A remark on a Diophantine equation of S. S. Pillai

Azizul Hoque

Czechoslovak Mathematical Journal (2024)

  • Volume: 74, Issue: 3, page 897-903
  • ISSN: 0011-4642

Abstract

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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 2 z 2 , the above equation has no solution in integers x and y with gcd ( x , y ) = 1 .

How to cite

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Hoque, Azizul. "A remark on a Diophantine equation of S. S. Pillai." Czechoslovak Mathematical Journal 74.3 (2024): 897-903. <http://eudml.org/doc/299312>.

@article{Hoque2024,
abstract = {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 $\gcd (x,y)=1$.},
author = {Hoque, Azizul},
journal = {Czechoslovak Mathematical Journal},
keywords = {Pillai's Diophantine equation; Lehmer sequence; primitive divisor},
language = {eng},
number = {3},
pages = {897-903},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A remark on a Diophantine equation of S. S. Pillai},
url = {http://eudml.org/doc/299312},
volume = {74},
year = {2024},
}

TY - JOUR
AU - Hoque, Azizul
TI - A remark on a Diophantine equation of S. S. Pillai
JO - Czechoslovak Mathematical Journal
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 74
IS - 3
SP - 897
EP - 903
AB - 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 $\gcd (x,y)=1$.
LA - eng
KW - Pillai's Diophantine equation; Lehmer sequence; primitive divisor
UR - http://eudml.org/doc/299312
ER -

References

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  1. Bilu, Y., Hanrot, G., Voutier, P. M., 10.1515/crll.2001.080, J. Reine Angew. Math. 539 (2001), 75-122. (2001) Zbl0995.11010MR1863855DOI10.1515/crll.2001.080
  2. Bosma, W., Cannon, J., Playoust, C., 10.1006/jsco.1996.0125, J. Symb. Comput. 24 (1997), 235-265. (1997) Zbl0898.68039MR1484478DOI10.1006/jsco.1996.0125
  3. Cohen, H., 10.1007/978-3-662-02945-9, Graduate Texts in Mathematics 138. Springer, Berlin (1993). (1993) Zbl0786.11071MR1228206DOI10.1007/978-3-662-02945-9
  4. Cohn, J. H. E., Square Fibonacci numbers, etc, Fibonacci Q. 2 (1964), 109-113. (1964) Zbl0126.07201MR0161819
  5. Hoque, A., 10.1007/s10998-023-00564-z, Period. Math. Hung. 88 (2024), 418-428. (2024) Zbl7880178MR4751334DOI10.1007/s10998-023-00564-z
  6. Hua, L. K., 10.1007/978-3-642-68130-1, Springer, Berlin (1982). (1982) Zbl0483.10001MR665428DOI10.1007/978-3-642-68130-1
  7. Le, M., 10.1216/rmjm/1187453105, Rocky Mt. J. Math. 37 (2007), 1181-1185. (2007) Zbl1146.11019MR2360292DOI10.1216/rmjm/1187453105
  8. Luca, F., Mignotte, M., 10.1216/rmjm/1022009287, Rocky Mt. J. Math. 30 (2000), 651-661. (2000) Zbl1014.11024MR1787004DOI10.1216/rmjm/1022009287
  9. Pillai, S. S., On the indeterminate equation x y - y x = a , Annamalai Univ. J. 1 (1932), 59-61. (1932) Zbl0005.05302
  10. Robbins, N., Fibonacci numbers of the form c x 2 , where 1 ≤ c ≤ 1000 , Fibonacci Q. 28 (1990), 306-315. (1990) Zbl0728.11013MR1077496
  11. Voutier, P. M., 10.1090/S0025-5718-1995-1284673-6, Math. Comput. 64 (1995), 869-888. (1995) Zbl0832.11009MR1284673DOI10.1090/S0025-5718-1995-1284673-6
  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. (2010) MR2766491
  13. Yuan, P., 10.1016/S0019-3577(05)80030-8, Indag. Math., New Ser. 16 (2005), 301-320. (2005) Zbl1088.11024MR2319301DOI10.1016/S0019-3577(05)80030-8

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