On the exponential Diophantine equation mx+(m+1)y=(1+m+m2)z

Murat Alan

doi:10.2478/auom-2021-0032

.blurhash-client-img { display: none !important; }

On the exponential Diophantine equation mx+(m+1)y=(1+m+m2)z

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

Volume 29 (2021): Issue 3 (November 2021)

By: Murat Alan

Open Access

|Nov 2021

Abstract

Let m > 1 be a positive integer. We show that the exponential Diophantine equation m^x + (m + 1)^y = (1 + m + m²)^z has only the positive integer solution (x, y, z) = (2, 1, 1) when m ≥ 2.

References

[1] Alan, M. (2018) On the exponential Diophantine equation (18m2 + 1)x + (7m2 − 1)y = (5m)z, Turk J Math, 42, 1990-1999.10.3906/mat-1801-76
Search in Google Scholar Back to article
[2] Alan, M. (2020) On the Exponential Diophantine Equation (m2 + m +1)x + my = (m + 1)z, Mediterr. J. Math., 17:189, 1-8.
Search in Google Scholar Back to article
[3] Bertók, C. (2016) The complete solution of the Diophantine equation (4m2 + 1)x + (5m2 − 1)y = (3m)z, Period Math Hung, 72, 37-42.10.1007/s10998-016-0111-x
Search in Google Scholar Back to article
[4] Bilu, Y., Hanrot, G., & Voutier, P.M. (2001) Existence of primitive divisors of Lucas and Lehmer numbers (with Appendix by Mignotte), J. Reine Angew. Math., 539, 75122.
Search in Google Scholar Back to article
[5] Birkhoff, G.D., & Vandiver, H.S. (1904) On the integral divisors of an−bn, Ann. of Math. Second Series, 5(4), 173-180.10.2307/2007263
Search in Google Scholar Back to article
[6] Cao, Z. (1999) A note on the Diophantine equation ax + by = cz, Acta Arith., 91, 85-93.
Search in Google Scholar Back to article
[7] Deng,N., Wu, D., & Yuan, P. (2019) The exponential Diophantine equation (3am2 − 1)x + (a(a − 3)m2 + 1)y = (am)z, Turk J Math, 43, 2561-2567.1410.3906/mat-1905-20
Search in Google Scholar Back to article
[8] Fu, R., & Yang H. (2017) On the exponential diophantine equation (am2 + 1)x + (bm2 − 1)y = (cm)z with c | m, Period Math. Hung., 75, 143-149.
Search in Google Scholar Back to article
[9] Hadano, T. (1976/1977) On the Diophantine equation ax+by = cz, Math. J. Okayama Univ., 19(1), 25-29.
Search in Google Scholar Back to article
[10] Hua, L.K. (1982) Introduction to Number Theory, Springer, Berlin, Germany.
Search in Google Scholar Back to article
[11] Jeśmanowicz, L. (1955/1956) Some remarks on Pythagorean numbers, Wiadom Mat 1, 196-202.
Search in Google Scholar Back to article
[12] Kzldere, E., Miyazaki, T. & Soydan G. (2018) On the Diophantine equation ((c + 1)m2 + 1)x + (cm2 − 1)y = (am)z, Turk J Math, 42, 2690-2698.10.3906/mat-1803-14
Search in Google Scholar Back to article
[13] Kzldere, E., Le, M., & Soydan, G. (2020) A note on the ternary purely exponential Diophantine equation Ax +By = Cz with A+B = C2, Studia Scientiarum Mathematicarum Hungarica, 57(2), 200-206.10.1556/012.2020.57.2.1457
Search in Google Scholar Back to article
[14] Le, M. (1995) Some exponential Diophantine equations I: the equation D1x2 − D2y2 = λkz, J. Number. Theory, 55(2), 209-221.10.1006/jnth.1995.1138
Search in Google Scholar Back to article
[15] Le, M, & Soydan, G., (2020) An application of Baker’s method to the Jeśmanowicz’ conjecture on primitive Pythagorean triples, Period Math Hung, 80, 74-80.10.1007/s10998-019-00295-0
Search in Google Scholar Back to article
[16] Le, M., Scott, R. & Styer, R. (2019) A Survey on the Ternary Purely Exponential Diophantine Equation ax + by = cz, Surveys in Mathematics and its Applications, 214, 109-140.
Search in Google Scholar Back to article
[17] Ma, M. & Chen, Y. (2017) Jeśmanowicz’ conjecture on Pythagorean triples, Bulletin of the Australian Mathematical Society, 96, 30-35.10.1017/S0004972717000107
Search in Google Scholar Back to article
[18] Mahler, K. (1933) Zur Approximation algebraischer Zahlen I: Uber den grossten Primtriler binarer formen, Math Ann 107, 691-730.10.1007/BF01448915
Search in Google Scholar Back to article
[19] Miyazaki, T. (2010) Exceptional cases of Terai’s conjecture on Diophantine equations, Arch Math, 95, 519-527.10.1007/s00013-010-0201-6
Search in Google Scholar Back to article
[20] Miyazaki, T. (2013) Generalizations of classical results on Jeśmanowicz’ conjecture concerning Pythagorean triples, Journal of Number Theory, 133, 583-595.10.1016/j.jnt.2012.08.018
Search in Google Scholar Back to article
[21] Miyazaki, T. & Terai, N. (2015) On Jeśmanowicz’ conjecture concerning primitive Pythagorean triples II, Acta Math Hungar, 147, 286-293.10.1007/s10474-015-0552-3
Search in Google Scholar Back to article
[22] Miyazaki, T. & Terai, N. (2019) A study on the exponential Diophantine equation ax + (a + b)y = bz, Publ. Math. Debrecen, 95, 19-37.10.5486/PMD.2019.8283
Search in Google Scholar Back to article
[23] Soydan, G,, Demirci, M., Cangül, I. N. & Togbé, A. (2017) On the conjecture of Jeśmanowicz, Int J Appl Math. Stat, 56, 46-72.
Search in Google Scholar Back to article
[24] Su, J. & Li, X. (2014) The Exponential Diophantine Equation (4m2 +1)x + (5m2 − 1)y = (3m)z, Abstr. Appl. Anal. Article ID 670175.
Search in Google Scholar Back to article
[25] Terai, N. (1994) The Diophantine equation ax + by = cz, Proc Japan Acad Ser A Math Sci, 70, 22-26.10.3792/pjaa.70.213
Search in Google Scholar Back to article
[26] Terai, N. (1999) Applications of a lower bound for linear forms in two logarithms to exponential Diophantine equations, Acta Arith 90, 17-35.10.4064/aa-90-1-17-35
Search in Google Scholar Back to article
[27] Terai, N. (2012) On the exponential Diophantine equation (4m2 + 1)x + (5m2 − 1)y = (3m)z, Int J Algebra 6, 1135-1146.
Search in Google Scholar Back to article
[28] Terai, N. (2014) On Jesmanowicz conjecture concerning primitive Pythagorean triples, J Number Theory, 141, 316-323.10.1016/j.jnt.2014.02.009
Search in Google Scholar Back to article
[29] Terai, N. & Hibino, T. (2015) On the exponential Diophantine equation (12m2 + 1)x + (13m2 − 1)y = (5m)z, Int J Algebra, 9, 261-272.10.12988/ija.2015.5529
Search in Google Scholar Back to article
[30] Terai, N. & Hibino, T. (2016) On the Exponential Diophantine Equation ax + lby = cz, International Journal of Algebra, 10, 393-403.10.12988/ija.2016.6747
Search in Google Scholar Back to article
[31] Terai, N. & Hibino, T. (2017) On the exponential Diophantine equation (3pm2 −1)x +(p(p−3)m2 +1)y = (pm)z, Period Math Hung, 74, 227-234.10.1007/s10998-016-0162-z
Search in Google Scholar Back to article
[32] Sierpinski, W. (1956) On the equation 3x + 4y = 5z (in Polish), Wiadom Mat 1, 194-195.
Search in Google Scholar Back to article
[33] Voutier, P.M. (1995) Primitive divisors of Lucas and Lehmer sequences, Math. Comp., 64, 869-888.
Search in Google Scholar Back to article
[34] Yuan, P.Z. and Han, Q. (2018) Jeśmanowicz’ conjecture and related equations, Acta Arith., 184, 37-49.
Search in Google Scholar Back to article
[35] Wiles, A. (1995) Modular Elliptic Curves and Fermat’s Last Theorem, Annals of Mathematics, 141(3), 443-551.10.2307/2118559
Search in Google Scholar Back to article
[36] Zsigmondy, K. (1892) Zur Theorie der Potenzreste, Monatsh. Math. Phys., 3, 265-284.
Search in Google Scholar Back to article