Computational Number Theory

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Here are computational Number Theory topics

            Second Powers    Third Powers    Fourth Powers    Fifth Powers    Sixth Powers    Seventh Powers
            Eighth Powers    Ninth Powers    Tenth Powers     Eleventh Powers Others          Publications



Update 2020.7.6

New!

           [401. 1/n=1/x1^2+1/x2^2+...1/xk^2  (2020.6.26)(2020.7.3)(2020.7.5)(2020.7.6)]

           [400. a^2X^6 + b^2Y^6 + 2abZ^3 = W^6  (2020.6.14)]

           [399. ax^4 + by^4 + cz^4 + dw^4 + eu^4= 0 Ⅱ  (2020.6.6)]

           [398. Table of solutions for ax^4 + by^4 + cz^4 + dw^4 = 0  (2020.5.28)]

           [397. Index of ax^4 + by^4 + cz^4 + dw^4 = 0, abcd=□  (2020.5.11)]

           [396. 4(m^2-1)X^4 + (1-m^2)(m^2+3)Y^4 = -(m^2-1)^2Z^4 + 4(m^2+3)W^4  (2020.5.11)]

           [395. (d^2-2dm^2+2m^4)X^4 + (d^2-2dm^2+2m^4)Y^4 = 2(-d+m^2)^2Z^4 + 2W^4  (2020.5.10)]

           [394. -dm^2(d+2)X^4 + (-2m^2+d+2)m^2(d-2m^2)Y^4 = d(d-2m^2)m^4Z^4 + (d+2)(2m^2-d-2)W^4  (2020.5.9)]

           [393. (m^4+1)X^4 + (m^4+1)Y^4 = 2Z^4 + 2W^4  (2020.5.7)]

           [392. 2mnX^4 + (m^2-n^2)Y^4 = 2mnZ^4 + (m^2-n^2)W^4  (2020.5.4)]

           [391. Index of ax^4 + by^4 + cz^4 + dw^4 = 0, abcd≠□  (2020.4.27)]

           [390. (m^2+1)^2(m^2-1)X^4 + (m^2-1)Y^4 + (m^2+1)Z^4 = 2m^2W^4  (2020.4.26)]

           [389. (d-2)(d+2)^2X^4 + (4d-8)Y^4 + Z^4 = (4d+8)W^4  (2020.4.25)]

           [388. pr^2X^4 + 4p(p+r)^2Y^4 + r^3Z^4 = r^2(p+r)W^4  (2020.4.24)]

           [387. 5X^4 + 45Y^4 + Z^4 = 6W^4  (2020.4.23)]

           [386. 2X^4 + 8Y^4 + Z^4 = 9W^4  (2020.4.22)]

           [385. 31X^4 + 31Y^4 + Z^4 = 2W^4  (2020.4.21)]

           [384. 7X^4 + 7Y^4 + 2Z^4 = 9W^4  (2020.4.20)]

           [383. 25X^4 + 4Y^4 + Z^4 = 5W^4  (2020.4.19)]

           [382. 79X^4 + 79Y^4 + 4Z^4 = 2W^4  (2020.4.18)]

           [381. 57X^4 + 57Y^4 = 8Z^4 + W^4 (2020.4.16)]

           [380. 18X^4 + 8Y^4 + Z^4 = 27W^4 (2020.4.15)]

           [379. 73X^4 + 73Y^4 + Z^4 = 2W^4 (2020.4.14)]

           [378. 75X^4 + 3Y^4 + 2Z^4 = 5W^4 (2020.4.13)]

           [377. pX^4 + pY^4 = 2Z^4 + W^4 revisited (2020.4.12)]

           [376. 33X^4 + 33Y^4 = 2Z^4 + W^4 (2020.4.9)]

           [375. 3X^4 + 3Y^4 = 2Z^4 + W^4 (2020.4.8)]

           [374. 9X^4 + 4Y^4 + 2Z^4 = 6W^4 (2020.4.6)]

           [373. d^2a^2X^4 + (-d+2a)^2a^2Y^4 + 2d(-d+2a)Z^4 = 4a^2W^4 (2020.4.4)]

           [372. 25X^4 + 25Y^4 + 8Z^4 = 25W^4 (2020.4.4)]

           [371. X^4 + 9Y^4 + 6Z^4 = W^4 (2020.4.4)]

           [370. 9X^4 + 36Y^4 + 4Z^4 = 9W^4 (2020.4.4)]

           [369. pX^4 + pY^4 + 2Z^4 = sW^4 (2020.4.2)]

           [368. 7X^4 + 7Y^4 + Z^4 = 8W^4 (2020.4.2)]

           [367. 47X^4 + 47Y^4 + 2Z^4 = 49W^4 (2020.4.1)]

           [366. 23X^4 + 23Y^4 + 2Z^4 = 25W^4 (2020.3.31)]

           [365. pX^4 + pY^4 = 2Z^4 + W^4 (2020.3.28)]

           [364. X^4 + 36Y^4 + 27Z^4 = 4W^4 (2020.3.27)]

           [363. 49X^4 + 49Y^4 + 8Z^4 = W^4 (2020.3.26)]

           [362. 3X^4 + 75Y^4 + 5Z^4 = 8W^4 (2020.3.25)]

           [361. 36X^4 + Y^4 + 3Z^4 = W^4 (2020.3.23)]

           [360. 165 = (-385495523231271884)^3 + 383344975542639445^3 + 98422560467622814^3 (2020.3.19)]

           [359. 25X^4 + Y^4 + 10Z^4 = W^4 (2020.3.18)]

           [358. 50X^4 + 2Y^4 + 5Z^4 = 2W^4 (2020.3.16)]

           [357. 27X^4 + 12Y^4 = 4Z^4 + 3W^4 (2020.3.14)]

           [356. 4X^4 + Y^4 + Z^4 = W^4 (2020.3.12)]

           [355. 31X^4 + 31Y^4 + 2Z^4 = W^4 (2020.3.10)]

           [354. 12X^4 + 12Y^4 = 8Z^4 + W^4 (2020.3.9)]

           [353. 19X^4 + 19Y^4 = 2Z^4 + W^4 (2020.3.8)]

           [352. 17X^4 + 17Y^4 = 18Z^4 + W^4 (2020.3.7)]

           [351. 9X^4 + Y^4 + 2Z^4 = 3W^4 (2020.3.6)]

           [350. 12X^4 + 3Y^4 + Z^4 = W^4 (2020.3.4)]

           [349. 9X^4 + 4Y^4 + 3Z^4 = W^4 (2020.3.2)]

           [348. 9X^4 + 9Y^4 = 2Z^4 + W^4 (2020.3.1)]

           [347. New solution: 4108 = 10378929361429825^3 + (-10007234208296573)^3 + 2*(-3869419050286010)^3 (2020.2.22)]
          
           [346. New solution: 5540 = (-9766714092174289)^3 + 9706646342356061^3 + 2*(2044177502818754)^3 (2020.2.22)]
          
           [345. New solution: 5540 = (-1274248491925945)^3 + 1094658539478551^3 + 2*(723458325298043)^3 (2020.2.22)]
          
           [344. New solution: 3676 = (36330027574363)^3 + (16201371125359)^3 + 2*(-29663565976595)^3 (2020.2.22)]
          
           [343. New solution: 3676 = (-1040743660046111)^3 + 1040742887519425^3 + 2*(10786916701381)^3 (2020.2.22)]

           [342. px^4 + py^4 = 2pu^4 + qv^4 (2020.2.22)]

           [341. y^2 = x^3 - px (2020.2.5)]

           [340. y^2 = x^3 + px (2020.2.5)]

           [339. x^4 + y^4 - 1 = z^2 Part 2 (2020.1.17)]

           [338. x^4 + y^4 - 1 = z^2 (2020.1.1)]

           [337. x1^n + x2^n + x3^n = y1^3 + y2^3 + y3^3 (2019.12.9)]

           [336. x1^6 + x2^6 + x3^6 = y1^3 + y2^3 + y3^3 (2019.12.6)]

           [335. x1^5 + x2^5 + x3^5 = y1^3 + y2^3 + y3^3 (2019.12.4)]

           [334. Primitive pythagorean triangles with same perimeter (2019.11.18)]

           [333. N-tuples of positive integers with the same sum and the same product (2019.10.26)]

           [332. Pythagorean triangles such that all triangles have same area (2019.10.2)]

           [331. (-74924259395610397)^3 + 72054089679353378^3 + 35961979615356503^3 = 906 (2019.9.24)]

           [330. 569936821221962380720^3 +(-569936821113563493509)^3 +(-472715493453327032)^3 = 3 (2019.9.19)]

           [329. (-80538738812075974)^3 + (80435758145817515)^3 + (12602123297335631)^3 = 42 (2019.9.14)]

           [328. Heron Triangles (2019.9.2)]

           [327. ax^4 + by^2 = c (2019.8.26)]

           [326. {a^4+b^4+c^4 = d^4+e^4+f^4 ,abc=def} (2019.8.19)]

           [325. {a^4+b^4+c^4+d^4 = e^4+f^4+g^4+h^4 ,abcd=efgh} Part 3 (2019.8.14)]

           [324. {a^4+b^4+c^4+d^4 = e^4+f^4+g^4+h^4 ,abcd=efgh} Part 2 (2019.8.12)]

           [323. {a^3+b^3+c^3 = d^3+e^3+f^3 , abc=def}  (2019.8.8)]

           [322. a^3x^6 + b^3y^6 + 2z^6 = 2w^6  (2019.7.30)]

           [321. x1^4 + 2nX2^4 = X3^8 + n^2X4^8  (2019.7.26)]

           [320. ax113 + bx123 + cx133 = ax213 + bx223 + cx233 = ... = axm13 + bxm23 + cxm33 (2019.7.24)]

           [319. Generalized Taxicab Number Part 5 (2019.7.17)]

           [318. Generalized Taxicab Number Part 4 (2019.7.9)]

           [317. Generalized Taxicab Number Part 3 (2019.7.2)]

           [316. X^3 + Y^3 + dZ^3 = 0 (2019.6.23)]

           [315. a/(b+c+d) +b/(a+c+d) +c/(a+b+d)+ d/(a+b+c) = n (2019.6.10)]

           [314. ax^3 + ay^3 + bz^3 + bw^3 = n (2019.5.21)]

           [313. ax^3 + by^3 + cz^3 = n (2019.5.14)]

           [312. x^5+y^5=cz^2 (2019.5.3)]

           [311. ax^3+by^3=cz^3 (2019.4.25)]

           [310. New solution: x^3+y^3+z^3=n (2019.3.24)(2019.3.18)(2019.3.17)]

           [309. 33 = 8866128975287528^3 -8778405442862239^3 -2736111468807040^3 (2019.3.13)]

           [308. ax^3 + by^3 = z^k (2019.2.27)]

           [307. ax^4+bx^2y^2+cy^4=dz^2 (2019.2.19)]

           [306. {ax^2+by^2=cs^2, ay^2+bx^2=ct^2} (2019.2.22)(2019.2.13)]

           [305. ax^4 + by^4 = cz^2 (2019.2.7)]

           [304. x^4 + y^4 = cz^2, 40000<=c<50000 (2019.1.30)(2019.1.28)]

           [303. x^4 + y^4 = cz^2, 30000<=c<40000 (2019.1.28)]

           [302. x^4 + y^4 = cz^2, 20000<=c<30000 (2019.1.23)]

           [301. x^4 + y^4 = cz^2, 10000<=c<20000 (2019.1.22)(2019.1.19)(2019.1.16)]

           [300. New solution: 7727 = 4677195736433^3 - 1373927732968^3 - 2*3680658132529^3 (2018.12.30)]

           [299. {ax^6 + by^6 + cz^6=u^6, bx^6 + cy^6 + az^6=v^6, cx^6 + ay^6 + bz^6=w^6} (2018.12.14)]

           [298. {ax^5 + by^5 + cz^5=u^5, bx^5 + cy^5 + az^5=v^5, cx^5 + ay^5 + bz^5=w^5} (2018.12.14)]

           [297. {ax^4 + by^4 + cz^4=u^4, bx^4 + cy^4 + az^4=v^4, cx^4 + ay^4 + bz^4=w^4} (2018.12.14)]

           [296. {ax^3 + by^3 + cz^3=u^3,bx^3 + cy^3 + az^3=v^3,cx^3 + ay^3 + bz^3=w^3} (2018.12.14)]

           [295. {ax^2 + by^2 + cz^2 = u^2, bx^2 + cy^2 + az^2 = v^2, cx^2 + ay^2 +bz^2 = w^2} (2018.12.13)]

           [294. ax^3 + by^3 + cz^3 + dw^3 = 0 (2018.11.28)]

           [293. Generalized Taxicab Number Part 2 (2018.11.17)(2011.2.1)]

           [292. 4x4 magic squares of squares (2018.11.6)]

           [291. A^3 + B^3 = n(C^3 + D^3)(2018.10.30)]

           [290. Magic hexagon(2018.10.25)]

           [289. m(x1^5+x2^5+x3^5+x4^5)=n(y1^5+y2^5+y3^5+y4^5)(2018.10.8)]

           [288. m(x1^4+x2^4+x3^4)=n(y1^4+y2^4+y3^4)(2018.10.6)]

           [287. m(x1^3+x2^3+x3^3+x4^3)=n(y1^3+y2^3+y3^3+y4^3) Part 2(2018.9.27)]

           [286. n^5=a^5+b^5+c^5-d^5-e^5-f^5-g^5 (2018.9.13)]

           [285. n^4+a^4+b^4=c^4+d^4+e^4 (2018.9.12)]

           [284. {A^2+B^2=C^2, Ap+Bq=Cr, Bp-Aq=C, p^2+q^2=r^2+1} (2018.9.10)]

           [283. m(x1^3+x2^3+x3^3+x4^3)=n(y1^3+y2^3+y3^3+y4^3) (2018.9.8)]

           [282. (x+y+z+w)(1/x+1/y+1/z+1/w) = n (2018.8.28)]

           [281. (x+y+z+w+s+t)(1/x+1/y+1/z+1/w+1/s+1/t) = n (2018.8.14)]

           [280. (x+y+z+w+s)(1/x+1/y+1/z+1/w+1/s) = n Part 2 (2018.8.1)]

           [279. (x+y+z+w+s)(1/x+1/y+1/z+1/w+1/s) = n (2018.7.26)(2018.7.22)]

           [278. x^4+mx^2y^2+y^4=z^2 (2018.7.14)]

           [277. { x^2+A=u^2, x^2-A=v^2 } Part 2 (2018.7.6)]

           [276. { x^2+A=u^2, x^2-A=v^2 } (2018.6.29)]

           [275. { A+x^2=u^2, A-x^2=v^2 } Part 2 (2018.6.26)]

           [274. { A+x^2=u^2, A-x^2=v^2 } (2018.6.23)]

           [273. New solution: 8968 = 1950774418348^3 + 987248618926^3 - 2*1612528225790^3 (2018.6.15)]

           [272. X^4 + Y^4 + Z^8 = 2W^4 (2018.6.6)]

           [271. New solution: 1084 = 10617017591719^3 + 2342686574875^3 - 2*8456801802575^3 (2018.5.23)]

           [270. New solution: 1975 = 36566656985380^3  - 35294370032999^3 -2*13506708241817^3 (2018.5.14)]

           [269. a^2X^8 + 2ab^4Y^4 + Z^8 = (a+b^4)^2W^4 (2018.5.5)]

           [268. New solution:  2372 = 3384145055^3 - 16870920115^3 + 2*13354335746^3 (2018.4.26)]
                              7862 = 10955159432^3 - 26911887070^3 + 2*20868465863^3

           [267. X^5 + aY^5 + aZ^5 = W^2 (2018.4.12)]

           [266. Sum of a square and a fourth power in n different ways (2018.4.4)(2018.3.28)(2018.3.24)]

           [265. x^n + y^n = z^(n-1) + w^(n-1) (2018.3.14)]

           [264. X^3+Y^3+Z^3-nXYZ = (n-3)(n^2+3n+9) (2018.3.6)]

           [263. x^7 + az^n = y^7 + aw^n, n=2,3,4,5,6,7 (2018.3.1)]

           [262. ax^4 + by^4 + cz^4 + dw^4 + eu^4= 0 (2018.2.2)]

           [261. New solution: 1121 = 975387209174^3+  493624309463^3 -2*806264112895^3(2018.1.8)]

           [260. X^3 + Y^3 + Z^3 + W^3 = 24n^2+2(2018.1.6)]

           [259. X^3 + Y^3 + Z^3 + W^3 = 3n^2+3n+1(2018.1.5)]

           [258. X^3 + Y^3 + Z^3 + W^3 = 1(2018.1.3)]

           [257. x(x+y)(x+2y)(x+3y) = u(u+v)(u+2v)(u+3v)(2017.12.20)]

           [256. ax1^4 + bx2^4 = ay1^4 + by2^4, 300<=a<400(2017.12.9)]

           [255. ax1^4 + bx2^4 = ay1^4 + by2^4, 200<=a<300(2017.11.17)]

           [254. ax1^4 + bx2^4 = ay1^4 + by2^4, 100<=a<200(2017.11.4)]

           [253. x^9 + az^8 = y^9 + aw^8(2017.10.24)]

           [252. x^9 + az^3 = y^9 + aw^3(2017.10.24)]

           [251. x^9 + az^2 = y^9 + aw^2(2017.10.24)]

           [250. x^8 + az^7 = y^8 + aw^7(2017.10.19)]

           [249. x^8 + az^2 = y^8 + aw^2(2017.10.19)]

           [248. x^8 + az^3 = y^8 + aw^3(2017.10.17)]

           [247. x^7 + az^3 = y^7 + aw^3(2017.10.12)]

           [246. x^7 + az^6 = y^7 + aw^6(2017.10.9)]

           [245. x^7 + az^2 = y^7 + aw^2(2017.10.9)]

           [244. x^6 + az^6 = y^6 + aw^6(2017.10.5)]

           [243. x^6 + az^4 = y^6 + aw^4(2017.9.26)]

           [242. x^5 + az^4 = y^5 + aw^4(2017.9.24)]

           [241. x^5 + az^3 = y^5 + aw^3(2017.9.24)]

           [240. x^5 + az^2 = y^5 + aw^2(2017.9.23)]

           [239. New solution:  788 =   4363964249747^3 - 4553874231847^3 + 2*1782603015764^3(2017.9.21)]
                              1975 =  4645912177207^3 - 4843138985528^3 + 2*1881489396298^3
                              2300 = -6452934831199^3 + 6730693690883^3 -2*2625897595264^3

           [238. New solution: 9463 = 13805037457^3 + 2*30636954619^3 - 39179981372^3(2017.9.15)]

           [237. x^6 + az^5 = y^6 + aw^5(2017.9.13)]

           [236. x^6 + az^2 = y^6 + aw^2(2017.9.11)]

           [235. x^6 + az^3 = y^6 + aw^3(2017.9.7)]

           [234. x1^7 + x2^7 + x3^7 + x4^7 = y1^7 + y2^7 + y3^7 + y4^7 + y5^7(2017.9.1)]

           [233. ax1^3 + bx2^3 + cx3^3 = ay1^n + by2^n + cy3^n(2017.8.21)]

           [232. New solution: 4819 = -7203186652517^3 + 7095691381372^3 + 2* 2019966858598^3(2017.8.9)]

           [231. Solutions of A^4 + hB^4 = C^4 + hD^4 Part 2(2017.8.2)(2017.7.5)(2013.9.12)(2013.8.17)]

           [230. Table of solutions of a^4 + b^4 + c^4 = d^4(2017.7.20)(2017.7.19)]

           [229. Table of solutions of a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4(2017.7.11)]

           [228. Solutions of A^4 + hB^4 = C^4 + hD^4 Part 2(2017.7.5)(2013.9.12)(2013.8.17)]

           [227. Solution of A^4 + hB^4 = C^4 + hD^4, 19000<=h<20000(2017.7.5)]

           [226. Solution of A^4 + hB^4 = C^4 + hD^4, 18000<=h<19000(2017.7.5)]

           [225. Solution of A^4 + hB^4 = C^4 + hD^4, 17000<=h<18000(2017.6.28)]

           [224. Solution of A^4 + hB^4 = C^4 + hD^4, 16000<=h<17000(2017.6.28)]

           [223. Solution of A^4 + hB^4 = C^4 + hD^4, 15000<=h<16000(2017.6.17)]

           [222. Solution of A^4 + hB^4 = C^4 + hD^4, 14000<=h<15000(2017.6.17)]

           [221. Solution of A^4 + hB^4 = C^4 + hD^4, 13000<=h<14000(2017.6.7)]

           [220. Solution of A^4 + hB^4 = C^4 + hD^4, 12000<=h<13000(2017.6.7)]

           [219. Solution of A^4 + hB^4 = C^4 + hD^4, 11000<=h<12000(2017.5.26)]

           [218. Solution of A^4 + hB^4 = C^4 + hD^4, 10000<=h<11000(2017.5.26)]

           [217. Solution of A^4 + hB^4 = C^4 + hD^4, 9000<=h<10000(2017.5.14)]

           [216. Solution of A^4 + hB^4 = C^4 + hD^4, 8000<=h<9000(2017.5.5)]

           [215. Solutions of A^4 + hB^4 = C^4 + hD^4 Part 2(2017.5.8)(2017.5.3)(2013.9.12)(2013.8.17)]

           [214. Solution of A^4 + hB^4 = C^4 + hD^4, 7000<=h<8000(2017.4.28)]

           [213. Solution of A^4 + hB^4 = C^4 + hD^4, 6000<=h<7000(2017.4.22)]

           [212. Solution of A^4 + hB^4 = C^4 + hD^4, 5000<=h<6000(2017.4.18)]

           [211. ax1^4 + bx2^4 = ay1^4 + by2^4(2017.4.5)(2017.4.1)]

           [210. a(1)x1^n + a(2)x2^n +...+ a(n)xn^n=0(2017.3.16)]

           [209. {x+y+z=n, x^3+y^3+z^3=n}(2017.3.1)]

           [208. X^n + Y^n + Z^n = Square for n=1,2,3(2017.2.26)]

           [207. {x^2+y^2+z^2=u^2, x^3+y^3+z^3=v^3}(2017.2.5)]

           [206. Solutions of A^4 + hB^4 = C^4 + hD^4 Part 3(2016.12.18)]

           [205. Solution of A^4 + hB^4 = C^4 + hD^4, 4000<=h<5000 (2016.11.24)]

           [204. Solution of A^4 + hB^4 = C^4 + hD^4, 3000<=h<4000 (2016.11.4)]

           [203. Solution of A^4 + hB^4 = C^4 + hD^4, 2000<=h<3000 (2016.10.24)]

           [202. Solution of A^4 + hB^4 = C^4 + hD^4, 1000<=h<2000 (2016.10.15)]

           [201. {y^2+z^2-x^2=p^2, x^2+z^2-y^2=q^2 , x^2+y^2-z^2=r^2} (2016.9.27)]

           [200. {x^2+yz=u^2, y^2+xz=v^2, z^2+xy=w^2} (2016.9.2)]

           [199. {x^2-y^2=u^2, x^2-z^2=v^2, y^2-z^2=w^2} (2016.8.12)]

           [198. Concordant number (2016.8.4)]

           [197. {a^2+b^2=x^2, a^2+c^2=y^2, b^2+c^2=z^2} (2016.7.22)]

           [196. X1^5 + X2^5 +...+ Xn^5=0, n is even and n > 5 (2016.7.12)]

           [195. X1^3 + X2^3 +...+ Xn^3=0, n is odd and n >= 5 (2016.7.1)]

           [194. X1^3 + X2^3 +...+ Xn^3=0, n is even and n > 3 (2016.6.16)]

           [193. {A + B = p^2, A + C = q^2, B + D = s^2, C + D = t^2, B + C = u^2, A + D = v^2} (2016.6.1)]

           [192. Congruent number (2016.5.6)]

           [191. x^4 - y^4 = cz^2 (2016.4.19)]

           [190. a + b + c = abc = N (2016.4.1)]

           [189. a^4 + b^4 + c^4 + d^4 = e^4 + f^4 + g^4 + h^4, abcd=efgh (2016.3.20)]

           [188. x^5 + y^3 = z^5 + w^3 (2016.1.29)]

           [187. px^n + qy^3 = qz^m + pw^3 (2016.1.3)]

           [186. X^4 + Y^4 + Z^4 + W^4 = T^n, n=5,6,7,9 (2015.11.19)]

           [185. X^4 + Y^4 + Z^4 + W^4 = T^3 (2015.11.4)]

           [184. X^4 + Y^4 + Z^4 + W^4 = T^2 Part 2 (2015.10.9)]

           [183. X^4 + Y^4 + Z^4 + W^4 = T^2 (2015.10.2)]

           [182. a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4 Part 3  (2015.8.27)(2015.8.25)(2015.8.13)(2015.8.4)]

           [181. a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4 Part 2  (2015.7.19)]

           [180. px^6 + qy^3 = qz^6 + pw^3  (2015.7.1)]

           [179. ax1^n + bx2^n + cx3^n = ay1^n + by2^n + cy3^n, n=2,6  (2015.6.1)]

           [178. a(1)x1^n + a(2)x2^n +...+ a(k)xk^n = a(1)y1^n + a(2)y2^n +...+ a(k)yk^n  (2015.5.1)]

           [177. a6x^6+a5x^5y+a4x^4y^2+a3x^3y^3+a2x^2y^4+a1xy^5+ a0y^6 = a6u^6+a5u^5v+a4u^4v^2+a3u^3v^3+a2u^2v^4+a1uv^5+ a0v^6  (2015.4.2)]

           [176. 4036 = 309317704627^3 - 315184953527^3 +   2*95027443732^3  (2015.3.22)]

           [175. a4x^4+a3x^3y+a2x^2y^2+a1xy^3+a0y^4 = a4u^4+a3u^3v+a2u^2v^2+a1uv^3+a0v^4  (2015.3.5)]
          
           [174. 9724 =  20018592691^3 -  26097822863^3 +   2*16957680280^3  (2015.2.17)]
          
           [173. Arithmetic progressions on quartic curves  (2015.2.7)]
          
          
           [172. Arithmetic progressions on cubic curves  (2015.2.7)]
          
           [171. Arithmetic progressions on quadratic curves  (2015.2.7)]
          
           [170. ax^8 + by^8 + cz^4 + dw^4 = 0  (2015.2.1)]
          
           [169. Happy New Year!  (2015.1.1)]
          
           [168. ax^6 + by^6 + cz^6 + dw^3 = 0  (2014.12.3)]
          
           [167. w^2 = x^6 + y^6 + z^6 Part 2  (2014.11.19)]
          
           [166. w^2 = x^6 + y^6 + z^6  (2014.11.18)]
          
           [165. X^n + Y^n = square for n=1, 2, 3  (2014.11.11)]
          
           [164. ax^6 + by^6 + cz^6 = w^2 and ax^6 + by^6 + cz^6 = w^3  (2014.10.10)]
          
           [163. x^3 + y^3 + z^3 = n  (2014.9.9)]
          
           [162. ax^5 + by^5 + cz^5 = w^2 Part 2  (2014.8.19)]
          
           [161. ax^6 + by^6 + cz^6 = w^2  (2014.8.7)]
          
           [160. ax^5 + by^5 + cz^5 = w^2  (2014.7.28)]
          
           [159. ax^4 + by^4 + cz^4 = w^2  (2014.7.22)]
          
           [158. x^2 + y^2 = z^n + w^n and x^3 + y^3 = z^n + w^n  (2014.7.11)]
          
           [157. Solution of ax^4 + by^4 + cz^4 + dw^4 = 0 Part 2  (2014.7.4)]
          
           [156. Equal Sums of Sixth Powers  (2014.6.1)]
          
           [155. Solution of ax^4 + by^4 + cz^4 + dw^4 = 0  (2014.5.1)]
          
           [154. Any rational number N is the sum and difference of four rational fourth powers
               in an infinite number of ways Part 2  (2014.4.5)]
               
           [153. Equal Sums of Eighth Powers  Part 5(2014.3.3)]
          
           [152. Solutions  of  x^3 + y^3 + z^3 = t^n  Part 2(2014.1.31)]
          
           [151. Happy New Year!  (2014.1.1)]
          
           [150. Solutions of sextic diophantine equation(2013.11.1)]
          
           [149. Solutions of x^3 + dy^3 + d^2z^3 - 3dxyz = 1(2013.9.28)]
          
           [148. Solutions of A^4 + hB^4 = C^4 + hD^4 Part 2(2013.9.12)(2013.8.17)]
          
           [147. Solutions of A^4 + hB^4 = C^4 + hD^4  (2013.8.1)]
          
           [146. Solution of X^k + Y^k = {U^2,V^4} for k=1,2  (2013.7.12)]
          
           [145. Three positive numbers such that the sum and difference of any two are squares  (2013.6.14)]
          
           [144. Four positive numbers such that the sum of any two is a cube  (2013.5.15)]
          
           [143. Solutions of x^4 + ax^2y^2 + y^4 = u^4 + au^2v^2 + v^4  (2013.5.1)]
          
           [142. Solutions of x^4 + ay^4 = z^4 + bt^4  (2013.4.1)]
          
           [141. Solutions of x^2 + y^4 = 5z^4  (2013.3.1)]
          
           [140. A parametric solution of X^k + Y^k + Z^k + W^k + S^k = {U^2,V^4} for k=2,4  (2013.2.1)]
          
           [139. Happy New Year!  (2013.1.1)]
          
           [138. Equal Sums of Seventh Powers  (2012.11.30)]
          
           [137. Many solutions of n=x^3 + y^3 + 2z^3  (2012.10.18)(2012.9.24)]
          
           [136. Solutions of ax^6 + by^3 = cz^2  (2012.9.10)]
          
           [135. Equal Sums of Eighth Powers  Part 4  (2012.8.10)]
          
           [134. New solution:  4388 = -18105946705^3 + 18108094211^3 - 2*1018372219^3  (2012.7.12)]
          
           [133. Solutions  of  x^4 + y^4 + z^4 = w^2  (2012.7.2)]
          
           [132. Equal Sums of Eighth Powers  Part 3  (2012.6.11)]
          
           [131. A collection of the parametric solutions for equal sums of sixth powers  (2012.5.3)]
          
           [130. A collection of the parametric solutions for equal sums of fifth powers  (2012.4.2)]
          
           [129. Equal Sums of Fifth Powers  (2012.3.2)]
          
           [128. Equal sums and equal products  (2012.2.1)]
          
           [127. Happy New Year!  (2012.1.3)]
          
           [126. Any non-zero rational number m is the sum of three rational cubes
               in an infinite number of non-trivial ways.  (2011.12.6)]
               
           [125. New parametric solutions of x^3 + y^3 + z^3 = t^2  (2011.11.16)]
          
           [124. Equal Sums of Eighth Powers et 1  (2011.10.11)]
          
           [123. Equal Sums of Fourth Powers et 1  (2011.9.13)]
          
           [122. Equal Sums of Fifth Powers et 1  (2011.8.23)]
          
           [121. Equal Sums of Sixth Powers et 1  (2011.8.1)]
          
           [120. Equal Sums of Seventh Powers et 1  (2011.7.18)]
          
           [119. Equal Sums of Eighth Powers  (2011.7.1)]
          
           [118. Solutions  of  x^4 + y^4 + z^4 = w^n  (2011.6.3)]
          
           [117. EQUAL SUMS OF ELEVENTH POWERS  (2011.5.6)]
          
           [116. Solutions  of  x^4 + y^4 + z^4 = nw^3  (2011.4.1)]
          
           [115. EQUAL SUMS OF SEVENTH POWERS  (2011.3.4)]
          
           [114. New solution:  3047 =  234115946^3  -  81713665^3 - 2* 183146068^3  (2011.2.16)]
          
           [113. Generalized Taxicab Number Part 2  (2011.2.1)]
          
           [112. Happy New Year!  (2011.1.1)]
          
          
          
          


Topics

 1: Second Powers

73. N=5,6,and 7 mod 8 is always a congruent number (2008.6.1)

145. Three positive numbers such that the sum and difference of any two are squares (2013.6.14)

158. x^2 + y^2 = z^n + w^n and x^3 + y^3 = z^n + w^n (2014.7.11)

171. Arithmetic progressions on quadratic curves (2015.2.7)

192. Congruent number (2016.5.6)

193. {A + B = p^2, A + C = q^2, B + D = s^2, C + D = t^2, B + C = u^2, A + D = v^2} (2016.6.1)

197. {a^2+b^2=x^2, a^2+c^2=y^2, b^2+c^2=z^2} (2016.7.22)

198. Concordant number (2016.8.4)

199. {x^2-y^2=u^2, x^2-z^2=v^2, y^2-z^2=w^2} (2016.8.12)

200. {x^2+yz=u^2, y^2+xz=v^2, z^2+xy=w^2} (2016.9.2)

201. {y^2+z^2-x^2=p^2, x^2+z^2-y^2=q^2 , x^2+y^2-z^2=r^2} (2016.9.27)

207. {x^2+y^2+z^2=u^2, x^3+y^3+z^3=v^3} (2017.2.5)

274. { A+x^2=u^2, A-x^2=v^2 } (2018.6.23)

275. { A+x^2=u^2, A-x^2=v^2 } Part2 (2018.6.26)

276. { x^2+A=u^2, x^2-A=v^2 } (2018.6.29)

277. { x^2+A=u^2, x^2-A=v^2 } Part 2 (2018.7.6)

284. {A^2+B^2=C^2, Ap+Bq=Cr, Bp-Aq=C, p^2+q^2=r^2+1} (2018.9.10)

295. {ax^2 + by^2 + cz^2 = u^2, bx^2 + cy^2 + az^2 = v^2, cx^2 + ay^2 +bz^2 = w^2} (2018.12.13)

306. {ax^2+by^2=cs^2, ay^2+bx^2=ct^2} (2019.2.13)

401. 1/n=1/x1^2+1/x2^2+...1/xk^2 (2020.6.25)

 2: Third Powers

1.  Cubic residue and x3+y3+z3=d(2005.4.2)

3.  Ramanujan's cubic identity(2005.4.26)(2006.5.10)

31. A13+A23+A33+...+An-13=An3(Ggeneralized Ramanujan's cubic identity) (2006.5.21)

32. 31116963 is expressed as a sum of three positive cubes by 20 ways (2006.5.27)

68. Sums of three cubes (2008.2.7)(2008.1.15)(2008.1.8)(2008.1.5)

71. 4 + 27 = 31!? (2008.5.12)

75. New solutions of n=x3+y3+2z3 (2008.7.31)(2008.7.13)(2008.6.22)

81. Every integer is a sum of four cubes of integers? (2008.11.1)

82. Table of the solutions for x3+by3+cz3=0. (2008.12.3)

102. Solutions of x3 + y3 + z3 = tn (2010.5.1) 103. 3956= 3481598477^3 - 1323591649^3 - 2* 2711779804^3 (2010.5.11) 8779= 64325766916^3 - 11325451823^3 - 2* 50962343435^3

104. Solutions of X^3 + Y^3 + Z^3 = mW^3 (2010.6.1)

114. New solution: 3047 = 234115946^3 - 81713665^3 - 2* 183146068^3 (2011.2.16)

125. New parametric solutions of x^3 + y^3 + z^3 = t^2 (2011.11.16)

126. Any non-zero rational number m is the sum of three rational cubes in an infinite number of non-trivial ways. (2011.12.6)

128. Equal sums and equal products (2012.2.1)

134. New solution: 4388 = -18105946705^3 + 18108094211^3 - 2*1018372219^3 (2012.7.12)

137. Many solutions of n=x^3 + y^3 + 2z^3 (2012.10.18)(2012.9.24)

144. Four positive numbers such that the sum of any two is a cube (2013.5.15)

149. Solutions of x^3 + dy^3 + d^2z^3 - 3dxyz = 1 (2013.9.28)

152. Solutions of x^3 + y^3 + z^3 = t^n Part 2 (2014.1.31)

158. x^2 + y^2 = z^n + w^n and x^3 + y^3 = z^n + w^n (2014.7.11)

163. x^3 + y^3 + z^3 = n (2017.2.15)(2014.9.9)

165. X^n + Y^n = square for n=1, 2, 3 (2014.11.1)

172. Arithmetic progressions on cubic curves (2015.2.7)

174. 9724 = 20018592691^3 - 26097822863^3 + 2*16957680280^3 (2015.2.17)

176. 4036 = 309317704627^3 - 315184953527^3 + 2*95027443732^3 (2015.3.22)

180. px^6 + qy^3 = qz^6 + pw^3 (2015.7.1)

187. px^n + qy^3 = qz^m + pw^3 (2016.1.3)

188. x^5 + y^3 = z^5 + w^3 (2016.1.29)

190. a + b + c = abc = N (2016.4.1)

194. X1^3 + X2^3 +...+ Xn^3=0, n is even and n > 3 (2016.6.16)

195. X1^3 + X2^3 +...+ Xn^3=0, n is odd and n >= 5 (2016.7.1)

207. {x^2+y^2+z^2=u^2, x^3+y^3+z^3=v^3} (2017.2.5)

208. X^n + Y^n + Z^n = Square for n=1,2,3 (2017.2.26)

209. {x+y+z=n, x^3+y^3+z^3=n} (2017.3.1)

232. New solution: 4819 = -7203186652517^3 + 7095691381372^3 + 2* 2019966858598^3 (2017.8.9)

233. ax1^3 + bx2^3 + cx3^3 = ay1^n + by2^n + cy3^n (2017.8.21)

238. New solution: 9463 = 13805037457^3 + 2*30636954619^3 - 39179981372^3(2017.9.15)

239. New solution: 788 = 4363964249747^3 - 4553874231847^3 + 2*1782603015764^3(2017.9.21) 1975 = 4645912177207^3 - 4843138985528^3 + 2*1881489396298^3 2300 = -6452934831199^3 + 6730693690883^3 -2*2625897595264^3

258. X^3 + Y^3 + Z^3 + W^3 = 1 (2018.1.3)

259. X^3 + Y^3 + Z^3 + W^3 = 3n^2+3n+1 (2018.1.5)

260. X^3 + Y^3 + Z^3 + W^3 = 24n^2+2 (2018.1.6)

261. New solution: 1121 = 975387209174^3+ 493624309463^3 -2*806264112895^3 (2018.1.8)

264. X^3+Y^3+Z^3-nXYZ = (n-3)(n^2+3n+9) (2018.3.6)

283. m(x1^3+x2^3+x3^3+x4^3)=n(y1^3+y2^3+y3^3+y4^3) (2018.9.8)

287. m(x1^3+x2^3+x3^3+x4^3)=n(y1^3+y2^3+y3^3+y4^3) Part 2 (2018.9.27)

291. A^3 + B^3 = n(C^3 + D^3) (2018.10.30)

294. ax^3 + by^3 + cz^3 + dw^3 = 0 (2018.11.28)

296. {ax^3 + by^3 + cz^3=u^3,bx^3 + cy^3 + az^3=v^3,cx^3 + ay^3 + bz^3=w^3} (2018.12.14)

300. New solution: 7727 = 4677195736433^3 - 1373927732968^3 - 2*3680658132529^3 (2018.12.30)

308. ax^3 + by^3 = z^k (2019.2.27)

309. 33 = 8866128975287528^3 -8778405442862239^3 -2736111468807040^3 (2019.3.13)

310. New solution: x^3+y^3+z^3=n (2019.3.24)(2019.3.18)(2019.3.17)

311. ax^3+by^3=cz^3 (2019.4.25)

313. ax^3 + by^3 + cz^3 = n (2019.5.14)

314. ax^3 + ay^3 + bz^3 + bw^3 = n (2019.5.21)

315. a/(b+c+d) +b/(a+c+d) +c/(a+b+d)+ d/(a+b+c) = n (2019.6.10)

316. X^3 + Y^3 + dZ^3 = 0 (2019.6.23)

320. ax113 + bx123 + cx133 = ax213 + bx223 + cx233 = ... = axm13 + bxm23 + cxm33 (2019.7.24)

323. {a^3+b^3+c^3 = d^3+e^3+f^3 , abc=def} (2019.8.8)

335. x1^5 + x2^5 + x3^5 = y1^3 + y2^3 + y3^3 (2019.12.4)

336. x1^6 + x2^6 + x3^6 = y1^3 + y2^3 + y3^3 (2019.12.6)

337. x1^n + x2^n + x3^n = y1^3 + y2^3 + y3^3 (2019.12.9)

340. y^2 = x^3+px (2020.2.5)

341. y^2 = x^3-px (2020.2.5)

347. New solution: 4108 = 10378929361429825^3 + (-10007234208296573)^3 + 2*(-3869419050286010)^3 (2020.2.22)]

346. New solution: 5540 = (-9766714092174289)^3 + 9706646342356061^3 + 2*(2044177502818754)^3 (2020.2.22)]

345. New solution: 5540 = (-1274248491925945)^3 + 1094658539478551^3 + 2*(723458325298043)^3 (2020.2.22)]

344. New solution: 3676 = (36330027574363)^3 + (16201371125359)^3 + 2*(-29663565976595)^3 (2020.2.22)]

343. New solution: 3676 = (-1040743660046111)^3 + 1040742887519425^3 + 2*(10786916701381)^3 (2020.2.22)]

 3: Fourth Powers

6.  A4+ B4+ C4= D4(Counterexample of Euler's conjecture) (2005.5.22)(2005.6.13)

8.  A4+ B4+ C4+ D4+ E4= F4(Ramanujan's identities) (2005.6.19)

9.  A4+ B4+ C4 = D4+ E4+ F4 (2005.6.25)

22. A4+ B4+ C4 = D4+ E4+ F4 PART 2  (2006.2.1)

23. A4+ B4+ C4+ D4 = E4+ F4 (2006.2.12)

24. A4+ B4+ C4 = D4+ E4 (2006.2.26)

25. A14+A24+A34+...+An4=B14+B24+B34+...+Bn4(May the Fourth with you!) (2006.3.19)

30. Representation by fourth powers (2006.5.2)

33. A4+ B4+ C4+ D4+ E4= F4(Ramanujan's identities, again) (2006.6.10)

34. A14+A24+A34+...+An-14=An4(Ggeneralized Ramanujan's 4th power identity) (2006.6.22)

35. a14+b14+c14 = a24+b24+c24 =..... = an4+bn4+cn4 (2006.6.22)

49. Current status of the solutions for A4=B4+C4+D4 (2007.1.26)

66. 58219814004+153558313604+1409765514=154345478014 (2007.10.24)

72. 49875884196554+24804526756004+5020388539764=50622976992574(2008.5.15)

76. 478867402721149764 + 88134256704402404 + 568278133081117854 = 629405169034106014 (2008.8.13)

35790871473754404 +148900264334684714 + 185659451142167204 = 202495067095797214 (2008.8.13)

77. 12597684734=11667058404+8593964554+5889033364 (2008.9.1)

78. Table of the solutions for A4+B4+C4=D4 (2008.9.1)

79. 1300643009914004+ 4408049425801604+ 5148181012992894= 5736463218719614 (2008.9.15)

80. Table of the solutions for A4+B4+C4=D4 Part 2 (2008.10.5) 8738221214=7693212804+6067108714+5584244404 17878823374=16629976634+12377969604+6863980004 18717138574=15935130804+15535564404+926224014

84. A4=B4+C4+D4+E4 (2009.2.10)

85. x4+y4+z4 = nw4 (2009.3.1)

86. A4+ B4+ C4+ D4+ E4= F4 (Ramanujan's identities, again x 2) (2009.3.16)

87. Representation by fourth powers (2009.5.1)

88. 9033964577482532388059482429398457291004947925005743028147465732645880^4 + 4417264698994538496943597489754952845854672497179047898864124209346920^4 + 1439965710648954492268506771833175267850201426615300442218292336336633^4 = 9161781830035436847832452398267266038227002962257243662070370888722169^4 (2009.6.1)

89. Representation by fourth powers Part 2 (2009.6.15)

91. Solutions of a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4 (2009.7.30)(2008.7.12)

93. Solutions of X4 + Y4 = 4481Z2. (2009.11.1)

97. Ramanujan's identity (4x^5-5x)^4 + (6x^4-3)^4 + (4x^4+1)^4 - (4x^5+x)^4 - (2x^4-1)^4 = 81 (2010.3.3)

98. 2 = (8x^5-2x)^4 + (8x^4+1)^4 + (8x^4-1)^4 - (8x^5+2x)^4 - (4x^2)^4 (2010.3.10)

99. 1 = (4x^4 + 1)^4 -(4x^4)^4 -(4x^3)^4 -(2x)^4 -(3x^2)^4 -(2x^2)^4 +(x^2)^4 (2010.3.26)

100. x^4 + ay^4 = z^4 + bt^4 (2010.3.31)

101. A4+ B4+ C4 = D4+ E4 Part 2 (2010.4.10)

108. Solutions of x^4 + y^4 + z^4 = nw^2 (2010.9.1)

105. Solutions of x^4 + y^4 = cz^2 (2010.7.1)

107. Any rational number N is the sum and difference of four rational fourth powers in an infinite number of ways. (2010.8.1)

110. x1^4 + x2^4 +...+ xn^4 = nw^2 (2010.11.1)

111. Generalized Taxicab Number (2010.12.3)

116. Solutions of x^4 + y^4 + z^4 = nw^3 (2011.4.1)

118. Solutions of x^4 + y^4 + z^4 = w^n (2011.6.3)

123. Equal Sums of Fourth Powers et 1 (2011.9.13)

133. Solutions of x^4 + y^4 + z^4 = w^2 (2012.7.2)

140. A parametric solution of X^k + Y^k + Z^k + W^k + S^k = {U^2,V^4} for k=2,4 (2013.2.1)

141. Solutions of x^2 + y^4 = 5z^4 (2013.3.1)

142. Solutions of x^4 + ay^4 = z^4 + bt^4 (2013.4.1)

143. Solutions of x^4 + ax^2y^2 + y^4 = u^4 + au^2v^2 + v^4 (2013.5.1)

146. Solution of X^k + Y^k = {U^2,V^4} for k=1,2 (2013.7.12)

147. Solutions of A^4 + hB^4 = C^4 + hD^4 (2013.8.1)

148. Solutions of A^4 + hB^4 = C^4 + hD^4 Part 2 (2013.8.17)

154. Any rational number N is the sum and difference of four rational fourth powers in an infinite number of ways Part 2 (2014.4.5)

155. Solution of ax^4 + by^4 + cz^4 + dw^4 = 0 (2014.5.1)

157. Solution of ax^4 + by^4 + cz^4 + dw^4 = 0 Part 2 (2014.7.4)

159. ax^4 + by^4 + cz^4 = w^2 (2014.7.22)

173. Arithmetic progressions on quartic curves (2015.2.7)

175. a4x^4+a3x^3y+a2x^2y^2+a1xy^3+a0y^4 = a4u^4+a3u^3v+a2u^2v^2+a1uv^3+a0v^4 (2015.3.5)

177. a6x^6+a5x^5y+a4x^4y^2+a3x^3y^3+a2x^2y^4+a1xy^5+ a0y^6 = a6u^6+a5u^5v+a4u^4v^2+a3u^3v^3+a2u^2v^4+a1uv^5+ a0v^6 (2015.4.2)

181. a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4 Part 2 (2015.7.19)

182. a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4 Part 3 (2015.8.27)(2015.8.25)(2015.8.13)(2015.8.4)

183. X^4 + Y^4 + Z^4 + W^4 = T^2 (2015.10.2)

184. X^4 + Y^4 + Z^4 + W^4 = T^2 Part 2 (2015.10.9)

185. X^4 + Y^4 + Z^4 + W^4 = T^3 (2015.11.4)

186. X^4 + Y^4 + Z^4 + W^4 = T^n, n=5,6,7,9 (2015.11.19)

189. a^4 + b^4 + c^4 + d^4 = e^4 + f^4 + g^4 + h^4, abcd=efgh (2016.3.20)

191. x^4 - y^4 = cz^2 (2016.4.19)

202. Solution of A^4 + hB^4 = C^4 + hD^4, 1000<=h<2000 (2016.10.15)

203. Solution of A^4 + hB^4 = C^4 + hD^4, 2000<=h<3000 (2016.10.24)

204. Solution of A^4 + hB^4 = C^4 + hD^4, 3000<=h<4000 (2016.11.4)

205. Solution of A^4 + hB^4 = C^4 + hD^4, 4000<=h<5000 (2016.11.24)

206. Solutions of A^4 + hB^4 = C^4 + hD^4 Part 3 (2016.12.18)

211. ax1^4 + bx2^4 = ay1^4 + by2^4(2017.4.5)(2017.4.1)

212. Solution of A^4 + hB^4 = C^4 + hD^4, 5000<=h<6000(2017.4.18)

213. Solution of A^4 + hB^4 = C^4 + hD^4, 6000<=h<7000(2017.4.22)

214. Solution of A^4 + hB^4 = C^4 + hD^4, 7000<=h<8000(2017.4.28)

216. Solution of A^4 + hB^4 = C^4 + hD^4, 8000<=h<9000(2017.5.5)

217. Solution of A^4 + hB^4 = C^4 + hD^4, 9000<=h<10000(2017.5.14)

218. Solution of A^4 + hB^4 = C^4 + hD^4, 10000<=h<11000(2017.5.26)

219. Solution of A^4 + hB^4 = C^4 + hD^4, 11000<=h<12000(2017.5.26)

220. Solution of A^4 + hB^4 = C^4 + hD^4, 12000<=h<13000(2017.6.7)

221. Solution of A^4 + hB^4 = C^4 + hD^4, 13000<=h<14000(2017.6.7)

222. Solution of A^4 + hB^4 = C^4 + hD^4, 14000<=h<15000(2017.6.17)

223. Solution of A^4 + hB^4 = C^4 + hD^4, 15000<=h<16000(2017.6.17)

224. Solution of A^4 + hB^4 = C^4 + hD^4, 16000<=h<17000(2017.6.28)

225. Solution of A^4 + hB^4 = C^4 + hD^4, 17000<=h<18000(2017.6.28)

226. Solution of A^4 + hB^4 = C^4 + hD^4, 18000<=h<19000(2017.7.5)

227. Solution of A^4 + hB^4 = C^4 + hD^4, 19000<=h<20000(2017.7.5)

229. Table of solutions of a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4(2017.7.11)

230. Table of solutions of a^4 + b^4 + c^4 = d^4(2017.7.19)

254. ax1^4 + bx2^4 = ay1^4 + by2^4, 100<={a,b}<200(2017.11.4)

255. ax1^4 + bx2^4 = ay1^4 + by2^4, 200<={a,b}<300(2017.11.17)

256. ax1^4 + bx2^4 = ay1^4 + by2^4, 300<={a,b}<400(2017.12.9)

257. x(x+y)(x+2y)(x+3y) = u(u+v)(u+2v)(u+3v)(2017.12.20)

262. ax^4 + by^4 + cz^4 + dw^4 + eu^4= 0 (2018.2.2)

266. Sum of a square and a fourth power in n different ways (2018.3.24)

278. x^4+mx^2y^2+y^4=z^2 (2018.7.14)

285. n^4+a^4+b^4=c^4+d^4+e^4 (2018.9.12)

288. m(x1^4+x2^4+x3^4)=n(y1^4+y2^4+y3^4)(2018.10.6)

297. ax^4 + by^4 + cz^4=u^4, bx^4 + cy^4 + az^4=v^4, cx^4 + ay^4 + bz^4=w^4(2018.12.14)

301. x^4 + y^4 = cz^2, 10000<=c<20000(2019.1.19)(2019.1.16)

302. x^4 + y^4 = cz^2, 20000<=c<30000(2019.1.23)

303. x^4 + y^4 = cz^2, 30000<=c<40000(2019.1.28)

304. x^4 + y^4 = cz^2, 40000<=c<50000(2019.1.30)(2019.1.28)

305. ax^4 + by^4 = cz^2(2019.2.7)

307. ax^4+bx^2y^2+cy^4=dz^2(2019.2.19)

317. Generalized Taxicab Number Part 3(2019.7.2)

318. Generalized Taxicab Number Part 4(2019.7.9)

324. {a^4+b^4+c^4+d^4 = e^4+f^4+g^4+h^4 ,abcd=efgh} Part 2(2019.8.12)

325. {a^4+b^4+c^4+d^4 = e^4+f^4+g^4+h^4 ,abcd=efgh} Part 3(2019.8.14)

326. {a^4+b^4+c^4 = d^4+e^4+f^4 ,abc=def} (2019.8.19)

327. ax^4 + by^2 = c (2019.8.26)

338. x^4 + y^4 - 1 = z^2 (2020.1.1)

339. x^4 + y^4 - 1 = z^2 Part 2(2020.1.17)

342. px^4 + py^4 = 2pu^4 + qv^4(2020.2.22)

348. 9X^4 + 9Y^4 = 2Z^4 + W^4(2020.3.1)

349. 9X^4 + 4Y^4 + 3Z^4 = W^4(2020.3.2)

350. 12X^4 + 3Y^4 + Z^4 = W^4(2020.3.4)

351. 9X^4 + Y^4 + 2Z^4 = 3W^4(2020.3.6)

352. 17X^4 + 17Y^4 = 18Z^4 + W^4(2020.3.7)

353. 19X^4 + 19Y^4 = 2Z^4 + W^4(2020.3.8)

354. 12X^4 + 12Y^4 = 8Z^4 + W^4(2020.3.9)

355. 31X^4 + 31Y^4 + 2Z^4 = W^4(2020.3.10)

356. 4X^4 + Y^4 + Z^4 = W^4(2020.3.12)

357. 27X^4 + 12Y^4 = 4Z^4 + 3W^4(2020.3.14)

358. 50X^4 + 2Y^4 + 5Z^4 = 2W^4(2020.3.16)

359. 25X^4 + Y^4 + 10Z^4 = W^4(2020.3.18)

361. 36X^4 + Y^4 + 3Z^4 = W^4(2020.3.23)

362. 3X^4 + 75Y^4 + 5Z^4 = 8W^4(2020.3.25)

363. 49X^4 + 49Y^4 + 8Z^4 = W^4(2020.3.26)

364. X^4 + 36Y^4 + 27Z^4 = 4W^4(2020.3.27)

365. pX^4 + pY^4 = 2Z^4 + W^4(2020.3.28)

366. 23X^4 + 23Y^4 + 2Z^4 = 25W^4(2020.3.31)

367. 47X^4 + 47Y^4 + 2Z^4 = 49W^4(2020.4.1)

368. 7X^4 + 7Y^4 + Z^4 = 8W^4(2020.4.2)

369. pX^4 + pY^4 + 2Z^4 = sW^4(2020.4.2)

370. 9X^4 + 36Y^4 + 4Z^4 = 9W^4(2020.4.4)

371. X^4 + 9Y^4 + 6Z^4 = W^4(2020.4.4)

372. 25X^4 + 25Y^4 + 8Z^4 = 25W^4(2020.4.4)

373. d^2a^2X^4 + (-d+2a)^2a^2Y^4 + 2d(-d+2a)Z^4 = 4a^2W^4(2020.4.4)

374. 9X^4 + 4Y^4 + 2Z^4 = 6W^4(2020.4.6)

375. 3X^4 + 3Y^4 = 2Z^4 + W^4(2020.4.8)

376. 33X^4 + 33Y^4 = 2Z^4 + W^4(2020.4.9)

377. pX^4 + pY^4 = 2Z^4 + W^4 revisited(2020.4.12)

378. 75X^4 + 3Y^4 + 2Z^4 = 5W^4(2020.4.13)

379. 73X^4 + 73Y^4 + Z^4 = 2W^4(2020.4.14)

380. 18X^4 + 8Y^4 + Z^4 = 27W^4(2020.4.15)

381. 57X^4 + 57Y^4 = 8Z^4 + W^4(2020.4.16)

382. 79X^4 + 79Y^4 + 4Z^4 = 2W^4 (2020.4.18)

383. 25X^4 + 4Y^4 + Z^4 = 5W^4 (2020.4.19)

384. 7X^4 + 7Y^4 + 2Z^4 = 9W^4 (2020.4.20)

385. 31X^4 + 31Y^4 + Z^4 = 2W^4 (2020.4.21)

386. 2X^4 + 8Y^4 + Z^4 = 9W^4 (2020.4.22)

387. 5X^4 + 45Y^4 + Z^4 = 6W^4 (2020.4.23)

388. pr^2X^4 + 4p(p+r)^2Y^4 + r^3Z^4 = r^2(p+r)W^4 (2020.4.24)

389. (d-2)(d+2)^2X^4 + (4d-8)Y^4 + Z^4 = (4d+8)W^4 (2020.4.25)

390. (m^2+1)^2(m^2-1)X^4 + (m^2-1)Y^4 + (m^2+1)Z^4 = 2m^2W^4 (2020.4.26)

391. Index of ax^4 + by^4 + cz^4 + dw^4 = 0, abcd≠□ (2020.4.27)

392. 2mnX^4 + (m^2-n^2)Y^4 = 2mnZ^4 + (m^2-n^2)W^4 (2020.5.4)

393. (m^4+1)X^4 + (m^4+1)Y^4 = 2Z^4 + 2W^4 (2020.5.7)

394. -dm^2(d+2)X^4 + (-2m^2+d+2)m^2(d-2m^2)Y^4 = d(d-2m^2)m^4Z^4 + (d+2)(2m^2-d-2)W^4 (2020.5.9)

395. (d^2-2dm^2+2m^4)X^4 + (d^2-2dm^2+2m^4)Y^4 = 2(-d+m^2)^2Z^4 + 2W^4 (2020.5.10)

396. 4(m^2-1)X^4 + (1-m^2)(m^2+3)Y^4 = -(m^2-1)^2Z^4 + 4(m^2+3)W^4 (2020.5.11)

398. Table of solutions for ax^4 + by^4 + cz^4 + dw^4 = 0 (2020.5.28)

399. ax^4 + by^4 + cz^4 + dw^4 + eu^4= 0 Ⅱ (2020.6.6)

 4: Fifth Powers

26. A5+ B5+ C5+ D5= E5+ F5+ G5 (2006.3.26)

27. A5+ B5+ C5+ D5= E5+ F5+ G5+ H5 (2006.3.26)

28. A15+A25+A35+...+An5=B15+B25+B35+...+Bn5 (2006.3.26)

36. A15+ A25+ A35+ A45 = B15+ B25+ B35+ B45+ B55+ B65 (2006.8.4)

39. A15+ A25+ A35+ A45+ A55+ A65 = B15+ B25+ B35+ B45+ B55+ B65 (2006.9.23)

40. A15+ A25+ A35+ A45+ A55+ A65+ A75 = B15 (2006.10.14)

41. A15+ A25+ A35+ A45+ A55+ A65+ A75 = 2 (2006.10.27)

70. Solutions of a15+a25+a35 = b15+b25+b35 (2010.2.12) (2008.5.1)

113. Generalized Taxicab Number Part 2 (2018.11.17)(2011.2.1)

122. Equal Sums of Fifth Powers et 1 (2011.8.23)

129. Equal Sums of Fifth Powers (2012.3.2)

130. A collection of the parametric solutions for equal sums of fifth powers (2012.4.2)

160. ax^5 + by^5 + cz^5 = w^2 (2014.7.28)

162. ax^5 + by^5 + cz^5 = w^2 Part 2 (2014.8.19)

196. X1^5 + X2^5 +...+ Xn^5=0, n is even and n > 5 (2016.7.12)

240. x^5 + az^2 = y^5 + aw^2 (2017.9.23)

241. x^5 + az^3 = y^5 + aw^3 (2017.9.24)

242. x^5 + az^4 = y^5 + aw^4 (2017.9.24)

267. X^5 + aY^5 + aZ^5 = W^2 (2018.4.12)

286. n^5=a^5+b^5+c^5-d^5-e^5-f^5-g^5 (2018.9.13)

289. m(x1^5+x2^5+x3^5+x4^5)=n(y1^5+y2^5+y3^5+y4^5) (2018.10.8)

298. ax^5 + by^5 + cz^5=u^5, bx^5 + cy^5 + az^5=v^5, cx^5 + ay^5 + bz^5=w^5(2018.12.14)

312. x^5+y^5=cz^2(2019.5.3)

319. Generalized Taxicab Number Part 5(2019.7.17)

 5: Sixth Powers

44. A16+ A26+ A36+ A46 = B16+ B26+ B36+ B46 (2006.12.1)

45. A16+ A26+ A36+ A46 = B16+ B26+ B36+ B46 Part 2 (2006.12.14)

47. A16+ A26+ ....+ A66 = B16+ B26+ ....+ B66 (2007.1.10)

48. A16+ A26+ ....+ A66 = B16+ B26+ ....+ B66 Part 2 (2007.1.21)

54. A16+ A26+ ....+ A46 = B16+ B26+ ....+ B46 (2007.3.28)

55. A16+ A26+ ....+ A46 = B16+ B26+ ....+ B46 Part 2 (2007.4.10)

56. A16+ A26+ ....+ A46 = B16+ B26+ ....+ B46 Part 3 (2007.5.1)

57. A16+ A26+ ....+ A76 = B16+ B26+ ....+ B76 (2007.5.20)

64. A16+ A26+ ....+ A56 = B16+ B26+ ....+ B56 (2007.10.15)

65. A16+ A26+ ....+ A66 = B16+ B26+ ....+ B66 (2007.10.15)

90. Solutions of a16+a26+a36 = b16+b26+b36 (2008.6.26)

121. Equal Sums of Sixth Powers et 1 (2011.8.1)

131. A collection of the parametric solutions for equal sums of sixth powers (2012.5.3)

136. Solutions of ax^6 + by^3 = cz^2 (2012.9.10)

150. Solutions of sextic diophantine equation (2013.11.1)

156. Equal Sums of Sixth Powers (2014.6.1)

161. ax^6 + by^6 + cz^6 = w^2 (2014.8.7)

164. ax^6 + by^6 + cz^6 = w^2 and ax^6 + by^6 + cz^6 = w^3 (2014.10.10)

166. w^2 = x^6 + y^6 + z^6 (2014.11.18)

167. w^2 = x^6 + y^6 + z^6 Part 2 (2014.11.19)

168. ax^6 + by^6 + cz^6 + dw^3 = 0 (2014.12.3)

179. ax1^n + bx2^n + cx3^n = ay1^n + by2^n + cy3^n, n=2,6 (2015.6.1)

235. x^6 + az^3 = y^6 + aw^3 (2017.9.7)

236. x^6 + az^2 = y^6 + aw^2 (2017.9.11)

237. x^6 + az^5 = y^6 + aw^5 (2017.9.13)

243. x^6 + az^4 = y^6 + aw^4 (2017.9.26)

244. x^6 + az^6 = y^6 + aw^6 (2017.10.5)

299. ax^6 + by^6 + cz^6=u^6, bx^6 + cy^6 + az^6=v^6, cx^6 + ay^6 + bz^6=w^6(2018.12.14))

322. a^3x^6+b^3y^6+ 2z^6 = 2w^6(2019.7.30))

400. a^2X^6 + b^2Y^6 + 2abZ^3 = W^6(2020.6.14))

 6: Seventh Powers

42. A17+ A27+ .... A87 = B17+ B27+ .... B87 (2006.11.10)

60. A17+ A27+ ....+ A57 = B17+ B27+ ....+ B57 (2007.6.15)

63. EQUAL SUMS OF SEVENTH POWERS (2007.8.20)(2007.10.15)

115. EQUAL SUMS OF SEVENTH POWERS (2011.3.4)

120. Equal Sums of Seventh Powers et 1 (2011.7.18)

138. Equal Sums of Seventh Powers (2012.11.30)

234. x1^7 + x2^7 + x3^7 + x4^7 = y1^7 + y2^7 + y3^7 + y4^7 + y5^7 (2017.9.1)

245. x^7 + az^2 = y^7 + aw^2 (2017.10.9)

246. x^7 + az^6 = y^7 + aw^6 (2017.10.9)

247. x^7 + az^3 = y^7 + aw^3 (2017.10.12)

263. x^7 + az^n = y^7 + aw^n,n=2,3,4,5,6,7 (2018.3.1)

 7: Eighth Powers

61. A18+ A28+ ....+ A78 = B18+ B28+ ....+ B78 (2007.7.16)

62. A18+ A28+ ....+ A68 = B18+ B28+ ....+ B68 (2007.8.5)

119. Equal Sums of Eighth Powers (2011.7.1)

124. Equal Sums of Eighth Powers et 1 (2011.10.11)

132. Equal Sums of Eighth Powers Part 3 (2012.6.11)

135. Equal Sums of Eighth Powers Part 4 (2012.8.10)

153. Equal Sums of Eighth Powers Part 5 (2014.3.3)

170. ax^8 + by^8 + cz^4 + dw^4 = 0 (2015.2.1)

248. x^8 + az^3 = y^8 + aw^3 (2017.10.17)

249. x^8 + az^2 = y^8 + aw^2 (2017.10.19)

250. x^8 + az^7 = y^8 + aw^7 (2017.10.19)

269. a^2X^8 + 2ab^4Y^4 + Z^8 = (a+b^4)^2W^4 (2018.5.5)

272. X^4 + Y^4 + Z^8 = 2W^4 (2018.6.6)

321. x1^4 + 2nX2^4 = X3^8 + n^2X4^8 (2019.7.26)

 8: Ninth Powers

106. ON EQUAL SUMS OF NINTH POWERS (2010.7.20)

109. ON EQUAL SUMS OF NINTH POWERS Part 2 (2010.10.1)

251. x^9 + az^2 = y^9 + aw^2 (2017.10.24)

252. x^9 + az^3 = y^9 + aw^3 (2017.10.24)

253. x^9 + az^8 = y^9 + aw^8 (2017.10.24)

 9: Tenth Powers

10: Eleventh Powers

117. EQUAL SUMS OF ELEVENTH POWERS (2011.5.6)

11: Others

29. Prime-generating polynomials(Fourth degree) (2006.3.31)

43. Solutions of 5.6.6, 6.8.8, 8.10.10, 9.10.10 and 10.12.12. (2006.11.18)

46. A Happy New Year! (2007.1.1)

50. A property of the solution for a1p-1+a2p-1+....+anp-1=bp-1 (2007.2.4)

67. A Happy New Year! (2008.1.1)

69. Sierpinski problem (2008.4.11)

74. Solve the Diophantine equation like Diophantus (2008.6.15)

83. A Happy New Year! (2009.1.1)]

92. Smallest number that is a sum of two n-th powers of positive rationals but not of two n-th powers of positive integers. (2009.9.6)

94. On factors of Mersenne numbers. (2009.12.10)

95. A Happy New Year! (2010.1.1)

96. Repunit (2010.1.13)

112. Happy New Year! (2011.1.1)

127. Happy New Year! (2012.1.3)

139. Happy New Year! (2013.1.1)

151. Happy New Year! (2014.1.1)

169. Happy New Year! (2015.1.1)

178. a(1)x1^n + a(2)x2^n +...+ a(k)xk^n = a(1)y1^n + a(2)y2^n +...+ a(k)yk^n (2015.5.1)

210. a(1)x1^n + a(2)x2^n +...+ a(n)xn^n=0 (2017.3.16)

265. x^n + y^n = z^(n-1) + w^(n-1) (2018.3.14)

279. (x+y+z+w+s)(1/x+1/y+1/z+1/w+1/s) = n (2018.7.22)

280. (x+y+z+w+s)(1/x+1/y+1/z+1/w+1/s) = n Part 2 (2018.8.1)

281. (x+y+z+w+s+t)(1/x+1/y+1/z+1/w+1/s+1/t) = n (2018.8.14)

282. (x+y+z+w)(1/x+1/y+1/z+1/w) = n (2018.8.28)

290. Magic hexagon (2018.10.25)

292. 4x4 magic squares of squares (2018.11.6)

328. Heron Triangles (2019.9.2)

332. Pythagorean triangles such that all triangles have same area (2019.10.2)

333. N-tuples of positive integers with the same sum and the same product (2019.10.26)

334. Primitive pythagorean triangles with same perimeter (2019.11.18) Publications 1. Solution of Polynomial equations(pdf) 2. Generalized Parametric Solutions to Multiple Sums of Powers(pdf) 3. Sum of Three Biquadatics a Multiple of a nth Power, n = (2,3,4,5,6,7,8 & 9)(pdf) 4. Methods for solving equations of the type (k+3) & (k+5) quartics-equal to a quartic for integer k (pdf) 5. Parametric Solutions to sums of six nth powers, n = (2,3,4,5,6,7,8 & 9)(pdf) 6. Parametric solutions to equation pa^n+qb^n=pc^n+qd^n(pdf)



















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