# Sum of powers

The following is a list of sum of powers: ap + bq = cr where gcd(a,b,c) = 1. If you have any comment please fill the form.

an + b3 = c2 (Finite number of solutions)

a3 + b2 = c7 (Finite number of solutions)

a3 + b2 = c9 (Finite number of solutions)

a3 + b3 = c2 (Infinite number of solutions)

a4 + b2 = c3 (Infinite number of solutions)

a4 + b3 = c2 (Infinite number of solutions)

a5 + b2 = c4 (Finite number of solutions)

a5 + b3 = c2 (Infinite number of solutions)

a5 + b4 = c2 (Finite number of solutions)

a7 + b3 = c2 (Finite number of solutions)

a8 + b2 = c3 (Finite number of solutions)

a8 + b3 = c2 (Finite number of solutions)

## an + b3 = c2

1. 1n + 23 = 32

## a3 + b2 = c7

1. 1 4143 + 2 213 4592 = 657
2. 9 2623 + 15 312 2832 = 1137

## a3 + b2 = c9

1. 73 + 132 = 29

## a3 + b3 = c2

### Formula 1:

• a = s4 + 6s2t2 - 3t4
• b = -s4 + 6s2t2 + 3t4
• c = 6st(s4 + 3t4)

### Formula 2:

• a = (1/4)(s4 + 6s2t2 - 3t4)
• b = (1/4)(-s4 + 6s2t2 + 3t4)
• c = (3/4)st(s4 + 3t4)

### Formula 3:

• a = s4 + 8st3
• b = -4s3t + 4t4
• c = s6 - 20s3t3 - 8t6

## a4 + b2 = c3

### Formula 1:

• a = (s2 + 3t2) (s4 - 18s2t2 + 9t4)
• b = 4st(s2 - 3t2) (s4 + 6s2t2 + 81t4) (3s4 + 2s2t2 + 3t4)
• c = (s4 - 2s2t2 + 9t4) (s4 + 30s2t2 + 9t4)

### Formula 2:

• a = 6st(s4 - 12t4)
• b = (s4 + 12t4) (s8 - 408s4t4 + 144t8)
• c = s8 + 168s4t4 + 144t8

### Formula 3:

• a = 6st(3s4 - 4t4)
• b = (3s4 + 4t4) (9s8 - 408s4t4 + 16t8)
• c = 9s8 + 168s4t4 + 16t8

### Formula 4:

• a = (3/2)st(s4 - 3t4)
• b = (1/8) (s4 + 3t4) (s8 - 102s4t4 + 9t8)
• c = (1/4) (s8 + 42s4t4 + 9t8)

See 55 solutions with both a and b both less than 1012

## a4 + b3 = c2

### Formula 1:

• a = 6 s4t + 72 st4
• b = s8 - 168 s4t4 + 144 t8
• c = s12 + 396 s8t4 - 4752 s4t8 - 1728 t12

### Formula 2:

• a = 18 s4t + 24 st4
• b = 9 s8 - 168 s4t4 + 16 t8
• c = 27 s12 + 1188 s8t4 - 1584 s4t8 - 64 t12

### Formula 3:

• a = -s6 + 40 s3t3 + 32 t6
• b = 8 s7t + 112 s4t4 - 256 st7
• c = s12 + 176 s9t3 + 5632 s3t9 - 1024 t12

### Formula 4:

• a = -s6 - 15 s4t2 + 45 s2t4 + 27 t6
• b = -s8 + 28 s6t2 + 42 s4t4 + 252 s2t6 - 81 t8
• c = 12 s11t - 44 s9t3 + 792 s7t4 + 2376 s4t7 - 1188 s3t9 + 2916 st11

### Formula 5:

• a = -s6 + 6 s4t + 15 s4t2 + 20 s3t3 - 15 s2t4 + 30 st4 + 17 t6
• b = 2 s8 + 8 s7t + 56 s4t3 - 28 s4t4 - 168 s3t4 - 112 s2t6 - 88 st7 + 42 t8
• c = 3 s12 + 12 s11t + 66 s10t2 - 44 s9t3 - 99 s8t4 + 792 s7t4 + 924 s6t6 + 2376 s4t7 + 1485 s4t8 - 1188 s3t9 - 2046 s2t10 - 156 st11 - 397 t12

### Formula 6:

• a = -5 s6 - 6 s4t + 15 s4t2 + 60 s3t3 + 45 s2t4 + 18 st4 + 9 t6
• b = 6 s8 + 56 s7t + 112 s6t2 + 168 s4t3 + 252 s4t4 + 168 s3t4 - 72 st7 - 18 t8
• c = 29 s12 + 156 s11t + 726 s10t2 + 2420 s9t3 + 4059 s8t4 + 3960 s7t4 + 2772 s6t6 + 2376 s4t7 + 3267 s4t8 + 3564 s3t9 + 1782 s2t10 + 324 st11 - 27 t12

### Formula 7:

• a = -7 s6 - 6 s4t + 30 s4t2 + 80 s3t3 + 60 s2t4 + 24 st4 + 8 t6
• b = 15 s8 + 104 s7t + 224 s6t2 + 336 s4t3 + 392 s4t4 + 224 s3t4 - 64 st7 - 16 t8
• c = 76 s12 + 516 s11t + 2244 s10t2 + 6072 s9t3 + 9504 s8t4 + 9504 s7t4 + 7392 s6t6 + 6336 s4t7 + 6336 s4t8 + 4928 s3t9 + 2112 s2t10 + 384 st11

See 1602 solutions with both a and b both less than 1012

## a5 + b2 = c4

1. 25 + 72 = 34

## a5 + b3 = c2

### Formula 1:

• a = -12s11t + 19008s6t6 + 248832st11
• b = s20 + 32832s15t5 + 10243584s10t10 - 680804352s5t15 + 429981696t20
• c = -s30 + 75168s25t5 + 207463680s20t10 + 4301966868480s10t20 - 32320864124928s5t25 - 8916100448256t30

### Formula 2:

• a = s12 + 440s9t3 - 10560s6t6 - 140800s3t9 + 102400t12
• b = -20s19t + 5700s16t4 + 182400s13t7 + 15808000s10t10 - 58368000s7t13 + 583680000s4t16 + 655360000st19
• c = s30 - 2900s27t3 + 156600s24t6 - 32016000s21t9 - 2433216000s18t12 - 778629120000s12t18 + 3278438400000s9t21 + 5131468800000s6t24 + 30408704000000s3t27 + 3355443200000t30

### Formula 3:

• a = s12 + 66s10t2 - 1485s8t4 - 5940s6t6 - 66825s4t8 + 133650s2t10 + 91125t12
• b = -s20 + 190s18t2 + 855s16t4 + 102600s14t6 + 1000350s12t8 - 10003500s10t10 + 45015750s8t12 + 207765000s6t14 + 77911875s4t16 + 779118750s2t18 - 184528125t20
• c = -30s29t + 1160s27t3 - 95526s25t5 - 1879200s23t7 - 24312150s21t9 + 162081000s19t11 - 6928962750s17t13 + 311803323750s13t17 - 328214025000s11t19 + 2215444668750s9t21 + 7705894500000s7t23 + 17627233668750s5t25 - 9632368125000s3t27 + 11210083593750st29

### Formula 4:

• a = -s12 + 66s10t2 + 1485s8t4 - 5940s6t6 + 66825s4t8 + 133650s2t10 - 91125t12
• b = s20 + 190s18t2 - 855s16t4 + 102600s14t6 - 1000350s12t8 - 10003500s10t10 - 45015750s8t12 + 207765000s6t14 - 77911875s4t16 + 779118750s2t18 + 184528125t20
• c = 30s29t + 1160s27t3 + 95526s25t5 - 1879200s23t7 + 24312150s21t9 + 162081000s19t11 + 6928962750s17t13 - 311803323750s13t17 - 328214025000s11t19 - 2215444668750s9t21 + 7705894500000s7t23 - 17627233668750s5t25 - 9632368125000s3t27 - 11210083593750st29

### Formula 5:

• a = s12 - 12s11t - 66s10t2 - 220s9t3 + 495s8t4 - 3960s7t5 + 3300s6t6 + 27720s5t7 + 32175s4t8 + 47300s3t9 - 67650s2t10 + 95700st11 + 57025t12
• b = 2s20 + 20s19t + 1140s17t3 - 3990s16t4 - 13680s15t5 + 91200s14t6 + 296400s13t7 + 1556100s12t8 + 642200s11t9 - 1580800s10t10 + 2223000s9t11 + 50758500s8t12 - 17214000s7t13 + 124488000s6t14 + 124602000s5t15 - 233343750s4t16 - 421087500s3t17 - 126160000s2t18 - 204347500st19 + 122051250t20
• c = -3s30 - 30s29t - 435s28t2 + 1160s27t3 - 3915s26t4 - 114318s25t5 + 267525s24t6 + 14857425s22t8 + 54927450s21t9 - 25962975s20t10 - 246123000s19t11 + 679589625s18t12 + 5047022250s17t13 + 16723607625s16t14 + 15511752000s15t15 + 76346904375s14t16 - 78271616250s13t17 + 111704574375s12t18 + 351565695000s11t19 - 513778010625s10t20 - 2039831906250s9t21 + 3497966859375s8t22 + 3404014200000s7t23 + 6263819746875s6t24 + 9457764018750s5t25 + 44594296875s4t26 - 5292677625000s3t27 - 6859171078125s2t28 + 1308568968750st29 - 1556004528125t30

### Formula 6:

• a = -3s12 - 12s11t + 132s10t2 + 1980s8t4 + 3168s7t5 - 3168s6t6 - 12672s5t7 + 39600s4t8 + 10560s3t9 + 61248s2t10 + 26112st11 - 27072t12
• b = 7s20 - 20s19t + 760s18t2 + 2280s17t3 + 2280s16t4 - 29184s15t5 + 127680s14t6 + 620160s13t7 - 829920s12t8 + 158080s11t9 - 9231872s10t10 - 6007040s9t11 - 24660480s8t12 - 60119040s7t13 + 49320960s6t14 + 73426944s5t15 + 21815040s4t16 - 73835520s3t17 + 143779840s2t18 + 33187840st19 + 29857792t20
• c = -10s30 + 390s29t + 870s28t2 + 14500s27t3 - 15660s26t4 + 363312s25t5 + 2871000s24t6 + 626400s23t7 - 28814400s22t8 + 8004000s21t9 + 27853920s20t10 + 701150400s19t11 - 1155777600s18t12 + 4379788800s17t13 + 18614102400s16t14 + 9927521280s15t15 + 74456409600s14t16 - 74456409600s13t17 - 200983641600s12t18 - 381579494400s11t19 - 108772439040s10t20 - 301001625600s9t21 - 1304716032000s8t22 - 26779852800s7t23 + 2317071974400s6t24 - 1085530152960s5t25 - 797590609920s4t26 - 1297945640960s3t27 + 46432665600s2t28 - 722915819520st29 - 109893615616t30

### Formula 7:

• a = 10s12 - 12s11t - 264s10t2 - 1540s9t3 - 990s8t4 - 3960s7t5 - 10560s6t6 + 3960s5t7 + 24750s4t8 + 47300s3t9 + 6600s2t10 + 7500st11 + 10150t12
• b = 41s20 + 740s19t + 1710s18t2 + 14820s17t3 + 61845s16t4 - 13680s15t5 - 216600s14t6 - 250800s13t7 + 2556450s12t8 + 10719800s11t9 + 20426900s10t10 + 11115000s9t11 + 17969250s8t12 + 29298000s7t13 + 30609000s6t14 + 58938000s5t15 + 60883125s4t16 - 5557500s3t17 - 23631250s2t18 - 21197500st19 - 624375t20
• c = -411s30 - 3810s29t - 60465s28t2 - 383380s27t3 - 614655s26t4 - 2801574s25t5 - 13004325s24t6 - 47449800s23t7 - 158028975s22t8 - 41720850s21t9 + 1905502275s20t10 + 6876436500s19t11 + 11686090125s18t12 + 7100048250s17t13 + 12360927375s16t14 + 50413194000s15t15 + 163600509375s14t16 + 355430126250s13t17 + 463427848125s12t18 + 385737772500s11t19 + 487027141875s10t20 + 383429118750s9t21 - 51944709375s8t22 - 181558125000s7t23 - 5074003125s6t24 + 370510706250s5t25 + 520780640625s4t26 + 336891187500s3t27 + 12290109375s2t28 - 17999531250st29 - 10367496875t30

### Formula 8:

• a = 19s12 + 60s11t + 528s10t2 + 440s9t3 - 3960s8t4 - 6336s7t5 - 10560s6t6 - 12672s5t7 - 31680s4t8 - 14080s3t9 + 16896s2t10 + 7680st11 + 5632t12
• b = -127s20 + 20s19t + 10260s18t2 + 31920s17t3 + 69540s16t4 + 207936s15t5 + 1130880s14t6 + 2663040s13t7 + 5690880s12t8 + 5058560s11t9 - 7840768s10t10 - 11381760s9t11 + 13278720s8t12 + 33853440s7t13 + 56033280s6t14 + 42491904s5t15 + 22179840s4t16 + 14008320s3t17 + 20234240s2t18 + 5242880st19 - 1032192t20
• c = -654s30 - 30630s29t - 113970s28t2 + 138620s27t3 - 203580s26t4 - 17200944s25t5 - 82945800s24t6 - 296913600s23t7 - 777988800s22t8 - 1805702400s21t9 - 2981329920s20t10 - 2958278400s19t11 - 7056326400s18t12 - 35038310400s17t13 - 74456409600s16t14 - 79420170240s15t15 + 210229862400s13t17 + 350383104000s12t18 + 263094681600s11t19 + 159823872000s10t20 + 252439756800s9t21 + 368824320000s8t22 + 466963660800s7t23 + 419069952000s6t24 + 349760913408s5t25 + 160101826560s4t26 + 13227786240s3t27 + 456130560s2t28 + 12960399360st29 + 2136997888t30

### Formula 9:

• a = 7s12 + 264s11t + 858s10t2 + 1320s9t3 + 1485s8t4 + 4752s7t5 + 27324s6t6 + 42768s5t7 + 31185s4t8 + 11880s3t9 - 1782s2t10 - 14904st11 - 4293t12
• b = 393s20 + 2440s19t + 7030s18t2 + 6840s17t3 - 110295s16t4 - 1187424s15t5 - 4247640s14t6 - 8290080s13t7 - 10070190s12t8 - 4801680s11t9 + 16992612s10t10 + 47483280s9t11 + 60621210s8t12 + 50971680s7t13 + 47462760s6t14 + 78452064s5t15 + 81097605s4t16 + 43769160s3t17 + 12881430s2t18 + 2653560st19 + 1426653t20
• c = 7792s30 + 72750s29t + 335820s28t2 + 1274840s27t3 + 8769600s26t4 + 77280534s25t5 + 393144300s24t6 + 1225238400s23t7 + 2442020400s22t8 + 1731565350s21t9 - 10804319460s20t10 - 53065319400s19t11 - 131393664000s18t12 - 218493292050s17t13 - 266996031300s16t14 - 251290382400s15t15 - 188467786800s14t16 - 123335536950s13t17 - 124721329500s12t18 - 340467348600s11t19 - 950857911360s10t20 - 1638371476350s9t21 - 1773668591100s8t22 - 1303266542400s7t23 - 750953689200s6t24 - 467651900574s5t25 - 295735106700s4t26 - 132724043640s3t27 - 35070382080s2t28 - 3787993350st29 + 1202316372t30

### Formula 10:

• a = 25s12 + 2200s9t3 - 10560s6t6 - 28160s3t9 + 4096t12
• b = -2500s19t + 142500s16t4 + 912000s13t7 + 15808000s10t10 - 11673600s7t13 + 23347200s4t16 + 5242880st19
• c = 3125s30 - 1812500s27t3 + 19575000s24t6 - 800400000s21t9 - 12166080000s18t12 - 155725824000s12t18 + 131137536000s9t21 + 41051750400s6t24 + 48653926400s3t27 + 1073741824t30

### Formula 11:

• a = -6s12 + 372s11t + 2112s10t2 + 5280s9t3 + 7920s8t4 + 6336s7t5 + 19008s6t6 + 50688s5t7 + 63360s4t8 + 42240s3t9 + 16896s2t10 - 3072st11 - 3072t12
• b = 1153s20 + 11360s19t + 53200s18t2 + 155040s17t3 + 392160s16t4 - 25536s15t5 - 5399040s14t6 - 22617600s13t7 - 51692160s12t8 - 80304640s11t9 - 84351488s10t10 - 40468480s9t11 + 37939200s8t12 + 96890880s7t13 + 95723520s6t14 + 71909376s5t15 + 61870080s4t16 + 46694400s3t17 + 21790720s2t18 + 5242880st19 + 851968t20
• c = 39151s30 + 578640s29t + 4130760s28t2 + 19156240s27t3 + 59069520s26t4 + 181781280s25t5 + 958809600s24t6 + 5076345600s23t7 + 19392091200s22t8 + 54568070400s21t9 + 115111607040s20t10 + 164280499200s19t11 + 73969766400s18t12 - 350383104000s17t13 - 1191302553600s16t14 - 2223764766720s15t15 - 2978256384000s14t16 - 3083371315200s13t17 - 2553903513600s12t18 - 1701509529600s11t19 - 940256133120s10t20 - 675358310400s9t21 - 944190259200s8t22 - 1252078387200s7t23 - 1137475584000s6t24 - 715121491968s5t25 - 340729528320s4t26 - 134406471680s3t27 - 41964011520s2t28 - 8556380160st29 - 587202560t30

### Formula 12:

• a = 64s12 + 384s11t + 2112s10t2 + 7040s9t3 + 7920s8t4 - 6336s7t5 - 32736s6t6 - 50688s5t7 - 61380s4t8 - 57640s3t9 - 24684s2t10 - 1464st11 + 2353t12
• b = -1024s20 - 10240s19t + 291840s17t3 + 1605120s16t4 + 4902912s15t5 + 11965440s14t6 + 27724800s13t7 + 61176960s12t8 + 117927680s11t9 + 173698304s10t10 + 172149120s9t11 + 104332800s8t12 + 39982080s7t13 + 38139840s6t14 + 70300608s5t15 + 75699420s4t16 + 47230200s3t17 + 20503280s2t18 + 6621140st19 + 894906t20
• c = -491520s29t - 7127040s28t2 - 45137920s27t3 - 160358400s26t4 - 410517504s25t5 - 1309593600s24t6 - 6253977600s23t7 - 27661824000s22t8 - 95586969600s21t9 - 261896002560s20t10 - 584432870400s19t11 - 1072074969600s18t12 - 1638041011200s17t13 - 2196464083200s16t14 - 2849198607360s15t15 - 3797276889600s14t16 - 4838571676800s13t17 - 5102636443200s12t18 - 3814863278400s11t19 - 1413894754080s10t20 + 699389520000s9t21 + 1600877638800s8t22 + 1501115608800s7t23 + 1055815261200s6t24 + 651072611880s5t25 + 359175395340s4t26 + 159392955780s3t27 + 48132773490s2t28 + 8829176700st29 + 888155653t30

### Formula 13:

• a = 64s12 + 768s11t + 2112s10t2 + 3520s9t3 + 7920s8t4 + 25344s7t5 + 55968s6t6 + 60192s5t7 + 33660s4t8 + 6160s3t9 - 8844s2t10 - 10308st11 - 2207t12
• b = 2048s20 + 10240s19t - 291840s17t3 - 2115840s16t4 - 8054784s15t5 - 17802240s14t6 - 23347200s13t7 - 11381760s12t8 + 28138240s11t9 + 86058752s10t10 + 126622080s9t11 + 131008800s8t12 + 124725120s7t13 + 129741120s6t14 + 121908864s5t15 + 78137880s4t16 + 31156200s3t17 + 7981520s2t18 + 1769020st19 + 442143t20
• c = 98304s30 + 983040s29t + 7127040s28t2 + 54640640s27t3 + 352788480s26t4 + 1571512320s25t5 + 4436582400s24t6 + 6253977600s23t7 - 8298547200s22t8 - 78682521600s21t9 - 259068349440s20t10 - 559537228800s19t11 - 884230694400s18t12 - 1073048256000s17t13 - 1042389734400s16t14 - 883549393920s15t15 - 837634608000s14t16 - 1189112659200s13t17 - 2085326942400s12t18 - 3274823793600s11t19 - 4055003448480s10t20 - 3830719202400s9t21 - 2794550176800s8t22 - 1667196799200s7t23 - 895510575000s6t24 - 451671056928s5t25 - 191974420980s4t26 - 58334973780s3t27 - 10541110530s2t28 - 501967410st29 + 184589480t30

### Formula 14

• a = 25s12 + 600s11t + 1650s10t2 + 2200s9t3 + 2475s8t4 + 7920s7t5 + 31020s6t6 + 39600s5t7 + 24255s4t8 + 7480s3t9 - 2046s2t10 - 7368st11 - 1763t12
• b = 1875s20 + 10000s19t + 23750s18t2 - 463125s16t4 - 3192000s15t5 - 9063000s14t6 - 14592000s13t7 - 14264250s12t8 - 2964000s11t9 + 23687300s10t10 + 49004800s9t11 + 51684750s8t12 + 38760000s7t13 + 34804200s6t14 + 45089280s5t15 + 37078215s4t16 + 16548240s3t17 + 4149030s2t18 + 789760st19 + 322343t20
• c = 81250s30 + 656250s29t + 2718750s28t2 + 10875000s27t3 + 73406250s26t4 + 484481250s25t5 + 1936293750s24t6 + 4854600000s23t7 + 7316156250s22t8 + 162581250s21t9 - 39972476250s20t10 - 135277605000s19t11 - 265610238750s18t12 - 363128973750s17t13 - 370827821250s16t14 - 294723288000s15t15 - 190503704250s14t16 - 121042991250s13t17 - 148321623750s12t18 - 341384607000s11t19 - 675919490850s10t20 - 884003881050s9t21 - 767542679550s8t22 - 472107657600s7t23 - 239161464450s6t24 - 128632685418s5t25 - 66582538110s4t26 - 24423970520s3t27 - 5299948650s2t28 - 427589970st29 + 128301258t30

### Formula 15:

• a = -55s12 - 348s11t + 462s10t2 + 660s9t3 + 4455s8t4 + 11880s7t5 + 10692s6t6 - 7128s5t7 + 4455s4t8 + 5940s3t9 + 8910s2t10 + 5508st11 - 567t12
• b = 1226s20 - 380s19t + 25840s18t2 + 160740s17t3 + 490770s16t4 + 347472s15t5 - 328320s14t6 + 2831760s13t7 + 4935060s12t8 + 6135480s11t9 - 1493856s10t10 - 10937160s9t11 - 13204620s8t12 - 24747120s7t13 - 20684160s6t14 - 3767472s5t15 + 5401890s4t16 - 277020s3t17 + 1108080s2t18 + 772740st19 + 287226t20
• c = -36599s30 + 240930s29t + 1657785s28t2 + 11693960s27t3 + 34087905s26t4 + 49546674s25t5 + 254142225s24t6 + 1307923200s23t7 + 2375837325s22t8 + 1812605850s21t9 - 2443371075s20t10 - 1782891000s19t11 + 5260879125s18t12 + 2309654250s17t13 + 19632061125s16t14 + 50258076480s15t15 + 106013130075s14t16 + 156594558150s13t17 + 139272151275s12t18 + 28882834200s11t19 - 78311866365s10t20 - 84387472650s9t21 - 64658162925s8t22 - 72150004800s7t23 - 27156143025s6t24 + 1161021438s5t25 + 2936802015s4t26 - 2717041320s3t27 - 1104511545s2t28 - 631233810st29 - 153743913t30

### Formula 16:

• a = 81s12 + 324s11t + 1782s10t2 + 5940s9t3 + 4455s8t4 - 7128s7t5 - 22572s6t6 - 26136s5t7 - 31185s4t8 - 31020s3t9 - 9834s2t10 + 804st11 + 1657t12
• b = -1458s20 - 14580s19t + 277020s17t3 + 1246590s16t4 + 2881008s15t5 + 5909760s14t6 + 12927600s13t7 + 27609660s12t8 + 52551720s11t9 + 71278272s10t10 + 53618760s9t11 + 15338700s8t12 - 3324240s7t13 + 10177920s6t14 + 28544688s5t15 + 28175670s4t16 + 14381100s3t17 + 6013120s2t18 + 2236540st19 + 251382t20
• c = 19683s30 - 590490s29t - 8562105s28t2 - 45664560s27t3 - 111307365s26t4 - 154117890s25t5 - 442375425s24t6 - 2968196400s23t7 - 13456775025s22t8 - 42084331650s21t9 - 102118323645s20t10 - 203055076800s19t11 - 323582560425s18t12 - 408808802250s17t13 - 459390230325s16t14 - 577967879520s15t15 - 844178628375s14t16 - 1119258449550s13t17 - 1068882324075s12t18 - 561383751600s11t19 + 62314471665s10t20 + 395258830650s9t21 + 394892797725s8t22 + 275540518800s7t23 + 175903201125s6t24 + 112283394858s5t25 + 67302682785s4t26 + 30539055520s3t27 + 8170447965s2t28 + 1348449810st29 + 168454745t30

### Formula 17:

• a = 125s12 + 1650s10t2 - 7425s8t4 - 5940s6t6 - 13365s4t8 + 5346s2t10 + 729t12
• b = -3125s20 + 118750s18t2 + 106875s16t4 + 2565000s14t6 + 5001750s12t8 - 10003500s10t10 + 9003150s8t12 + 8310600s6t14 + 623295s4t16 + 1246590s2t18 - 59049t20
• c = -2343750s29t + 18125000s27t3 - 298518750s25t5 - 1174500000s23t7 - 3039018750s21t9 + 4052025000s19t11 - 34644813750s17t13 + 62360664750s13t17 - 13128561000s11t19 + 17723557350s9t21 + 12329431200s7t23 + 5640714774s5t25 - 616471560s3t27 + 143489070st29

### Formula 18:

• a = -125s12 + 1650s10t2 + 7425s8t4 - 5940s6t6 + 13365s4t8 + 5346s2t10 - 729t12
• b = 3125s20 + 118750s18t2 - 106875s16t4 + 2565000s14t6 - 5001750s12t8 - 10003500s10t10 - 9003150s8t12 + 8310600s6t14 - 623295s4t16 + 1246590s2t18 + 59049t20
• c = 2343750s29t + 18125000s27t3 + 298518750s25t5 - 1174500000s23t7 + 3039018750s21t9 + 4052025000s19t11 + 34644813750s17t13 - 62360664750s13t17 - 13128561000s11t19 - 17723557350s9t21 + 12329431200s7t23 - 5640714774s5t25 - 616471560s3t27 - 143489070st29

### Formula 19:

• a = 162s12 + 324s11t - 5940s9t3 - 8910s8t4 - 7128s7t5 - 14256s6t6 - 11880s5t7 - 2970s4t8 + 11220s3t9 + 5808s2t10 + 1116st11 + 710t12
• b = 729s20 + 43740s19t + 138510s18t2 + 277020s17t3 + 1315845s16t4 + 2881008s15t5 + 3139560s14t6 - 369360s13t7 - 4201470s12t8 - 1867320s11t9 + 18166356s10t10 + 30143880s9t11 + 23541570s8t12 + 16046640s7t13 + 9377640s6t14 + 4807152s5t15 + 5447205s4t16 + 2876220s3t17 + 570190s2t18 - 280260st19 - 53831t20
• c = -334611s30 - 1771470s29t - 8562105s28t2 - 102745260s27t3 - 402418935s26t4 - 880184394s25t5 - 1641070125s24t6 - 2854035000s23t7 - 4923210375s22t8 - 12763878750s21t9 - 27022954725s20t10 - 31070927700s19t11 + 15936614325s18t12 + 101162856150s17t13 + 153130076775s16t14 + 138209710320s15t15 + 90307481175s14t16 + 67903834950s13t17 + 127210623525s12t18 + 179245377900s11t19 + 156621301515s10t20 + 113032588050s9t21 + 70044654825s8t22 + 31858547400s7t23 + 7215286275s6t24 - 490350618s5t25 - 177854535s4t26 + 1342327060s3t27 + 657823095s2t28 + 103017570st29 - 4942947t30

### Formula 20:

• a = -972s11t + 19008s6t6 + 3072st11
• b = 6561s20 + 2659392s15t5 + 10243584s10t10 - 8404992s5t15 + 65536t20
• c = -531441s30 + 493177248s25t5 + 16804558080s20t10 + 53110702080s10t20 - 4926210048s5t25 - 16777216t30

### Formula 21:

• a = 185s12 + 144s11t - 2046s10t2 - 9680s9t3 - 13365s8t4 - 15840s7t5 - 20724s6t6 - 9504s5t7 + 8415s4t8 + 16720s3t9 + 6930s2t10 + 1776st11 + 701t12
• b = 5879s20 + 77680s19t + 282910s18t2 + 962160s17t3 + 2793855s16t4 + 4009152s15t5 + 1780680s14t6 - 1842240s13t7 + 4972110s12t8 + 30786080s11t9 + 60768916s10t10 + 63350560s9t11 + 46734870s8t12 + 31427520s7t13 + 20695560s6t14 + 14923968s5t15 + 9746715s4t16 + 3114480s3t17 - 42370s2t18 - 377360st19 - 51701t20
• c = -647992s30 - 6865710s29t - 60148320s28t2 - 327777720s27t3 - 1008911160s26t4 - 2330215830s25t5 - 5138568000s24t6 - 11560212000s23t7 - 24474231000s22t8 - 37074027750s21t9 - 11987750880s20t10 + 100080815400s19t11 + 281880469800s18t12 + 418321156050s17t13 + 437431406400s16t14 + 414474013440s15t15 + 475822992600s14t16 + 632485984950s13t17 + 739119775200s12t18 + 669407736600s11t19 + 482938228440s10t20 + 287513985150s9t21 + 131883508800s8t22 + 40068928800s7t23 + 7215449400s6t24 + 4046295006s5t25 + 5076470880s4t26 + 3161731960s3t27 + 847439160s2t28 + 79084230st29 - 5574720t30

### Formula 22:

• a = -100s12 - 1500s11t - 3300s10t2 - 3300s9t3 + 11880s7t5 + 35640s6t6 + 35640s5t7 + 17820s4t8 + 5940s3t9 + 3564s2t10 + 3564st11 + 648t12
• b = 10625s20 + 47500s19t + 118750s18t2 + 427500s17t3 + 2458125s16t4 + 9918000s15t5 + 21033000s14t6 + 27702000s13t7 + 28343250s12t8 + 33345000s11t9 + 48683700s10t10 + 54685800s9t11 + 39013650s8t12 + 12927600s7t13 - 10157400s6t14 - 21053520s5t15 - 14751315s4t16 - 5263380s3t17 - 969570s2t18 - 14580st19 + 53217t20
• c = -1090625s30 - 7031250s29t - 17671875s28t2 + 34437500s27t3 + 648421875s26t4 + 3636056250s25t5 + 11556590625s24t6 + 25545375000s23t7 + 49468471875s22t8 + 105690318750s21t9 + 234308345625s20t10 + 428299042500s19t11 + 589603404375s18t12 + 612058376250s17t13 + 490801528125s16t14 + 314112978000s15t15 + 176688550125s14t16 + 131650292250s13t17 + 197475438375s12t18 + 347906866500s11t19 + 472956410025s10t20 + 443380679550s9t21 + 281935847475s8t22 + 116558789400s7t23 + 21219750225s6t24 - 9832721382s5t25 - 9084393405s4t26 - 3230767620s3t27 - 642157875s2t28 - 92706930st29 - 16277841t30

### Formula 23:

• a = -192s12 - 384s11t + 2112s10t2 + 7920s8t4 + 6336s7t5 - 3168s6t6 - 6336s5t7 + 9900s4t8 + 1320s3t9 + 3828s2t10 + 816st11 - 423t12
• b = 7168s20 - 10240s19t + 194560s18t2 + 291840s17t3 + 145920s16t4 - 933888s15t5 + 2042880s14t6 + 4961280s13t7 - 3319680s12t8 + 316160s11t9 - 9231872s10t10 - 3003520s9t11 - 6165120s8t12 - 7514880s7t13 + 3082560s6t14 + 2294592s5t15 + 340860s4t16 - 576840s3t17 + 561640s2t18 + 64820st19 + 29158t20
• c = -327680s30 + 6389760s29t + 7127040s28t2 + 59392000s27t3 - 32071680s26t4 + 372031488s25t5 + 1469952000s24t6 + 160358400s23t7 - 3688243200s22t8 + 512256000s21t9 + 891325440s20t10 + 11218406400s19t11 - 9246220800s18t12 + 17519155200s17t13 + 37228204800s16t14 + 9927521280s15t15 + 37228204800s14t16 - 18614102400s13t17 - 25122955200s12t18 - 23848718400s11t19 - 3399138720s10t20 - 4703150400s9t21 - 10193094000s8t22 - 104608800s7t23 + 4525531200s6t24 - 1060088040s5t25 - 389448540s4t26 - 316881260s3t27 + 5668050s2t28 - 44123280st29 - 3353687t30

### Formula 24:

• a = 395s12 + 1836s11t + 6072s10t2 + 5720s9t3 - 11880s8t4 - 31680s7t5 - 40128s6t6 - 38016s5t7 - 31680s4t8 - 14080s3t9 + 1536st11 + 512t12
• b = -12931s20 + 38060s19t + 852340s18t2 + 3456480s17t3 + 8569380s16t4 + 18666816s15t5 + 39982080s14t6 + 71829120s13t7 + 97693440s12t8 + 89789440s11t9 + 45021184s10t10 + 8852480s9t11 + 13278720s8t12 + 33853440s7t13 + 39690240s6t14 + 27549696s5t15 + 12840960s4t16 + 4669440s3t17 + 1556480s2t18 + 327680st19 + 16384t20
• c = -2730128s30 - 44424330s29t - 222407670s28t2 - 660279540s27t3 - 2555226540s26t4 - 12752601984s25t5 - 49309216200s24t6 - 139926484800s23t7 - 306268257600s22t8 - 539008569600s21t9 - 784750579200s20t10 - 992175840000s19t11 - 1208091744000s18t12 - 1532926080000s17t13 - 1861410240000s16t14 - 1826663915520s15t15 - 1191302553600s14t16 - 210229862400s13t17 + 552826675200s12t18 + 784366387200s11t19 + 637820190720s10t20 + 413083238400s9t21 + 250800537600s8t22 + 148812595200s7t23 + 80393011200s6t24 + 36125540352s5t25 + 12315525120s4t26 + 2888826880s3t27 + 456130560s2t28 + 62914560st29 + 6291456t30

### Formula 25:

• a = 537s12 + 2460s11t + 8778s10t2 + 27060s9t3 + 44055s8t4 + 32472s7t5 - 5940s6t6 - 32472s5t7 - 35145s4t8 - 27060s3t9 - 12342s2t10 - 2460st11 + 57t12
• b = -29396s20 - 387860s19t - 1429940s18t2 - 1075020s17t3 + 8603580s16t4 + 35858928s15t5 + 78801360s14t6 + 128815440s13t7 + 182078520s12t8 + 242642920s11t9 + 300571336s10t10 + 308265880s9t11 + 235430520s8t12 + 125450160s7t13 + 45969360s6t14 + 14994192s5t15 + 8193180s4t16 + 4814220s3t17 + 1853260s2t18 + 453460st19 + 52684t20
• c = 4387861s30 + 1976910s29t - 450850965s28t2 - 4115300680s27t3 - 18483783795s26t4 - 52053249618s25t5 - 102595034925s24t6 - 163810490400s23t7 - 288536145975s22t8 - 675364413450s21t9 - 1649574184905s20t10 - 3469944105000s19t11 - 6043269274575s18t12 - 8709877262250s17t13 - 10416189942225s16t14 - 10430102044800s15t15 - 8985230820225s14t16 - 7025882685750s13t17 - 5285702680575s12t18 - 3824469279000s11t19 - 2458531261305s10t20 - 1260330950550s9t21 - 452454063975s8t22 - 77491317600s7t23 + 23954991075s6t24 + 25048268658s5t25 + 11603147805s4t26 + 3900051080s3t27 + 974522235s2t28 + 156118290st29 + 12092581t30

### Formula 26:

• a = -359s12 - 1692s11t + 66s10t2 + 4620s9t3 + 16335s8t4 + 30888s7t5 + 27324s6t6 + 7128s5t7 + 4455s4t8 + 5940s3t9 + 5346s2t10 + 2268st11 + 81t12
• b = 20240s20 + 73980s19t + 562780s18t2 + 2664180s17t3 + 6887880s16t4 + 9699120s15t5 + 10218960s14t6 + 18426960s13t7 + 30944160s12t8 + 32277960s11t9 + 12004200s10t10 - 16272360s9t11 - 33611760s8t12 - 37305360s7t13 - 26224560s6t14 - 9972720s5t15 - 1108080s4t16 + 277020s3t17 + 277020s2t18 + 131220st19 + 29160t20
• c = -1525899s30 + 16253970s29t + 183283335s28t2 + 976109840s27t3 + 3225987405s26t4 + 8310854394s25t5 + 22076391375s24t6 + 56106334800s23t7 + 106221134025s22t8 + 135194463450s21t9 + 108742574115s20t10 + 59127148800s19t11 + 50324799825s18t12 + 103934441250s17t13 + 231658321275s16t14 + 427193650080s15t15 + 624299543775s14t16 + 699825237750s13t17 + 570253634325s12t18 + 299331190800s11t19 + 55336884615s10t20 - 59005588050s9t21 - 71878871475s8t22 - 47719465200s7t23 - 21048508125s6t24 - 5798257506s5t25 - 1018890495s4t26 - 273987360s3t27 - 111307365s2t28 - 33657930st29 - 4979799t30

### Formula 27:

• a = -295s12 + 204s11t + 3630s10t2 + 5500s9t3 + 12375s8t4 + 3960s7t5 - 4092s6t6 + 3960s5t7 + 12375s4t8 + 5500s3t9 + 3630s2t10 + 204st11 - 295t12
• b = 16514s20 + 83100s19t + 494000s18t2 + 541500s17t3 + 401850s16t4 + 2155056s15t5 + 5836800s14t6 + 342000s13t7 - 11485500s12t8 - 17438200s11t9 - 22360416s10t10 - 17438200s9t11 - 11485500s8t12 + 342000s7t13 + 5836800s6t14 + 2155056s5t15 + 401850s4t16 + 541500s3t17 + 494000s2t18 + 83100st19 + 16514t20
• c = 1506463s30 + 25129050s29t + 80181375s28t2 + 295423000s27t3 + 908299575s26t4 + 2205784602s25t5 + 969060375s24t6 - 3460860000s23t7 + 33766875s22t8 + 15646319250s21t9 + 32981112315s20t10 + 58931451000s19t11 + 100774111875s18t12 + 134729831250s17t13 + 90889171875s16t14 - 90889171875s14t16 - 134729831250s13t17 - 100774111875s12t18 - 58931451000s11t19 - 32981112315s10t20 - 15646319250s9t21 - 33766875s8t22 + 3460860000s7t23 - 969060375s6t24 - 2205784602s5t25 - 908299575s4t26 - 295423000s3t27 - 80181375s2t28 - 25129050st29 - 1506463t30

See 61 solutions with both a and b both less than 1012.

## a5 + b4 = c2

1. 35 + 114 = 1222

## a7 + b3 = c2

1. 27 + 173 = 712
2. 177 + 76 2713 = 21 063 9282

## a8 + b2 = c3

1. 338 + 1 549 0342 = 15 6133

## a8 + b3 = c2

1. 438 + 96 2223 = 30 042 9072

## Credits

The case x3 + y3 = z2 has been solved by Louis J. Mordell in his book "Diophantine equations", Academic Press, London-New York 1969.

Don Zagier found six parameterizations for x4 + y3 = z2. The last one was found by Johnny Edwards.

Some parametrizations of x5 + y3 = z2 are due to Don Zagier, Frits Beukers, and Steve Thiboutot. This is described in F.Beukers, Duke Math.J. 91 (1998), p61-88. A complete parameterization of this case was found by Johnny Edwards in 2001.

It was proved by Darmon and Granville in 1993 that if 1/p + 1/q + 1/r < 1 then the equation Axp + Byq + Czr = 0 has only finitely many solutions. See H. Darmon, A. Granville, On the equations zm = F(x,y) and Axp + Byq = Czr. Preprint 28 Volume II (1994), University of Georgia.

There are no other solutions for the exponent triple (2,3,8). It has been shown by Nils Bruin in Compositio Mathematicae 118 (1999) 305-321.

Last updated February 25th, 2011.