x + by = cz"> x + by = cz">

Surveys in Mathematics and its Applications


ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 14 (2019), 109 -- 140

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

A SURVEY ON THE TERNARY PURELY EXPONENTIAL DIOPHANTINE EQUATION ax + by = cz

Maohua Le, Reese Scott and Robert Styer

Abstract. Let a, b, c be fixed coprime positive integers with \min\a,b,c\>1. In this survey, we consider some unsolved problems and related works concerning the positive integer solutions (x,y,z) of the ternary purely exponential diophantine equation ax + by = cz.

2010 Mathematics Subject Classification: 11D61.
Keywords: ternary purely exponential diophantine equation; Jesmanowicz conjecture, Terai-Jesmanowicz conjecture.

Full text

References

  1. X.-F. An. On Jesmanowicz' conjecture concerning Pythagorean numbers. Master's thesis, Chongqing: Chongqing Normal Univ., 2015. (in Chinese).

  2. Y.~An. On Jesmanowicz' conjecture concerning Pythagorean numbers. Master's thesis, Chongqing: Southwest Univ., 2014. (in Chinese).

  3. M.~A. Bennett and N.~Billerey. Sums of two S-units via Frey-Hellegouarch curves. Math. Comp., 86(305), (2017), 1375--1401. MR3614021 Zbl 06693286.

  4. M.~A. Bennett and C.~Skinner. Ternary diophantine equations via Galois representations and modular forms. Canad. J. Math., 56(1), (2004), 23--54. MR2031121 Zbl 1053.11025.

  5. C.~Bertók. The complete solution of the diophantine equation (4m2+1)x + (5m2-1)y = (3m)z. Period. Math. Hung., 72(1), (2016), 37--42. MR3470802 Zbl 1389.11085.

  6. F.~Beukers and H.~P. Schlickewei. The equation x+y=1 in finitely generated groups. Acta Arith., 78(2), (1996), 189--199. MR3470802 Zbl 0880.11034.

  7. Y.~Bilu, G.~Hanrot, and P.~M. Voutier. Existence of primitive divisors of Lucas and Lehmer numbers, with an appendix by M. Mignotte. J. Reine Angew. Math., 539, (2001), 75--122. MR1863855 Zbl 0995.11010.

  8. Y.-S. Cao. On the equation ax - by = (2p)z. Northeast Math. J., 5(4), (1989), 477--484. (in Chinese). MR1053528 Zbl 0713.11024.

  9. Z.-F. Cao. The equation ax - by = (2ps)z and Hugh Edgar's problem. Chinese Sci. Bull., 30(14), (1985), 1116--1117. (in Chinese). MR0871578

  10. Z.-F. Cao. The equation x2 + 2m = yn and Hugh Edgar's problem. Chinese Sci. Bull., 31(7), (1986), 555--556. (in Chinese).

  11. Z.-F. Cao. On the diophantine equation ax + by = cz I. Chinese Sci. Bull., 31(22), (1986), 1688--1690. (in Chinese). MR0886632

  12. Z.-F. Cao. Exponential diophantine equations of the form ax + by = cz. J. Harbin Inst. Tech., 4, (1987), 113--121. (in Chinese). MR0951789 Zbl 0971.11517.

  13. Z.-F. Cao. On the diophantine equation ax + by = cz II. Chinese Sci. Bull., 33(3), (1988), 237. (in Chinese).

  14. Z.-F. Cao. Introduction to diophantine equations. Harbin Inst. Tech. Press, Harbin, 1989. (in Chinese). Zbl 0849.11029.

  15. Z.-F. Cao. A note on the diophantine equation ax + by = cz. Acta Arith., 91(1), (1999), 85--93. MR1726477 Zbl 0946.11009.

  16. Z.-F. Cao and C.-S. Cao. On the equation ax + by = cz. Pure Appl. Math., 4(4), (1988), 98--100. (in Chinese). Zbl 0921.11017.

  17. Z.-F. Cao and X.-L. Dong. On the Terai-Jesmanowicz conjecture. Publ. Math. Debrecen, 61(3--4), (2002), 253--265. MR1943694 Zbl 1012.11025.

  18. Z.-F. Cao and X.-L. Dong. An application of a lower bound for linear forms in two logarithms to the Terai-Jesmanowicz conjecture. Acta Arith., 110(2), (2003), 153--164. MR2008082 Zbl 1031.11020.

  19. Z.-F. Cao, R.-Z. Tong, and Z.-J. Wang. A conjecture on the exponential diophantine equations. Chinese J. Nature, 14(11), (1991), 872--873. (in Chinese).

  20. Z.-F. Cao and D.-Z. Wang. On Hugh Edgar's problem. Chinese Sci. Bull., 32(14), (1987), 1043--1046. MR0940140

  21. A.~P. Chaves and D.~Marques. A diophantine equation related to the sum of consecutive k-generalized Fibonacci numbers. Fibonacci Quart., 52(1), (2014), 70--74. MR3181100 Zbl 1290.11021.

  22. A.~P. Chaves and D.~Marques. A diophantine equation related to the sum of powers of two consecutive generalized Fibonacci numbers. J. Number Theory, 156(1), (2015), 1--14. MR3360324 Zbl 1395.11031.

  23. H.~Che. On the diophantine equation (21n)x + (220n)y = (221n)z. Master's thesis, Chongqing: Southwest Univ., 2011. (in Chinese).

  24. F.-J. Chen. On the diophantine equation (na)x + (nb)y = (nc)z. Adv. Math. China, 47(3), (2018), 388--392. Zbl 07028020.

  25. J.-M. Chen. On the diophantine equation ax + by = cz. J. Zhejiang Normal Univ., Nat. Sci., 17(4), (1994), 17--20. (in Chinese).

  26. J.-P. Chen. The positive integer solutions of the diophantine equation ax + by = cz. J. Hainan Univ., Nat. Sci., 30(4), (2012), 309--315. (in Chinese).

  27. L.-M. Chen. The odd integer solutions of the equation nx + (3n2 -1)y = (4n2 -1)z. Adv. Math. Beijing, 39(4), (2010), 507--511. MR2760537

  28. Z.~Cheng, C.-F. Sun, and X.-N. Du. On the diophantine equation (20n)x + (21n)y = (29n)z. Math. Appl., 26(1), (2013), 129--133. MR3077060 Zbl 1274.11088.

  29. M.~Cipu and M.~Mignotte. Bounds for counterexamples to Terai's conjecture. Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 53(101)(3), (2010), 231--237. MR2732411 Zbl 1212.11046.

  30. V.~A. Dem'janenko. On Jesmanowicz' problem for Pythagorean numbers. Izv. Vyss Ucebn. Zayed. Mat., 48(1), (1965), 52--56. (in Russian). MR0191865 Zbl 0166.05103.

  31. M.-J. Deng. On Jesmanowicz' conjecture. J. Harbin Inst. Tech., 25(2), (1993), 14--17. (in Chinese). MR1236132 Zbl 0971.11514.

  32. M.-J. Deng. A note on the diophantine equation (a2-b2)x + (2ab)y = (a2 + b2)z. J. Nat. Sci. Heilongjiang Univ., 19(3), (2002), 8--10. MR1935472 Zbl 1076.11508.

  33. M.-J. Deng. On the diophantine equation (15n)x + (112n)y = (113n)z. J. Nat. Sci. Heilongjiang Univ., 24(5), (2007), 617--620. (in Chinese). Zbl 1150.11015.

  34. M.-J. Deng. A note on the diophantine equation (na)x + (nb)y = (nc)z. Bull. Aust. Math. Soc., 89(2), (2014), 316--321. MR3182668 Zbl 1372.11046.

  35. M.-J. Deng and G.~L. Cohen. On the conjecture of Jesmanowicz concerning Pythagorean triples. Bull. Aust. Math. Soc., 57(4), (1998), 515--524. MR1623283 Zbl 0916.11020.

  36. M.-J. Deng and G.~L. Cohen. A note on a conjecture of Jesmanowicz. Colloq. Math., 86(1), (2000), 25--30. MR1799885 Zbl 0960.11026.

  37. M.-J. Deng and J.~Guo. A note on Jesmanowicz' conjecture concerning primitive Pythagorean triples II. Acta Math. Hung., 153(2), (2017), 436--448. MR3720980 Zbl 1399.11099.

  38. M.-J. Deng and D.-M. Huang. A note on Jesmanowicz conjecture concerning primitive Pythagorean triples. Bull. Aust. Math. Soc., 95(1), (2017), 5--13. MR3592539 Zbl 06717752.

  39. X.-L. Dong and Z.-F. Cao. The Terai-Jesmanowicz conjecture on the equation ax + by = cz. Chinese Ann. Math., Ser. A, 21A(6), (2000), 709--714. (in Chinese). Zbl 0987.11021.

  40. Q.~Feng. The shuffle variant of Jesmanowicz' conjecture on primitive Pythagorean numbers. Math. Pract. Theory, 45(16), (2015), 312--315. (in Chinese). Zbl 1349.11072.

  41. Q.~Feng and D.~Han. On the diophantine system a2+b2 = cr and ax + by = cz for b a prime. Int. J. Appl. Math. Stat., 52(7), (2014), 65--73. MR3256469 Zbl 1339.11047.

  42. C.-Y. Fu. On Jesmanowicz' conjecture concerning Pythagorean numbers. Master's thesis, Haikou: Hainan Univ., 2016. (in Chinese).

  43. C.-Y. Fu and M.-J. Deng. On the diophantine equation (n(72r - 4))x + (n(4dot 7r))y = (n(72r+4))z. J. Nat. Sci. Heilongjiang Univ., 32(5), (2015), 596--599. (in Chinese). Zbl 1349.11073.

  44. R.-Q. Fu and H.~Yang. On the exponential diophantine equation ((am2+1)x + (bm2-1)y = (cm)z with c \mid m. Period. Math. Hung., 75(2), (2017), 143--149. MR3718506 Zbl 06850216.

  45. Y.~Fujita and T.~Miyazaki. Jesmanowicz' conjecture with congruence relations. Colloq. Math., 128(2), (2012), 211--222. MR3002349 Zbl 1318.11049.

  46. Y.~Fujita and T.~Miyazaki. Jesmanowicz' conjecture with congruence relations II. Canad. Math. Bull., 57(3), (2014), 495--505. MR3239111 Zbl 1291.11071.

  47. A.~O. Gel'fond. Sur la divisibilité de la différence des puissance de deux nombres entiers par une puissance d'un idéal premier. Mat. Sb., 7(49), (1940), 7--25. MR0001755 Zbl 0023.10405.

  48. C.~A. Gómez Ruiz and F.~Luca. An exponential diophantine equation related to the sum of powers of two consecutive k-generalized Fibonacci numbers. Colloq. Math., 137(2), (2014), 171--188. MR3286304 Zbl 1338.11016.

  49. S.-S. Gou. On Jesmanowicz' conjecture concerning Pythagorean numbers. Master's thesis, Chongqing: Southwest Univ., 2016. (in Chinese).

  50. S.~S. Gou and H.~Zhang. On the diophantine equation (16n)x + (63n)y = (65n)z. J. Southwest China Normal Univ., Nat. Sci., 40(4), (2015), 4--7. (in Chinese).

  51. W.-J. Guan. The Jesmanowicz conjecture on Pythagorean numbers. Basic Sci. J. Text. Univ., 24(4), (2011), 557--559.

  52. W.-J. Guan and S.~Che. On the diophantine equation 2y ny-x = (b+2)x - bx. J. Northwest Univ., Nat. Sci., 44(4), (2014), 534--536. (in Chinese). MR3309624 Zbl 1313.11072.

  53. X.-G. Guan. On a class of pure exponential diophantine equations. J. Hebei North Univ., Nat. Sci., 28(4), (2012), 5--8. (in Chinese).

  54. X.-G. Guan. On the exponential diophantine equation ax + by = 2z. J. Zhoukou Normal Univ., 31(2), (2014), 26--30. (in Chinese).

  55. X.-G. Guan. On Terai's conjecture concerning the diophantine equation ax + by = cz. Adv. Math. China, 44(6), (2015), 837--844. (in Chinese). MR3493559 Zbl 1349.11074.

  56. X.-G. Guan. On the pure exponential diophantine equation ax+by = (m2 + 1)z. Adv. Math. China, 45(5), (2016), 687--699. (in Chinese). Zbl 1374.11055.

  57. Y.-D. Guo and M.-H. Le. A note on Jesmanowicz' conjecture concerning Pythagorean numbers. Commen. Math. Univ. St. Pauli, 44(2), (1995), 225--228. MR1366530 Zbl 0849.11036.

  58. R.~K. Guy. Unsolved Problems in Number Theory. Science Press, Beijing, 3rd edition, 2007. MR2076335 Zbl 1058.11001.

  59. T.~Hadano. On the diophantine equation ax + by = cz. Math. J. Okayama Univ., 19(1), (1976/1977), 25--29. MR0432538 Zbl 0349.10012.

  60. Q.~Han and P.-Z. Yuan. A note on Jesmanowicz' conjecture. Acta Math. Hungar., 156(1), (2018), 220--225. MR3856913 Zbl 07011148.

  61. B.~He and A.~Togbé. The exponential diophantine equation nx + (n+1)y = (n+2)z revisited. Glasgow Math. J., 51(5), (2009), 659--667. MR2534015 Zbl 1194.11044.

  62. B.~He and A.~Togbé. On the positive integer solutions of the exponential diophantine equation ax + (3a2-1)y = (4a2-1)z. Adv. Math. China, 40(2), (2011), 227--234. MR2841128

  63. B.~He, A.~Togbé, and S.-C. Yang. On the solutions of the exponential diophantine equation ax + by = (m2+1)z. Quaes. Math., 36(1), (2013), 119--135. MR3043675 Zbl 1274.11086.

  64. B.~He and S.-C. Yang. The positive integer solutions of the diophantine equation (8a3-3a)x + (3a2-1)y = (4a2-1)z. J. Sichuan Univ., Nat. Sci., 47(1), (2010), 13--16. (in Chinese). MR2643276 Zbl 1240.11058.

  65. N.~Hirata-Kohno. S-unit equations and integer solutions to exponential diophantine equations. In Analytic number theory and surrounding areas, volume 1511, pages 92--97. Kyoto RIMS Kokyuroku, 2006.

  66. N.~Hirata-Kohno and F.~Luca. On the diophantine equation Fnx + Fn+1y = Fmy. Rocky Mt. J. Math., 45(2), (2015), 509--538. MR3356626 Zbl 1332.11042.

  67. Y.-Z. Hu. The diophantine equation (8a3 - 3a)2x + (3a2-1)y = (4a2-1)z. J. Sichuan Univ, Nat. Sci., 44(2), (2007), 225--228. (in Chinese). MR2340286 Zbl 1164.11318.

  68. Y.-Z. Hu. On the exponential diophantine equation a2x + (3a2-1)y = (4a2-1)z. Adv. Math. China, 36(4), (2007), 429--434. MR2381768 Zbl 1131.11328.

  69. Y.-Z. Hu and M.-H. Le. A note on ternary purely exponential diophantine equations. Acta Arith., 171(2), (2015), 173--182. MR3414305 Zbl 1379.11030.

  70. Y.-Z. Hu and M.-H. Le. An upper bound for the number of solutions of ternary purely exponential diophantine equations. J. Number Theory, 183, (2018), 62--73. MR3715228 Zbl 06802525.

  71. Y.-Z. Hu and M.-H. Le. An upper bound for the number of solutions of ternary purely exponential diophantine equations II. arXiv 1808:06557, 2018.

  72. Y.-Z. Hu and P.-Z. Yuan. The exponential diophantine equation ax + by = cz. Acta Math. Sinica, Chinese Ser., 48(6), (2005), 1175--1178. (in Chinese). MR2205060 Zbl 1124.11307.

  73. Y.-Z. Hu and P.-Z. Yuan. The simultaneous diophantine equations a2 + b2 = c3 and ax + by = cz. Acta Math. Sinica, Chinese Ser., 52(5), (2009), 1027--1032. (in Chinese). MR2583775 Zbl 1125.11309.

  74. Y.-Z. Hu and P.-Z. Yuan. Jesmanowicz' conjecture concerning Pythagorean numbers. Acta Math. Sinica, Chinese Ser., 53(2), (2010), 297--300. (in Chinese). MR2666062 Zbl 1224.11049.

  75. L.~Jesmanowicz. Several remarks on Pythagorean numbers. Wiadom. Math., 1(2), (1955/1956), 196--202. (in Polish). MR0110662 Zbl 0074.27205.

  76. Z.~Ke. On Jesmanowicz' conjecture. J. Sichuan Univ., Nat. Sci., 4(2), (1958), 81--90. (in Chinese).

  77. Z.~Ke. On Pythagorean numbers. J. Sichuan Univ., Nat. Sci., 4(1), (1958), 73--80. (in Chinese).

  78. Z.~Ke. On the diophantine equation (a2-b2)x + (2ab)y = (a2+b2)z. J. Sichuan Univ., Nat. Sci., 5(3), (1959), 25--34. (in Chinese).

  79. Z.~Ke. On the Pythagorean numbers (2n+1), 2n(n+1), 2n(n+1)+1 I. J. Sichuan Univ., Nat. Sci., 9(2), (1963), 9--14. (in Chinese).

  80. Z.~Ke. On the Pythagorean numbers (2n+1), 2n(n+1), 2n(n+1)+1 III. J. Sichuan Univ., Nat. Sci., 10(4), (1964), 11--26. (in Chinese).

  81. Z.~Ke and Q.~Sun. On the Pythagorean numbers (2n+1), 2n(n+1), 2n(n+1)+1 II. J. Sichuan Univ., Nat. Sci., 10(3), (1964), 1--12. (in Chinese).

  82. Z.~Ke and Q.~Sun. Diophantine equations. Shanghai Edu. Publ. House, Shanghai, 1980. (in Chinese). Zbl 0603.10016.

  83. M.-H. Le. On the diophantine equation ax + by = cz. J. Changchun Teachers College, Nat. Sci., 2(1), (1985), 50--62. (in Chinese).

  84. M.-H. Le. On two problems of Hall and Edgar. Northeast Math. J., 4(4), (1988), 432--434. (in Chinese). MR0987066 Zbl 0695.10113.

  85. M.-H. Le. A note on Jesmanowicz' conjecture. Colloq. Math., 69(1), (1995), 47--51. MR1341681 Zbl 0835.11015.

  86. M.-H. Le. On Jesmanowicz' conjecture concerning Pythagorean numbers. Proc. Japan Acad., Ser. A, 72A(5), (1996), 97--98. MR1404479 Zbl 0876.11013.

  87. M.-H. Le. A note on the diophantine equation (m3 - 3m)x + (3m2-1)y = (m2 + 1)z. Proc. Japan Acad., Ser. A, 73A(7), (1997), 148--149. MR1487581 Zbl 0910.11010.

  88. M.-H. Le. A note on Jesmanowicz' conjecture concerning Pythagorean numbers. Bull. Aust. Math. Soc., 59, (1999), 477--480. MR1697985 Zbl 0940.11021.

  89. M.-H. Le. An upper bound for the number of solutions of the exponential diophantine equation ax + by = cz. Proc. Japan Acad., Ser. A, 75A(6), (1999), 90--91. MR1712652 Zbl 0939.11018.

  90. M.-H. Le. On Terai's conjecture concerning Pythagorean numbers. Bull. Aust. Math. Soc., 61, (2000), 329--334. MR1748713 Zbl 0979.11021.

  91. M.-H. Le. On the exponential diophantine equation (m3 - 3m)x + (3m2-1)y = (m2 + 1)z. Publ. Math. Debrecen, 58(3--4), (2001), 461--466. MR1831054 Zbl 1062.11020.

  92. M.-H. Le. On Cohn's conjecture concerning the diophantine equation x2 + 2m = yn. Archiv der Mathematik, 78, (2002), 26--35. MR1887313 Zbl 1006.11013.

  93. M.-H. Le. A conjecture concerning the exponential diophantine equation ax + by = cz. Acta Arith., 106(4), (2003), 345--353. MR1957910 Zbl 1023.11013.

  94. M.-H. Le. On Terai's conjecture concerning the exponential diophantine equation ax + by = cz. Acta Math. Sinica, Chinese Ser., 46(2), (2003), 245--250. (in Chinese). MR1988161 Zbl 1136.11306.

  95. M.-H. Le. A note on the exponential diophantine equation ax + by = cz. Proc. Japan Acad., Ser. A, 80A(4), (2004), 21--23. MR2055070 Zbl 1050.11040.

  96. M.-H. Le. A conjecture concerning the pure exponential diophantine equation ax + by = cz. Acta Math. Sinica, English Ser., 20(4), (2005), 943--948. MR2156975 Zbl 1159.11308.

  97. M.-H. Le. An open problem concerning the diophantine equation ax + by = cz. Publ. Math. Debrecen, 68(3-4), (2006), 283--295. MR2212322 Zbl 1111.11019.

  98. M.-H. Le. On the diophantine system a2 + b2 = c3 and ax + by = cz for b an odd prime. Acta Math. Sinica, English Ser., 24(6), (2008), 917--924. MR2212322 Zbl 1218.11037.

  99. M.-H. Le. A note on the diophantine system a2 + b2 = cr and ax + by = cz. Acta Math. Sinica, Chinese Ser., 51(4), (2008), 677--684. (in Chinese). MR2454004 Zbl 1174.11041.

  100. M.-H. Le. A note on Jesmanowicz' conjecture concerning primitive Pythagorean triplets. Acta Arith., 138(2), (2009), 137--144. MR2520132 Zbl 1297.11015.

  101. M.-H. Le. The pure exponential diophantine equation ax + by = cz for generalized Pythagorean triplets. Acta Math. Sinica, Chinese Ser., 53(6), (2010), 1239--1248. (in Chinese). MR2789638 Zbl 1240.11060.

  102. M.-H. Le. An application of Baker's method to Jesmanowicz' conjecture on primitive Pythagorean triples. arXiv:1811.00654

  103. M.-H. Le. An application of the BHV theorem to a new conjecture on exponential diophantine equations. arXiv:1811.00609.

  104. M.-H. Le, A.~Togbé, and H.-L. Zhu. On a pure ternary exponential diophantine equation. Publ. Math. Debrecen, 85(3--4), (2014), 395--411. MR3291838 Zbl 1340.11041.

  105. B.~Leszczyński. On the equation nx + (n+1)y = (n+2)z. Wiadom Mat., 3(1), (1959), 37--39. (in Polish) MR0115973 Zbl 0095.26205.

  106. S.-Z. Li. On Jesmanowicz' conjecture concerning Pythagorean numbers. Master's thesis, Chongqing: Southwest Univ., 2011. (in Chinese).

  107. Y.-M. Li. On the diophantine equation (48n)x + (55n)y = (73n)z. J. Chongqing Tech. Busin. Univ., Nat. Ser., 2, (2018), 27--30. (in Chinese).

  108. Z.~Li. On Jesmanowicz' conjecture concerning Pythagorean numbers. J. Nat. Sci. Hailongjiang Univ., 20(3), (2003), 54--55. (in Chinese). MR2025903 Zbl 1058.11026.

  109. C.-N. Lin. On the diophantine equation (51n)x + (1300n)y = (1301)z. Master's thesis, Chongqing: Southwest Univ., 2017. (in Chinese).

  110. M.-Y. Lin. The positive integer solutions of an exponential diophantine equation. J. Math. Wuhan, 26(4), (2006), 409--414. (in Chinese). MR2241997 Zbl 1118.11017.

  111. D.-R. Ling and J.-X. Weng. On the diophantine equation (195n)x + (28n)y = (197n)z. Pure Appl. Math., 29(4), (2013), 342--349. (in Chinese). MR3154606 Zbl 1299.11034.

  112. B.-L. Liu. A new conjecture on primitive Pythagorean numbers. Math. Pract. Theory, 43(9), (2013), 253--255. (in Chinese). Zbl 1235.11091.

  113. B.-L. Liu. A note on ternary exponential diophantine equations. J. Inner Mongolia Univ, Nat. Sci., 43(4), (2014), 401--402. (in Chinese). MR3243647 Zbl 1313.11073.

  114. B.-L. Liu. On the diophantine equation (143n)x + (24n)y = (145n)z. Math. Pract. Theory, 47(20), (2017), 178--182. (in Chinese). MR3727287 Zbl 1399.11102.

  115. H.-L. Liu. On Jesmanowicz' conjecture concerning Pythagorean numbers. Master's thesis, Chongqing: Southwest Univ., 2017. (in Chinese).

  116. Z.-W. Liu. On the ternary pure exponential diophantine equation ax + by = cz. Pure Appl. Math., 23(1), (2007), 28--30. (in Chinese).

  117. W.-D. Lu. On the Pythagorean numbers 4n2-1, 4n and 4n2+1. J. Sichuan Univ. Nat. Sci., 5(2), (1959), 39--42. (in Chinese).

  118. W.-Y. Lu, L.~Gao, and H.-F. Hao. On the integer solutions of the diophantine equation (44n)x + (117n)y = (125n)z. Pure Appl. Math., 30(6), (2014), 627--633. (in Chinese). Zbl 1324.11035.

  119. W.-Y. Lu, L.~Gao, X.-H. Wang, and H.-F. Hao. On the diophantine equation (91n)x + (4140n)y = (4141n)z. J. Guizhou Normal Univ., Nat. Sci., 33(2), (2015), 48--53. (in Chinese).

  120. F.~Luca. On the equation x\sp 2+2\sp a-3\sp b=y\sp n. Int. J. Math. Math. Sci., 29(4), (2002), 239--244. MR1897992 Zbl 1085.11021.

  121. F.~Luca. On the system of diophantine equations a2 + b2 = (m2+1)r and ax + by = (m2+1)z. Acta Arith., 153(4), (2012), 373--392. MR2925378 Zbl 1272.11047.

  122. F.~Luca and R.~Oyono. An exponential diophantine equation related to powers of two consecutive Fibonacci numbers. Proc. Japan Acad., Ser. A, 87A(4), (2011), 45--50. MR2803898 Zbl 1253.11046.

  123. J.~Ma. On the diophantine equation (57n)x + (1624n)y = (1625n)z. Master's thesis, Chongqing: Chongqing Southwest Univ., 2013. (in Chinese).

  124. M.-M. Ma. On Jesmanowicz' conjecture. Master's thesis, Nanjing: Nanjing Normal Univ., 2015. (in Chinese).

  125. M.-M. Ma and Y.-G. Chen. Jesmanowicz' conjecture on Pythagorean triples. Bull. Aust. Math. Soc., 96(1), (2017), 30--35. MR3668397 Zbl 06757135.

  126. M.-M. Ma and J.-D. Wu. On the diophantine equation (65n)x + (72n)y = (97n)z. J. Nanjing Normal Univ., Nat. Sci., 37(4), (2014), 28--30. (in Chinese). MR3307760 Zbl 1324.11036.

  127. M.-M. Ma and J.-D. Wu. On the diophantine equation (an)x + (bn)y = (cn)z. Bull. Korean Math. Soc., 52(4), (2015), 1133--1138. MR3385756 Zbl 1335.11027.

  128. K.~Mahler. Zur Approximation algebraischer Zahlen I: Über den grössten Primteiler binärer Formen. Math. Ann., 107, (1933), 691--730. MR1512822

  129. A.~Makowski. On the diophantine equation 2x + 11y = 5z. Norsk. Mat. Tidsskr., 7(1), (1959), 81--96. MR0109803 Zbl 0084.27104.

  130. A.~Makowski. On the equation nx + (n+1)y = (n+2)z. Wiadom. Mat., 9(3), (1967), 221--224. MR0213292 Zbl 0153.06603.

  131. T.~Miyazaki. On the conjecture of Jesmanowicz concerning Pythagorean triples. Bull. Aust. Math. Soc., 80(3), (2009), 413--422. MR2569916 Zbl 1225.11038.

  132. T.~Miyazaki. Exceptional cases of Terai's conjecture on diophantine equations. Arch. Math. Basel, 95(6), (2010), 519--527. MR2745461 Zbl 1210.11047.

  133. T.~Miyazaki. Generalizations of classical results on Jesmanowicz' conjecture concerning Pythagorean triples. In T.~Komatsu et~al., editor, Diophantine Analysis and Related Fields, AIP Conf. Proc., 1264, (2010), 41--51. New York. MR2731813

  134. T.~Miyazaki. Jesmanowicz' conjecture on exponential diophantine equations. Funct. Approx. Comment. Math., 45(2), (2011), 207--229. MR2895155 Zbl 1266.11064.

  135. T.~Miyazaki. The shuffle variant of Jesmanowicz' conjecture concerning Pythagorean triples. J. Aust. Math. Soc., 90(3), (2011), 355--370. MR2833306 Zbl 1225.11039.

  136. T.~Miyazaki. Terai's conjecture on exponential diophantine equations. Int. J. Number Theory, 7(4), (2011), 981--999. MR2812648 Zbl 1221.11092.

  137. T.~Miyazaki. Generalizations of classical results on Jesmanowicz' conjecture concerning Pythagorean triples. J. Number Theory, 133(2), (2013), 583--589. MR2994375 Zbl 1309.11029.

  138. T.~Miyazaki. The shuffle variant of Terai's conjecture on exponential diophantine equations. Publ. Math. Debrecen, 83(1--2), (2013), 43--62. MR3081225 Zbl 1274.11087.

  139. T.~Miyazaki. A note on the article by F. Luca \lq\lq On the system of diophantine equations a2+b2= (m2+1)r and ax + by = (m2+1)z''. Acta Arith., 164(1), (2014), 31--42. MR3223317 Zbl 1300.11027.

  140. T.~Miyazaki. A remark on Jesmanowicz' conjecture for the non-coprimality case. Acta Math. Sinica, English Ser., 31(8), (2015), 1255--1260. MR3367686 Zbl 1330.11021.

  141. T.~Miyazaki. Upper bounds for solutions of an exponential diophantine equation. Rocky Mt. J. Math., 45(1), (2015), 303--344. MR3334214 Zbl 1378.11046.

  142. T.~Miyazaki and F.~Luca. On the system of diophantine equations (m2-1)r + b2 = c2 and (m2-1)x + by = cz. J. Number Theory, 153, (2015), 321--345. MR3327578 Zbl 1365.11033.

  143. T.~Miyazaki and N.~Terai. On the exponential diophantine equation (m2+1)x + (cm2-1)y = (am)z. Bull. Aust. Math. Soc., 90(1), (2014), 9--19. MR3227125 Zbl 1334.11019.

  144. T.~Miyazaki and N.~Terai. On Jesmanowicz' conjecture concerning Pythagorean triples II. Acta Math. Hung., 147(2), (2015), 286--293. MR3420578 Zbl 1374.11056.

  145. T.~Miyazaki and A.~Togbé. The diophantine equation (2am-1)x + (2m)y = (2am+1)z. Int. J. Number Theory, 8(8), (2012), 2035--2044. MR2978854 Zbl 1290.11068.

  146. T.~Miyazaki, A.~Togbé, and P.-Z. Yuan. On the diophantine equation ax + by = (a+2)z. Acta Math. Hung., 149(1), (2016), 1--9. MR3498942 Zbl 1389.11088.

  147. T.~Miyazaki, P.-Z. Yuan, and D.-Y. Wu. Generalizations of classical results on Jesmanowicz' conjecture concerning Pythagorean triples II. J. Number Theory, 141(1), (2014), 184--201. MR3195395 Zbl 1310.11040.

  148. L.~J. Mordell. Diophantine Equations. Academic Press, London, 1969. MR0249355 Zbl 0188.34503.

  149. T.~Nagell. Sur une classe d'équations exponentielles. Ark. Mat., 3(4), (1958), 569--582. MR0103858 Zbl 0083.03902.

  150. X.-W. Pan. A note on the exponential diophantine equation (am2+1)x + (bm2-1)y = (cm)z. Colloq. Math., 149(2), (2017), 265--273. MR3697141 Zbl 06789358.

  151. M.~Perisastri. A note on the equation ax - by = 10z. Math. Stud., 37(2), (1969), 211--212. MR0263738 Zbl 0207.35302.

  152. Z.~Rábai. A note on the shuffle variant of Jesmanowicz's conjecture. Tokyo J. Math., 40(1), (2017), 153--163. MR3689983 Zbl 1390.11071.

  153. D.-M. Rao. A note on the diophantine equation (2n+1)x + (2n(n+1))y = (2n(n+1)+1)z. J Sichuan Univ., Nat. Sci., 6(1), (1960), 79--80. (in Chinese).

  154. J.-H. Ren and J.-K. Zhang. On the diophantine equation ax + by = cz. J. Northwest Univ., Nat. Sci., 19(1), (1989), 12--22. (in Chinese). MR1018048 Zbl 0695.10016.

  155. R.~Scott. On the equation px-qy = c and ax+by = cz. J. Number Theory, 44(1), (1993), 153--165. MR1225949 Zbl 0786.11020.

  156. R.~Scott and R.~Styer. On px - qy = c and related three term exponential diophantine equations with prime bases. J. Number Theory, 105(2), (2004), 212--234. MR2040155 Zbl 1080.11032.

  157. R.~Scott and R.~Styer. Number of solutions to ax + by = cz. Publ. Math. Debrecen, 88(1--2), (2016), 131--138. MR3452168 Zbl 1374.11057.

  158. T.~N. Shorey and R.~Tijdeman. Exponential Diophantine Equations. Cambridge Univ. Press, Cambridge, 1986. MR0891406 Zbl 1156.11015.

  159. W.~Sierpinski. On the equation 3x + 4y = 5z. Wiadom. Math., 1(2), 1955/1956), 194--195. (in Polish). MR0105384 Zbl 0074.27204.

  160. G.~Soydan, M.~Demirci, I.~N. Cangul, and A.~Togbé. On the conjecture of Jesmanowicz. Int. J. Appl. Math. Stat., 56(6), (2017), 46--72. MR3685484

  161. L.-J. Su and X.-X. Li. The exponential diophantine equation (4m2+1)x + (5m2-1)y = (3m)z. Abst. App. Anal., pages 1--5, April 2014. MR3198228 Zbl 07022844.

  162. C.-F. Sun and Z.~Cheng. A note on Jesmanowicz' conjecture. J. Math. Wuhan, 33(5), (2013), 788--794. MR3154659

  163. C.-F. Sun and Z.~Cheng. A conjecture of Jesmanowicz concerning Pythagorean triples. Adv. Math. China, 43(2):267--275, 2014. MR3210692 Zbl 1324.11039.

  164. C.-F. Sun and Z.~Cheng. On Jesmanowicz' conjecture concerning Pythagorean triples. J. Math. Res. Appl., 35(2), (2015), 143--148. MR3242773 Zbl 1324.11039.

  165. C.-F. Sun and M.~Tang. On the diophantine equation (an)x + (bn)y = (cn)z. Chinese Math. Ann., Ser. A, 39(1), (2018), 87--94. (in Chinese). MR3821058 Zbl 06960844.

  166. H.-N. Sun. On the diophantine equation (35n)x + (612n)y = (613n)z. Master's thesis, Chongqing: Southwest Univ., 2015. (in Chinese).

  167. Q.~Sun and X.-M. Zhou. On the diophantine equation ax + by = cz. Chinese Sci. Bull., 29(1), (1984), 61. (in Chinese). MR0838063 Zbl 0565.10017.

  168. K.~Takakuwa. A remark on Jesmanowicz' conjecture. Proc. Japan Acad., Ser. A, 72A(6), (1996), 109--110. MR1404483 Zbl 0863.11025.

  169. K.~Takakuwa and Y.~Asaeda. On a conjecture on Pythagorean numbers. Proc. Japan Acad., Ser. A, 69A(7), (1993), 252--255. MR1249222 Zbl 0796.11009.

  170. K.~Takakuwa and Y.~Asaeda. On a conjecture on Pythagorean numbers II. Proc. Japan Acad., Ser. A, 69A(8), (1993), 287--290. MR1249439 Zbl 0796.11010.

  171. K.~Takakuwa and Y.~Asaeda. On a conjecture on Pythagorean numbers III. Proc. Japan Acad., Ser. A, 69A(9), (1993), 345--349. MR1261610 Zbl 0822.11025.

  172. G.~Tang. On the diophantine equation (45n)x + (28n)y = (53n)z. J. Southwest Nation. Univ., Nat. Sci., 40(1). (2014), 101--104. (in Chinese).

  173. M.~Tang and J.-X. Weng. Jesmanowicz' conjecture with Fermat numbers. Taiwan J. Math., 18(3), (2014), 925--930. MR3213395 Zbl 1357.11041.

  174. M.~Tang and Q.-H. Yang. The diophantine equation (bn)x + (2n)y = ((b+2)n)z. Colloq. Math., 132(1), (2013), 95--100. MR3106090 Zbl 1355.11026.

  175. M.~Tang and Z.-J. Yang. Jesmanowicz' conjecture revisited. Bull. Aust. Math. Soc., 88(3), (2013), 486--491. MR3189299 Zbl 1310.11041.

  176. N.~Terai. The diophantine equation ax+by=cz. Proc. Japan Acad, Ser. A, 70A(1), (1994), 22--26. MR1272664 Zbl 0812.11024.

  177. N.~Terai. The diophantine equation ax+by=cz II. Proc. Japan Acad, Ser. A, 71A(6), (1995), 109--111. MR1344658 Zbl 0842.11009.

  178. N.~Terai. The diophantine equation ax+by=cz III. Proc. Japan Acad, Ser. A, 72A(1), (1996), 20--22. MR1382780 Zbl 0858.11017.

  179. N.~Terai. Applications of a lower bound for linear forms in two logarithms to exponential diophantine equations. Acta Arith., 90(1), (1999), 17--35. MR1708700 Zbl 0933.11013.

  180. N.~Terai. On the exponential diophantine equation ax+ly=cz. Proc. Japan Acad, Ser. A, 77A(9), (2001), 151--154. MR1869111 Zbl 1009.11026.

  181. N.~Terai. On an exponential diophantine equation concerning Fibonacci numbers. In Abstracts and short communications and poster sessions, page~55, Beijing, 2002. Int. Conf. Math., Higher Edu. Press.

  182. N.~Terai. On the exponential diophantine equation (4m2+1)x + (5m2-1)y = (3m)z. Int. J. Algebra, 6(21--24), (2012), 1135--1146. MR2974671 Zbl 1271.11039.

  183. N.~Terai. On Jesmanowicz' conjecture concerning primitive Pythagorean triples. J. Number Theory, 141(2), (2014), 316--323. MR3195402 Zbl 1309.11030.

  184. N.~Terai and T.~Hibino. On the exponential diophantine equation (12m2+1)x + (13m2-1)y = (5m)z. Int. J. Algebra, 9, (2015), 261--272.

  185. N.~Terai and T.~Hibino. The exponential diophantine equation (3pm2-1)x + (p(p-3)m)y = (pm)z. Period. Math. Hung., 74(2), (2017), 227--234. MR3645689 Zbl 1399.11104.

  186. N.~Terai and K.~Takakuwa. A note on the diophantine equation ax + by = cz. Proc. Japan Acad., Ser. A, 73A(9), (1997), 161--164. MR1606004 Zbl 0910.11011.

  187. M.~Toyoizumi. On the equation ax - by = (2p)z. Math. Stud., 46(2--4), (1978), 113--115. Zbl 0526.10014.

  188. S.~Uchiyama. On the diophantine equation 2x = 3y + 13z. Math. J. Okayama Univ., 19(1), (1976/1977), 31--38. MR0432539 Zbl 0349.10013.

  189. M.~Waldschmidt. Perfect powers: Pillai's works and their developments. arXiv:0908:4031v1, 27 August 2009.

  190. J.-H. Wang and M.-J. Deng. On the diophantine equation (a2-b2)x + (2ab)y = (a2+b2)z. J. Nat. Sci. Heilongjiang Univ., 13(4), (1996), 23--25. (in Chinese). MR1445399 Zbl 1076.11511.

  191. J.-P. Wang, T.-T. Wang, and W.-P. Zhang. A note on the exponential diophantine equation (4m2+1)x + (5m2-1)y = (3m)z. Colloq. Math., 139(1), (2015), 121--126. MR3332737 Zbl 1364.11087.

  192. L.-L. Wang. On the diophantine equation (39n)x + (760n)y = (761n)z. Master's thesis, Chongqing: Southwest Univ., 2011. (in Chinese).

  193. T.-T. Wang, X.-H. Wang, and Y.-Z. Jiang. An application of the Baker method to Jesmanowicz' conjecture on Pythagorean triples. Rev. R. Acad. Cienc. Exactas Fis. Nat., Ser. A Mat., RASCOM, 112(2), (2018), 385--390. MR3775276 Zbl 06859078.

  194. X.-H. Wang and S.~Gou. On Miyazaki's conjecture on primitive Pythagorean numbers. Math. Pract. Theory, 44(8), (2014), 287--290. (in Chinese). MR3237472 Zbl 1340.11043.

  195. J.~Y. Xia and P.-Z. Yuan. On the Terai-Jesmanowicz conjecture. Acta Math. Sinica, English Ser., 24(12), (2008), 2061--2064. MR2453085 Zbl 1234.11035.

  196. J.-J. Xing. On Jesmanowicz' conjecture concerning Pythagorean numbers. Master's thesis, Chongqing: Southwest Univ., 2015. (in Chinese).

  197. H.~Yang and R.-Q. Fu. A kind of an exponential diophantine system and its integer solutions. J. Northwest Univ., Nat. Sci., 43(4), (2013), 524--526. (in Chinese). MR3183801 Zbl 1299.11036.

  198. H.~Yang and R.-Q. Fu. A note on Jesmanowicz' conjecture concerning primitive Pythagorean triples. J. Number Theory, 156(1), (2015), 183--194. MR3360336 Zbl 1395.11066.

  199. H.~Yang and R.-Q. Fu. Fermat primes and Jesmanowicz' conjecture. Adv. Math. China, 46(6), (2017), 857--866. (in Chinese). MR3778513 Zbl 1399.11107.

  200. H.~Yang, R.-Z. Ren, and R.-Q. Fu. On Jesmanowicz' conjecture concerning Pythagorean numbers. Math. J. Wuhan, 37(3), (2017), 506--512. (in Chinese). Zbl 1399.11108.

  201. S.-C. Yang and B.~He. The solutions of a class of exponential diophantine equations. Adv. Math. China, 41(5), (2012), 565--573. (in Chinese). MR3058640 Zbl 1274.11097

  202. X.-Z. Yang. On the diophantine equation ax + by = cz. J. Sichuan Univ., Nat. Sci., 1985(4), (1985), 151--158. (in Chinese). MR0843518

  203. Z.-J. Yang and M.~Tang. On the diophantine equation ((8n)x + (15n)y = (17n)z. Bull. Aust. Math. Soc., 86(2), (2012), 348--352. MR2979995 Zbl 1272.11048

  204. Z.-J. Yang and J.-X. Weng. On the diophantine equation (12n)x + (35n)y = (37n)z. Pure Appl. Math., 28(5), (2012), 698--704. (in Chinese). MR3053137 Zbl 1274.11098

  205. Y.-H. Yu and Z.-P. Li. The exceptional solutions of the exponential diophantine equation (bn)x + (2n)y = ((b+2)n)z. Math. Pract. Theory, 44(18), (2014), 290--293. MR3328405 Zbl 1324.11034

  206. P.-Z. Yuan and Q.~Han. Jesmanowicz' conjecture and related equations. Acta Arith., 184(1), (2018), 37--49. MR3826639 Zbl 06921720

  207. X.-W. Zhang and W.-P. Zhang. The exponential diophantine equation ((22m-1)n)x + (2m+1 n)y = ((22m+1)n)z. Bull. Math. Soc. Math. Roum., Nouv. Sér., 57(3), (2014), 337--344. MR3241772 Zbl 1340.11044

  208. C.-Y. Zheng. A note on coprime cases of Jesmanowicz' conjecture. J. Huaihai Engin. College, Nat. Ser., 3, (2017), 1--3. (in Chinese).

  209. X.-E. Zhou. On the diophantine equation px + qy = 2z (200 J. Nat. Sci. Hainan Univ., 32(3), (2014), 197--199. (in Chinese).

  210. M.-H. Zhu and X.-X. Li. The exponential diophantine equation 4x + by = (b+4)z. J. Math. Wuhan, 36(4), (2016), 782--786. (in Chinese). Zbl 1363.11062.




Maohua Le
Institute of Mathematics, Lingnan Normal College
Zhangjiang, Guangdong 524048,
China.
e-mail: lemaohua2008@163.com

Reese Scott
Somerville, MA, USA.

Robert Styer
Department of Mathematics and Statistics,
Villanova University,
800 Lancaster Avenue
Villanova, PA 19085 USA.
e-mail: robert.styer@villanova.edu
http://www41.homepage.villanova.edu/robert.styer/

http://www.utgjiu.ro/math/sma