Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
Pillai's Diophantine equation; Lehmer sequence; primitive divisor
Pillai's Diophantine equation; Lehmer sequence; primitive divisor
Summary:
S. S. Pillai proved that for a fixed positive integer $a$, the exponential Diophantine equation $x^y-y^x= a$, $\min (x,y)>1$, has only finitely many solutions in integers $x$ and $y$. We prove that when $a$ is of the form $2z^2$, the above equation has no solution in integers $x$ and $y$ with $\gcd (x,y)=1$.
References:
[1] Bilu, Y., Hanrot, G., Voutier, P. M.: Existence of primitive divisors of Lucas and Lehmer numbers. J. Reine Angew. Math. 539 (2001), 75-122. DOI 10.1515/crll.2001.080 | MR 1863855 | Zbl 0995.11010
[2] Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24 (1997), 235-265. DOI 10.1006/jsco.1996.0125 | MR 1484478 | Zbl 0898.68039
[3] Cohen, H.: A Course in Computational Number Theory. Graduate Texts in Mathematics 138. Springer, Berlin (1993). DOI 10.1007/978-3-662-02945-9 | MR 1228206 | Zbl 0786.11071
[4] Cohn, J. H. E.: Square Fibonacci numbers, etc. Fibonacci Q. 2 (1964), 109-113. MR 0161819 | Zbl 0126.07201
[5] Hoque, A.: On a class of Lebesgue-Ramanujan-Nagell equations. Period. Math. Hung. 88 (2024), 418-428. DOI 10.1007/s10998-023-00564-z | MR 4751334 | Zbl 7880178
[6] Hua, L. K.: Introduction to Number Theory. Springer, Berlin (1982). DOI 10.1007/978-3-642-68130-1 | MR 665428 | Zbl 0483.10001
[7] Le, M.: On the Diophantine equation $y^x-x^y=z^2$. Rocky Mt. J. Math. 37 (2007), 1181-1185. DOI 10.1216/rmjm/1187453105 | MR 2360292 | Zbl 1146.11019
[8] Luca, F., Mignotte, M.: On the equation $y^x \pm x^y = z^2$. Rocky Mt. J. Math. 30 (2000), 651-661. DOI 10.1216/rmjm/1022009287 | MR 1787004 | Zbl 1014.11024
[9] Pillai, S. S.: On the indeterminate equation $x^y-y^x = a$. Annamalai Univ. J. 1 (1932), 59-61. Zbl 0005.05302
[10] Robbins, N.: Fibonacci numbers of the form $cx^2$, where $1 \leq c \leq 1000$. Fibonacci Q. 28 (1990), 306-315. MR 1077496 | Zbl 0728.11013
[11] Voutier, P. M.: Primitive divisors of Lucas and Lehmer sequences. Math. Comput. 64 (1995), 869-888. DOI 10.1090/S0025-5718-1995-1284673-6 | MR 1284673 | Zbl 0832.11009
[12] Waldschmidt, M.: Perfect powers: Pillai's works and their developments. Collected works of S. Sivasankaranarayana Pillai. Volume 1 Ramanujan Mathematical Society, Mysore (2010), xxii--xlvii. MR 2766491
[13] Yuan, P.: On the Diophantine equation $ax^2 + by^2 = ck^n$. Indag. Math., New Ser. 16 (2005), 301-320. DOI 10.1016/S0019-3577(05)80030-8 | MR 2319301 | Zbl 1088.11024
Search
Partner of
