a^2-ab+b^2-c=0的整数解

yangchuanju · 发表于 2024-10-31 08:43

本帖最后由 yangchuanju 于 2024-10-31 11:16 编辑

不定方程a^2-ab+b^2-c=0，c=2^k-1的整数解
k c 整数解数整数解(a,b)
1 1 6 (-1,-1),(-1,0),(0,-1),(0,1),(1,0),(1,1)
2 3 6 (-2,-1),(-1,-2),(-1,1),(1,-1),(1,2),(2,1)
3 7 12 (-3,-2),(-3,-1),(-2,-3),(-2,1),(-1,-3),(-1,2),(1,-2),(1,3),(2,-1),(2,3),(3,1),(3,2)
4 15 0
5 31 12 (-6,-5),(-6,-1),(-5,-6),(-5,1),(-1,-6),(-1,5),(1,-5),(1,6),(5,-1),(5,6),(6,1),(6,5)
6 63 12 (-9,-6),(-9,-3),(-6,-9),(-6,3),(-3,-9),(-3,6),(3,-6),(3,9),(6,-3),(6,9),(9,3),(9,6)
7 127 12 (-13,-7),(-13,-6),(-7,-13),(-7,6),(-6,-13),(-6,7),(6,-7),(6,13),(7,-6),(7,13),(13,6),(13,7)
8 255 0
9 511 24 (-26,-15),(-26,-11),(-25,-19),(-25,-6),(-19,-25),(-19,6),(-15,-26),(-15,11),(-11,-26),(-11,15),(-6,-25),(-6,19),
(6,-19),(6,25),(11,-15),(11,26),(15,-11),(15,26),(19,-6),(19,25),(25,6),(25,19),(26,11),(26,15)
10 1023 0?
11 2047 0
13 8191 12 (-91,-90),(-91,-1),(-90,-91),(-90,1),(-1,-91),(-1,90),(1,-90),(1,91),(90,-1),(90,91),(91,1),(91,90)
15 32767 48 (-209,-107),(-209,-102),(-198,-157),(-198,-41),(-197,-159),(-197,-38),(-193,-166),(-193,-27),
(-166,-193),(-166,27),(-159,-197),(-159,38),(-157,-198),(-157,41),(-107,-209),(-107,102),
(-102,-209),(-102,107),(-41,-198),(-41,157),(-38,-197),(-38,159),(-27,-193),(-27,166),
(27,-166),(27,193),(38,-159),(38,197),(41,-157),(41,198),(102,-107),(102,209),
(107,-102),(107,209),(157,-41),(157,198),(159,-38),(159,197),(166,-27),(166,193),
(193,27),(193,166),(197,38),(197,159),(198,41),(198,157),(209,102),(209,107)
17 131071 12 (-365,-359),(-365,-6),(-359,-365),(-359,6),(-6,-365),(-6,359),(6,-359),(6,365),
(359,-6),(359,365),(365,6),(365,359)
19 524287 12 (-762,-679),(-762,-83),(-679,-762),(-679,83),(-83,-762),(-83,679),(83,-679),(83,762),
(679,-83),(679,762),(762,83),(762,679)
21 2097151 72 (-1671,-890),(-1671,-781),(-1670,-909),(-1670,-761),(-1626,-1151),(-1626,-475),(-1585,-1254),(-1585,-331),
(-1575,-1274),(-1575,-301),(-1519,-1365),(-1519,-154),(-1365,-1519),(-1365,154),(-1274,-1575),(-1274,301),
(-1254,-1585),(-1254,331),(-1151,-1626),(-1151,475),(-909,-1670),(-909,761),(-890,-1671),(-890,781),
(-781,-1671),(-781,890),(-761,-1670),(-761,909),(-475,-1626),(-475,1151),(-331,-1585),(-331,1254),
(-301,-1575),(-301,1274),(-154,-1519),(-154,1365),(154,-1365),(154,1519),(301,-1274),(301,1575),
(331,-1254),(331,1585),(475,-1151),(475,1626),(761,-909),(761,1670),(781,-890),(781,1671),
(890,-781),(890,1671),(909,-761),(909,1670),(1151,-475),(1151,1626),(1254,-331),(1254,1585),
(1274,-301),(1274,1575),(1365,-154),(1365,1519),(1519,154),(1519,1365),(1575,301),(1575,1274),
(1585,331),(1585,1254),(1626,475),(1626,1151),(1670,761),(1670,909),(1671,781),(1671,890)
23 8388607 0
25 33554431 48 (-6649,-3955),(-6649,-2694),(-6630,-4081),(-6630,-2549),(-6581,-4326),(-6581,-2255),(-6551,-4445),(-6551,-2106),
(-4445,-6551),(-4445,2106),(-4326,-6581),(-4326,2255),(-4081,-6630),(-4081,2549),(-3955,-6649),(-3955,2694),
(-2694,-6649),(-2694,3955),(-2549,-6630),(-2549,4081),(-2255,-6581),(-2255,4326),(-2106,-6551),(-2106,4445),
(2106,-4445),(2106,6551),(2255,-4326),(2255,6581),(2549,-4081),(2549,6630),(2694,-3955),(2694,6649),
(3955,-2694),(3955,6649),(4081,-2549),(4081,6630),(4326,-2255),(4326,6581),(4445,-2106),(4445,6551),
(6551,2106),(6551,4445),(6581,2255),(6581,4326),(6630,2549),(6630,4081),(6649,2694),(6649,3955)
27 134217727 48 (-13327,-7669),(-13327,-5658),(-13323,-7706),(-13323,-5617),(-12819,-9722),(-12819,-3097),(-12806,-9753),(-12806,-3053),
(-9753,-12806),(-9753,3053),(-9722,-12819),(-9722,3097),(-7706,-13323),(-7706,5617),(-7669,-13327),(-7669,5658),
(-5658,-13327),(-5658,7669),(-5617,-13323),(-5617,7706),(-3097,-12819),(-3097,9722),(-3053,-12806),(-3053,9753),
(3053,-9753),(3053,12806),(3097,-9722),(3097,12819),(5617,-7706),(5617,13323),(5658,-7669),(5658,13327),
(7669,-5658),(7669,13327),(7706,-5617),(7706,13323),(9722,-3097),(9722,12819),(9753,-3053),(9753,12806),
(12806,3053),(12806,9753),(12819,3097),(12819,9722),(13323,5617),(13323,7706),(13327,5658),(13327,7669)
29 536870911 0
31 2147483647 12 (-48511,-43813),(-48511,-4698),(-43813,-48511),(-43813,4698),(-4698,-48511),(-4698,43813),
(4698,-43813),(4698,48511),(43813,-4698),(43813,48511),(48511,4698),(48511,43813)
33 8589934591 0
35 34359738367 0
37 1.37439E+11 24 (-421799,-274170),(-421799,-147629),(-374039,-367325),(-374039,-6714),(-367325,-374039),(-367325,6714),
(-274170,-421799),(-274170,147629),(-147629,-421799),(-147629,274170),(-6714,-374039),(-6714,367325),
(6714,-367325),(6714,374039),(147629,-274170),(147629,421799),(274170,-147629),(274170,421799),
(367325,-6714),(367325,374039),(374039,6714),(374039,367325),(421799,147629),(421799,274170)
39 5.49756E+11 96 略
41 2.19902E+12 0?
43 8.79609E+12 0?
45 3.51844E+13 384 略
47 1.40737E+14 0?
49 5.6295E+14 24 (-27039174,-17342185),(-27039174,-9696989),(-26541431,-19154106),(-26541431,-7387325),(-19154106,-26541431),(-19154106,7387325),
(-17342185,-27039174),(-17342185,9696989),(-9696989,-27039174),(-9696989,17342185),(-7387325,-26541431),(-7387325,19154106),
(7387325,-19154106),(7387325,26541431),(9696989,-17342185),(9696989,27039174),(17342185,-9696989),(17342185,27039174),
(19154106,-7387325),(19154106,26541431),(26541431,7387325),(26541431,19154106),(27039174,9696989),(27039174,17342185)

yangchuanju · 发表于 2024-10-31 11:11

不定方程(r^2-2)/(2^k-1)=c总有两个整数解，
式中r——非0整数，k——奇数
r^2-2=c*(2^k-1)，令c=2则r^2-2=2*2^k-2，r^2=2^(k+1)，
r=±2^(k+1)/2，c=2。

yangchuanju · 发表于 2024-10-31 11:50

例当r等于227时，有2个整数解——
计算网站WolframAlpha
（网址：https://www.wolframalpha.com/inp ... %5E2-2%5E89%2B1%3D0）
给出
a = ± 20769187434139310514121985316880384, b = 2
整数解中的a即r，b即c。
然2^227-1是一个二合梅森数，等于
2^227-1<69> = 26986333437777017<17> · 7992177738...31<52>
它的两个素因子都有2个整数解
p1=26986333437777017
a11=9098601234699638
b11=3067646985795826
a12=17887732203077379
b12=11856777954173567
p2=7992177738205979626491506950867720953545660121688631
a21=20769187434139310514121985316880384
b21=53972666875554034
a22=7992177738205979605722319516728410439423674804808247
b22=7992177738205979584953132082589099979274356363481897
2^114=20769187434139310514121985316880383

按照剩余定理在求a时应有4组整数解——
p1*d1+y1=p2*f1+z1
p1*d2+y2=p2*f2+z2
p1*d3+y1=p2*f3+z2
p1*d4+y2=p2*f4+z1
式中p1，p2两个素因子；y1，y2，z1，z2四个单个素因子的整数解(y1=a11,y2=a12,z1=a21,z2=a22)；
d1，d2，d3，d4；f1，f2，f3，f4待求数；
4等式的左端或右端模（除以）二合梅森数的余数就是要求二合梅森数的整数解。

解4个剩余定理方程，得4组整数解，汇总如下——
a1= 20769187434139310514121985316880384
a2= 215679573337205118357336120696157024619909721241014065726843565113343
a3= 162186295080328207630990647431934508905618293183239919333967314328232
a4= 53493278256876910726345473264222536483478862197084660514861567665495
b1= 2
b2= 215679573337205118357336120696157003850722287101703551604858248232961
b3= 121960526464680918133721770108942695726441576048276334493486130753986
b4= 13267509641229621229076595941230723304302145062121075674380384091249
4整数解各不相同，不是重解；计算软件仅给出a=±20769187434139310514121985316880384，b=2一对，且含一个负整数解，
应该是计算软件没有把全部整数解计算出来的原因，不定方程(a^2-2)/(2^227-1)=b还是有4组正整数解的。

yangchuanju · 发表于 2024-10-31 11:56

同样，3素因子2^193-1应有8组整数解，计算网站仅能给出1组最小整数解——
a——
12554203470773361527671578846256876507176182213740981125119
8970180594342698998938557401015428467103217008067318559963
9819109390112017323496942173364426034706100331656462421843
6235086513681354794763920728122977994633135125982799856687
6319116957092006732907658118292354837571575762945269169104
2735094080661344204174636673050906797498610557271606603948
3584022876430662528733021445399904365101493880860750465828
158456325028528675187087900672

b——
12554203470773361527671578846098420182147653538553893224449
6409338520157472071505936166448673823480137544522410708137
7679890599148195969129667701582140783291824394063910120217
3096676258561249929322661943472013461932894914164667088737
3180706701971901867466399333641390304871335551127136401154
595875289697522849807362201268621546084334619679054302322
1023180802245435601300400210833149721478414417315842614002
2

计算软件仅给出一组最小整数解——a = ± 158456325028528675187087900672, b = 2

yangchuanju · 发表于 2024-10-31 14:45

不定方程a^2+ab+b^2=p和a^2-ab+b^2=p整数解对照——
不定方程a^2+ab+b^2=p有整数解的素数p 不定方程a^2-ab+b^2=p有整数解的素数p
p 整解数正整解数整解数正整解数整数解
3 5 1 6 2 (-2,-1),(-1,-2),(-1,1),(1,-1),(1,2),(2,1)
5 0 0 0 0
7 12 2 12 4 (-3,-2),(-3,-1),(-2,-3),(-2,1),(-1,-3),(-1,2),(1,-2),(1,3),(2,-1),(2,3),(3,1),(3,2)
11 0 0 0 0
13 12 2 12 4 (-4,-3),(-4,-1),(-3,-4),(-3,1),(-1,-4),(-1,3),(1,-3),(1,4),(3,-1),(3,4),(4,1),(4,3)
17 0 0 0 0
19 12 2 12 4 (-5,-3),(-5,-2),(-3,-5),(-3,2),(-2,-5),(-2,3),(2,-3),(2,5),(3,-2),(3,5),(5,2),(5,3)
23 0 0 0 0
29 0 0 0 0
31 12 2 12 4 (-6,-5),(-6,-1),(-5,-6),(-5,1),(-1,-6),(-1,5),(1,-5),(1,6),(5,-1),(5,6),(6,1),(6,5)
37 12 2 12 4 (-7,-4),(-7,-3),(-4,-7),(-4,3),(-3,-7),(-3,4),(3,-4),(3,7),(4,-3),(4,7),(7,3),(7,4)
41 0 0 0 0
43 12 2 12 4 (-7,-6),(-7,-1),(-6,-7),(-6,1),(-1,-7),(-1,6),(1,-6),(1,7),(6,-1),(6,7),(7,1),(7,6)
47 0 0 0 0
53 0 0 0 0
59 0 0 0 0
61 12 2 12 4 (-9,-5),(-9,-4),(-5,-9),(-5,4),(-4,-9),(-4,5),(4,-5),(4,9),(5,-4),(5,9),(9,4),(9,5)
67 12 2 12 4 (-9,-7),(-9,-2),(-7,-9),(-7,2),(-2,-9),(-2,7),(2,-7),(2,9),(7,-2),(7,9),(9,2),(9,7)
71 0 0 0 0
73 12 2 12 4 (-9,-8),(-9,-1),(-8,-9),(-8,1),(-1,-9),(-1,8),(1,-8),(1,9),(8,-1),(8,9),(9,1),(9,8)
79 12 2 12 4 (-10,-7),(-10,-3),(-7,-10),(-7,3),(-3,-10),(-3,7),(3,-7),(3,10),(7,-3),(7,10),(10,3),(10,7)
83 0 0 0 0
89 0 0 0 0
97 12 2 12 4 (-11,-8),(-11,-3),(-8,-11),(-8,3),(-3,-11),(-3,8),(3,-8),(3,11),(8,-3),(8,11),(11,3),(11,8)
101 0 0 0 0
103 12 2 12 4 (-11,-9),(-11,-2),(-9,-11),(-9,2),(-2,-11),(-2,9),(2,-9),(2,11),(9,-2),(9,11),(11,2),(11,9)
107 0 0 0 0
109 12 2 12 4 (-12,-7),(-12,-5),(-7,-12),(-7,5),(-5,-12),(-5,7),(5,-7),(5,12),(7,-5),(7,12),(12,5),(12,7)
113 0 0 0 0
127 12 2 12 4 (-13,-7),(-13,-6),(-7,-13),(-7,6),(-6,-13),(-6,7),(6,-7),(6,13),(7,-6),(7,13),(13,6),(13,7)
131 0 0 0 0
137 0 0 0 0
139 12 2 12 4 (-13,-10),(-13,-3),(-10,-13),(-10,3),(-3,-13),(-3,10),(3,-10),(3,13),(10,-3),(10,13),(13,3),(13,10)
149 0 0 0 0
151 12 2 12 4 (-14,-9),(-14,-5),(-9,-14),(-9,5),(-5,-14),(-5,9),(5,-9),(5,14),(9,-5),(9,14),(14,5),(14,9)
157 12 2 12 4 (-13,-12),(-13,-1),(-12,-13),(-12,1),(-1,-13),(-1,12),(1,-12),(1,13),(12,-1),(12,13),(13,1),(13,12)
163 12 2 12 4 (-14,-11),(-14,-3),(-11,-14),(-11,3),(-3,-14),(-3,11),(3,-11),(3,14),(11,-3),(11,14),(14,3),(14,11)
167 0 0 0 0
173 0 0 0 0
179 0 0 0 0
181 12 2 12 4 (-15,-11),(-15,-4),(-11,-15),(-11,4),(-4,-15),(-4,11),(4,-11),(4,15),(11,-4),(11,15),(15,4),(15,11)
191 0 0 0 0
193 12 2 12 4 (-16,-9),(-16,-7),(-9,-16),(-9,7),(-7,-16),(-7,9),(7,-9),(7,16),(9,-7),(9,16),(16,7),(16,9)
197 0 0 0 0
199 12 2 12 4 (-15,-13),(-15,-2),(-13,-15),(-13,2),(-2,-15),(-2,13),(2,-13),(2,15),(13,-2),(13,15),(15,2),(15,13)
211 12 2 12 4 (-15,-14),(-15,-1),(-14,-15),(-14,1),(-1,-15),(-1,14),(1,-14),(1,15),(14,-1),(14,15),(15,1),(15,14)
223 12 2 12 4 (-17,-11),(-17,-6),(-11,-17),(-11,6),(-6,-17),(-6,11),(6,-11),(6,17),(11,-6),(11,17),(17,6),(17,11)
227 0 0 0 0
229 12 2 12 4 (-17,-12),(-17,-5),(-12,-17),(-12,5),(-5,-17),(-5,12),(5,-12),(5,17),(12,-5),(12,17),(17,5),(17,12)
233 0 0 0 0
239 0 0 0 0
241 12 2 12 4 (-16,-15),(-16,-1),(-15,-16),(-15,1),(-1,-16),(-1,15),(1,-15),(1,16),(15,-1),(15,16),(16,1),(16,15)
251 0 0 0 0
257 0 0 0 0

yangchuanju · 发表于 2024-11-1 08:41

经比对，容易发现，如果不定方程a^2+abb+b^2-p=0有整数解，则不定方程a^2-ab+b^2-p也有整数解，整数解组数相同，都是12组，但正整数解组数不同；
如果前一个无整数解，后一个也一定无整数解。

yangchuanju · 发表于 2024-11-1 08:47

本帖最后由 yangchuanju 于 2024-11-1 17:52 编辑

a^2-ab+b^2-97=0的整数解——
(-11,-8),(-11,-3),(-8,-11),(-8,3),(-3,-11),(-3,8),
(3,-8),(3,11),(8,-3),(8,11),(11,3),(11,8)
对照：a^2+ab+b^2-97=0的整数解——
(-11,3),(-11,8),(-8,-3),(-8,11),(-3,-8),(-3,11),
(3,-11),(3,8),(8,-11),(8,3),(11,-8),(11,-3)
经等比，两类整数解数值绝对值相同，仅b相差一个正负号，
即(s,t)若是第2个不定方程的一组整数解，则(s,-t)便是第1个不定方程的一组整数解！

yangchuanju · 发表于 2024-11-2 12:19

a^2+ab+b^2-19=0 12解 a^2+ab+b^2-73=0 12解

a=-5,b=2 a=-9,b=1
a=-5,b=3 a=-9,b=8
a=-3,b=-2 a=-8,b=-1
a=-3,b=5 a=-8,b=9
a=-2,b=-3 a=-1,b=-8
a=-2,b=5 a=-1,b=9
a=2,b=-5 a=1,b=-9
a=2,b=3 a=1,b=8
a=3,b=-5 a=8,b=-9
a=3,b=2 a=8,b=1
a=5,b=-3 a=9,b=-8
a=5,b=-2 a=9,b=-1
12整数解中只要3个a，3个b，且数字相同——2,3,5或1,8,9；加上相反数共6个，各取2次得12解；
当a取s1时b只能取s2,s3；当a取s2时b只能取s1,s3；当a取s3时b只能取s1,s2。
12解：±s1,±s2；±s1,±s3；±s2,±s1；±s2,±s3；±s3,±s1；±s3,±s2；6*4=24，除以2才是12。
各对组合中要么取++和--，要么取+-和-+，不能全取；12组整数解中正整数解只有2组，即++只取2次。

yangchuanju · 发表于 2024-11-2 17:18

a^2+ab+b^2-(2^37-1)=0  24解
a^2+ab+b^2-1387=0  24解
a^2+ab+b^2-3*19*73*(2*1387-1)^2=0  24解

序号 a^2+ab+b^2-(2^37-1)=0 a^2+ab+b^2-1387=0 -3*19*73*(2*1387-1)^2=0 a b
1 a=-421799, b=147629 a=-43, b=21 a=-196883, b=44368 -s6 +s2
2 a=-421799, b=274170 a=-43, b=22 a=-196883, b=152515 -s6 +s3
3 a=-374039, b=6714 a=-42, b=13 a=-180245, b=2773 -s5 +s1
4 a=-374039, b=367325 a=-42, b=29 a=-180245, b=177472 -s5 +s4
5 a=-367325, b=-6714 a=-29, b=-13 a=-177472, b=-2773 -s4 -s1
6 a=-367325, b=374039 a=-29, b=42 a=-177472, b=180245 -s4 +s5
7 a=-274170, b=-147629 a=-22, b=-21 a=-152515, b=-44368 -s3 -s2
8 a=-274170, b=421799 a=-22, b=43 a=-152515, b=196883 -s3 +s6
9 a=-147629, b=-274170 a=-21, b=-22 a=-44368, b=-152515 -s2 -s3
10 a=-147629, b=421799 a=-21, b=43 a=-44368, b=196883 -s2 +s6
11 a=-6714, b=-367325 a=-13, b=-29 a=-2773, b=-177472 -s1 -s4
12 a=-6714, b=374039 a=-13, b=42 a=-2773, b=180245 -s1 +s5
13 a=6714, b=-374039 a=13, b=-42 a=2773, b=-180245 +s1 -s5
14 a=6714, b=367325 a=13, b=29 a=2773, b=177472 +s1 +s4
15 a=147629, b=-421799 a=21, b=-43 a=44368, b=-196883 +s2 -s6
16 a=147629, b=274170 a=21, b=22 a=44368, b=152515 +s2 +s3
17 a=274170, b=-421799 a=22, b=-43 a=152515, b=-196883 +s3 -s6
18 a=274170, b=147629 a=22, b=21 a=152515, b=44368 +s3 +s2
19 a=367325, b=-374039 a=29, b=-42 a=177472, b=-180245 +s4 -s5
20 a=367325, b=6714 a=29, b=13 a=177472, b=2773 +s4 +s1
21 a=374039, b=-367325 a=42, b=-29 a=180245, b=-177472 +s5 -s4
22 a=374039, b=-6714 a=42, b=-13 a=180245, b=-2773 +s5 -s1
23 a=421799, b=-274170 a=43, b=-22 a=196883, b=-152515 +s6 -s3
24 a=421799, b=-147629 a=43, b=-21 a=196883, b=-44368 +s6 -s2

12整数解中只要6个a，6个b，且数字相同；加上相反数共12个，各取2次得24解；
当a取s1时b只能取s4,s5；当a取s2时b只能取s3,s6；当a取s3时b只能取s2,s6；当a取s4时b只能取s1,s5；当a取s5时b只能取s1,s4；当a取s6时b只能取s2,s3。
各对组合中要么取++和--，要么取+-和-+，不能全取；24组整数解中正整数解只有4组，即++只取4次。

yangchuanju · 发表于 2024-11-2 17:24

a^2+ab+b^2-19^2=0  18解
a^2+ab+b^2-19^2*(2*1387-1)^2=0  18解
a^2+ab+b^2-3*19^2*(2*1387-1)^2=0  18解
a^2+ab+b^2-3*19^2=0  18解

序号 a^2+ab+b^2-19^2=0 -19^2*(2*1387-1)^2=0 -3*19^2*(2*1387-1)^2=0 -3*19^2=0
1 a=-21, b=5 a=-58233, b=13865 a=-105374, b=52687 a=-38, b=19
2 a=-21, b=16 a=-58233, b=44368 a=-102601, b=30503 a=-37, b=11
3 a=-19, b=0 a=-52687, b=0 a=-102601, b=72098 a=-37, b=26
4 a=-19, b=19 a=-52687, b=52687 a=-72098, b=-30503 a=-26, b=-11
5 a=-16, b=-5 a=-44368, b=-13865 a=-72098, b=102601 a=-26, b=37
6 a=-16, b=21 a=-44368, b=58233 a=-52687, b=-52687 a=-19, b=-19
7 a=-5, b=-16 a=-13865, b=-44368 a=-52687, b=105374 a=-19, b=38
8 a=-5, b=21 a=-13865, b=58233 a=-30503, b=-72098 a=-11, b=-26
9 a=0, b=-19 a=0, b=-52687 a=-30503, b=102601 a=-11, b=37
10 a=0, b=19 a=0, b=52687 a=30503, b=-102601 a=11, b=-37
11 a=5, b=-21 a=13865, b=-58233 a=30503, b=72098 a=11, b=26
12 a=5, b=16 a=13865, b=44368 a=52687, b=-105374 a=19, b=-38
13 a=16, b=-21 a=44368, b=-58233 a=52687, b=52687 a=19, b=19
14 a=16, b=5 a=44368, b=13865 a=72098, b=-102601 a=26, b=-37
15 a=19, b=-19 a=52687, b=-52687 a=72098, b=30503 a=26, b=11
16 a=19, b=0 a=52687, b=0 a=102601, b=-72098 a=37, b=-26
17 a=21, b=-16 a=58233, b=-44368 a=102601, b=-30503 a=37, b=-11
18 a=21, b=-5 a=58233, b=-13865 a=105374, b=-52687 a=38, b=-19

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a^2-ab+b^2-c=0的整数解

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