.6.6.
2. Theorem
There are many parameter solutions of A16+ A26+ A36+ A46+ A56+ A66 = B16+ B26+ B36+ B46+ B56+ B66.
A16+ A26+ A36+ A46+ A56+ A66 = B16+ B26+ B36+ B46+ B56+ B66.................(1)
Proof.
A1=a1a+b1b-c1
A2=a3a+c3
A3=a4a+c4
A4=a1a-b2b+c2
A5=a2a-b1b+c2
A6=a2a+b2b+c2
B1=a1a+b1b+c1
B2=a3a-c3
B3=a4a-c4
B4=a1a-b2b-c2
B5=a2a-b1b-c2
B6=a2a+b2b-c2
A16+ A26+ A36+ A46+ A56+ A66 = B16+ B26+ B36+ B46+ B56+ B66.............................(2)
a2=ma1,a3=na1,a4=pa1,c1=qc2,c3=rc2,c4=tc2
As for this equation (2), the factorization is done as follows.
|(m,n,p,q,r,t)|<6
(m,n,p,q,r,t)
(3 1 -1 -2 -6 3) 240a1ac2(-19c22+b22b2+2a1b2ba+4a2a12)(21c22+b22b2+2a1b2ba+6a2a12)
(3 1 -1 -1 -5 3) 240a1ac2(14c22+6a12a2+2b2ba1a+b22b2)(-12c22+4a12a2+2b2ba1a+b22b2)
(3 1 -1 2 -2 3) 240a1ac2(5c22+b22b2+2a1ab2b+6a12a2)(-3c22+b22b2+2a1ab2b+4a12a2)
(3 1 -1 3 -3 1) 240a1ac2(6c22+b22b2+2a1ab2b+6a2a12)(-4c22+b22b2+2a1ab2b+4a2a12)
(3 2 -4 3 2 2) -240a1ac2(5c22+15a2a12+2ba1ab2+b22b2)(3c22+5a2a12-2ba1ab2-b22b2)
(3 3 -5 3 2 2) -240a1ac2(3c22-b22b2-2a1ab2b+12a12a2)(5c22+b22b2+2a1ab2b+22a12a2)
(3 4 -6 3 2 2) -240a1ac2(3c22+21a12a2-2b2a1ba-b22b2)(5c22+31a12a2+2b2a1ba+b22b2)
Take c2=1
Case 1. 4a2a12+2a1b2ba+b22b2=19.....................................(3)
We can find infinitely many rational solutions of (3),then we obtain infinitely parameter solutions of (2).
Take x=2a1a,y=b2b
Then (3) becomes to x2+xy+y2=19........................................(4)
(x,y)=(3,2) is a solution of (4).
We obtain parameter solution of (3) by using (a,b)=(3/(2a1),2/b2).
A1=7k2b22-4a12
A2=-9k2b22-32kb2a1-68a12
A3=3k2b22+20kb2a1+44a12
A4=15k2b22+20kb2a1-28a12
A5=11k2b22-20kb2a1-52a12
A6= k2b22-44kb2a1-36a12
B1=-k2b22-16kb2a1-36a12
B2=15k2b22+16kb2a1+28a12
B3=-9k2b22-4kb2a1-4a12
B4=11k2b22+12kb2a1-44a12
B5=7k2b22-28kb2a1-68a12
B6=-3k2b22-52kb2a1-52a12
Case 2. 4a2a12+2a1b2ba+b22b2=12
A1=4k2b22-4kb2a1-8a12
A2=-8k2b22-28kb2a1-56a12
A3=4k2b22+20kb2a1+40a12
A4=12k2b22+12kb2a1-24a12
A5=8k2b22-20kb2a1-40a12
A6=-12a1(3kb2+2a1)
B1=-12(kb2+2a1)a1
B2=12k2b22+12kb2a1+24a12
B3=-8k2b22-4kb2a1-8a12
B4=8k2b22+4kb2a1-40a12
B5=4k2b22-28kb2a1-56a12
B6=-4k2b22-44kb2a1-40a12
Case 3. 5a2a12-2ba1ab2-b22b2=-3
A1=-4k2b22-20kb2a1+32a12
A2=8k2b22-8kb2a1-16a12
A3=-4k2b22+40kb2a1-28a12
A4=12k2b22-24kb2a1+12a12
A5=8k2b22-20kb2a1-4a12
A6= -24a12
B1=8k2b22+4kb2a1-28a12
B2=-24(kb2-a1)a1
B3=-12k2b22+24kb2a1+12a12
B4=8k2b22-32kb2a1+32a12
B5=4k2b22-28kb2a1+16a12
B6=-4k2b22-8kb2a1-4a12
Case 4. 12a2a12-2ba1ab2-b22b2=-3
A1=-5k2b22-20kb2a1+76a12
A2=7k2b22-16kb2a1-36a12
A3=-k2b22+48kb2a1-68a12
A4=9k2b22-28kb2a1+28a12
A5= 5k2b22-20kb2a1-12a12
A6=-k2b22+4kb2a1-60a12
B1=7k2b22+4kb2a1-68a12
B2=-k2b22-32kb2a1+60a12
B3=-9k2b22+32kb2a1+28a12
B4=5k2b22-36kb2a1+76a12
B5= k2b22-28kb2a1+36a12
B6=-5k2b22-4kb2a1-12a12
Case 5. 21a2a12-2ba1ab2-b22b2=-3
A1=8k2b22+30kb2a1-198a12
A2=-10k2b22+36kb2a1+90a12
A3=12(-7kb2+15a1)a1
A4=-12k2b22+48kb2a1-72a12
A5=-6k2b22+30kb2a1+36a12
A6=2k2b22-12kb2a1+162a12
B1=-10k2b22-6kb2a1+180a12
B2=2k2b22+60kb2a1-162a12
B3=12k2b22-60kb2a1-72a12
B4=-6k2b22+60kb2a1-198a12
B5=-6(-7kb2+15a1)a1
B6= 8k2b22+36a12
3. Example
Case 1: (a1,b2)=(1,1)
k
1 36+ 1096+ 676+ 76+ 616+ 796 = 536+ 596+ 176+ 216+ 896+ 1076
2 16+ 76+ 46+ 36+ 26+ 56 = 36+ 56+ 26+ 16+ 46+ 76
3 596+ 2456+ 1316+ 1676+ 136+ 1596 = 936+ 2116+ 976+ 916+ 896+ 2356
4 276+ 856+ 436+ 736+ 116+ 496 = 296+ 836+ 416+ 456+ 176+ 776
5 576+ 1516+ 736+ 1496+ 416+ 776 = 476+ 1616+ 836+ 976+ 116+ 1296
6 316+ 736+ 346+ 796+ 286+ 336 = 216+ 836+ 446+ 536+ 26+ 596
7 3396+ 7336+ 3316+ 8476+ 3476+ 2956 = 1976+ 8756+ 4736+ 5796+ 796+ 5636
8 376+ 756+ 336+ 916+ 416+ 276 = 196+ 936+ 516+ 636+ 136+ 556
9 5636+10856+ 4676+13676+ 6596+ 3516 = 2616+13876+ 7696+ 9556+ 2476+ 7636
10 876+ 1616+ 686+ 2096+ 1066+ 476 = 376+ 2116+ 1186+ 1476+ 446+ 1096
Case 2: (a1,b2)=(1,1)
k
1 26+ 236+ 166+ 06+ 136+ 156 = 96+ 126+ 56+ 76+ 206+ 226
2 06+ 36+ 26+ 16+ 16+ 26 = 16+ 26+ 16+ 06+ 26+ 36
3 46+ 536+ 346+ 306+ 76+ 336 = 156+ 426+ 236+ 116+ 266+ 526
4 56+ 376+ 236+ 276+ 16+ 216 = 96+ 336+ 196+ 136+ 136+ 356
5 66+ 336+ 206+ 286+ 56+ 176 = 76+ 326+ 196+ 156+ 86+ 306
6 76+ 326+ 196+ 306+ 86+ 156 = 66+ 336+ 206+ 176+ 56+ 286
7 406+ 1616+ 946+ 1626+ 536+ 696 = 276+ 1746+ 1076+ 956+ 146+ 1366
8 96+ 336+ 196+ 356+ 136+ 136 = 56+ 376+ 236+ 216+ 16+ 276
9 706+ 2396+ 1366+ 2646+ 1076+ 876 = 336+ 2766+ 1736+ 1616+ 46+ 1906
10 226+ 716+ 406+ 816+ 356+ 246 = 96+ 846+ 536+ 506+ 46+ 556
Case 3: (a1,b2)=(1,1)
k
1 16+ 26+ 16+ 06+ 26+ 36 = 26+ 06+ 36+ 16+ 16+ 26
2 26+ 06+ 36+ 16+ 16+ 26 = 16+ 26+ 16+ 06+ 26+ 36
3 86+ 46+ 76+ 66+ 16+ 36 = 76+ 66+ 36+ 16+ 46+ 86
4 286+ 206+ 176+ 276+ 116+ 66 = 296+ 186+ 216+ 86+ 86+ 256
5 76+ 66+ 36+ 86+ 46+ 16 = 86+ 46+ 76+ 36+ 16+ 66
6 586+ 566+ 176+ 756+ 416+ 66 = 716+ 306+ 696+ 326+ 26+ 496
7 386+ 406+ 76+ 546+ 316+ 36 = 496+ 186+ 516+ 256+ 26+ 326
8 326+ 366+ 36+ 496+ 296+ 26 = 436+ 146+ 476+ 246+ 46+ 276
9 596+ 706+ 16+ 966+ 586+ 36 = 826+ 246+ 936+ 496+ 116+ 506
10 1426+ 1766+ 76+ 2436+ 1496+ 66 = 2036+ 546+ 2376+ 1286+ 346+ 1216
Case 4: (a1,b2)=(1,1)
k
1 176+ 156+ 76+ 36+ 96+ 196 = 196+ 96+ 176+ 156+ 36+ 76
2 26+ 56+ 36+ 16+ 46+ 76 = 46+ 16+ 76+ 36+ 26+ 56
3 296+ 216+ 676+ 256+ 276+ 576 = 76+ 456+ 436+ 136+ 396+ 696
4 76+ 16+ 96+ 56+ 16+ 56 = 56+ 76+ 16+ 16+ 56+ 96
5 1496+ 596+ 1476+ 1136+ 136+ 656 = 1276+ 1256+ 376+ 216+ 796+ 1576
6 286+ 156+ 236+ 236+ 66+ 96 = 266+ 216+ 136+ 56+ 126+ 276
7 1036+ 656+ 736+ 916+ 316+ 276 = 1016+ 716+ 636+ 236+ 376+ 956
8 1016+ 716+ 636+ 956+ 376+ 236 = 1036+ 656+ 736+ 276+ 316+ 916
9 5096+ 3876+ 2836+ 5056+ 2136+ 1056 = 5356+ 3096+ 4136+ 1576+ 1356+ 4536
10 266+ 216+ 136+ 276+ 126+ 56 = 286+ 156+ 236+ 96+ 66+ 236
Case 5: (a1,b2)=(1,1)
k
1 406+ 296+ 246+ 96+ 156+ 386 = 416+ 256+ 306+ 366+ 126+ 116
2 536+ 616+ 66+ 126+ 366+ 736 = 646+ 176+ 726+ 516+ 36+ 346
3 16+ 36+ 26+ 16+ 26+ 46 = 26+ 16+ 46+ 26+ 16+ 36
4 256+ 376+ 786+ 366+ 306+ 736 = 26+ 556+ 606+ 276+ 396+ 826
5 386+ 56+ 606+ 336+ 96+ 386 = 256+ 476+ 186+ 126+ 306+ 596
6 56+ 16+ 66+ 46+ 06+ 36 = 46+ 56+ 06+ 16+ 36+ 66
7 1016+ 376+ 1026+ 816+ 126+ 446 = 886+ 896+ 246+ 186+ 516+ 1076
8 2776+ 1316+ 2466+ 2286+ 546+ 976 = 2546+ 2236+ 1086+ 516+ 1236+ 2746
9 206+ 116+ 166+ 176+ 56+ 66 = 196+ 156+ 106+ 46+ 86+ 196
10 416+ 256+ 306+ 366+ 126+ 116 = 406+ 296+ 246+ 96+ 156+ 386
HOME