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1
Content available remote

A note on the diophantine equation $x² + b^y = c^z$

100%
Le M.
Acta Arithmetica
|
1995
|
tom 71
|
nr 3
253-257
2
Content available remote

A note on perfect powers of the form $x^{m-1} + ... + x + 1$

100%
Le M.
Acta Arithmetica
|
1995
|
tom 69
|
nr 1
91-98
3
Content available remote

A note on Jeśmanowicz' conjecture

100%
Le M.
Colloquium Mathematicum
|
1996
|
tom 69
|
nr 1
47-51
4
Content available remote

On Terai's conjecture concerning Pythagorean numbers

100%
Le M.
Acta Arithmetica
|
2001
|
tom 100
|
nr 1
41-45
5
Content available remote

Upper bounds for class numbers of real quadratic fields

100%
Le M.
Acta Arithmetica
|
1994
|
tom 68
|
nr 2
141-144
6
Content available remote

The solvability of the diophantine equation $D_1x^2-D_2y^4=1$

100%
Le M.
Colloquium Mathematicum
|
1995
|
tom 68
|
nr 2
165-170
7
Content available remote

A note on the number of solutions of the generalized Ramanujan-Nagell equation $x²-D = k^n$

100%
Le M.
Acta Arithmetica
|
1996-1997
|
tom 78
|
nr 1
11-18
8
Content available remote

On the diophantine equation $(x^m + 1)(x^n + 1) = y²$

100%
Le M.
Acta Arithmetica
|
1997
|
tom 82
|
nr 1
17-26
EN
1. Introduction. Let ℤ, ℕ, ℚ be the sets of integers, positive integers and rational numbers respectively. In [7], Ribenboim proved that the equation    (1) $(x^m + 1)(x^n + 1) = y²$, x,y,m,n ∈ ℕ, x > 1, n > m ≥ 1, has no solution (x,y,m,n) with 2|x and (1) has only finitely many solutions (x,y,m,n) with 2∤x. Moreover, all solutions of (1) with 2∤x satisfy max(x,m,n) < C, where C is an effectively computable constant. In this paper we completely determine all solutions of (1) as follows.   Theorem. Equation (1) has only the solution (x,y,m,n)=(7,20,1,2).
9
Content available remote

A note on the diophantine equation $(x^m-1)/(x-1) = y^n$

100%
Le M.
Acta Arithmetica
|
1993
|
tom 64
|
nr 1
19-28
10
Content available remote

On the diophantine equation $D₁x² + D₂ = 2^{n+2}$

100%
Le M.
Acta Arithmetica
|
1993
|
tom 64
|
nr 1
29-41
11
Content available remote

A note on primes p with $σ(p^m)=z^n$

100%
Le M.
Colloquium Mathematicum
|
1991
|
tom 62
|
nr 2
193-196
12
Content available remote

A correction to the paper "Upper bounds for class numbers of real quadratic fields" (Acta Arith. 68 (1994), 141-144)

100%
Le M.
Acta Arithmetica
|
1995
|
tom 72
|
nr 4
399
13
Content available remote

A note on the integer solutions ofhyperelliptic equations

100%
Le M.
Colloquium Mathematicum
|
1995
|
tom 68
|
nr 2
171-177
14
Content available remote

Lower bounds for the solutions in the second case of Fermat's equation with prime power exponents

100%
Le M.
Colloquium Mathematicum
|
1993
|
tom 65
|
nr 2
227-229
15
Content available remote

On the diophantine equation D₁x⁴ -D₂y² = 1

100%
Le M.
Acta Arithmetica
|
1996
|
tom 76
|
nr 1
1-9
16
Content available remote

A conjecture concerning the exponential diophantine equation $a^x + b^y = c^z$

100%
Le M.
Acta Arithmetica
|
2003
|
tom 106
|
nr 4
345-353
17
Content available remote

A note on Jeśmanowicz' conjecture concerning primitive Pythagorean triplets

100%
Le M.
Acta Arithmetica
|
2009
|
tom 138
|
nr 2
137-144
18
Content available remote

On the number of solutions of the generalized Ramanujan-Nagell equation $x²-D = 2^{n+2}$

100%
Le M.
Acta Arithmetica
|
1991-1992
|
tom 60
|
nr 2
149-167
19
Content available remote

A note on the diophantine equation ${k\choose 2}-1=q^n+1$

100%
Le M.
Colloquium Mathematicum
|
1998
|
tom 76
|
nr 1
31-34
EN
In this note we prove that the equation ${k\choose 2}-1=q^n+1$, $q\ge 2, n\ge 3$, has only finitely many positive integer solutions $(k,q,n)$. Moreover, all solutions $(k,q,n)$ satisfy $k\lt10^{10^{182}}$, $q\lt10^{10^{165}}$ and $n\lt 2\cdot 10^{17}$.
20
Content available remote

A note on ternary purely exponential diophantine equations

64%
Hu Y., Le M.
Acta Arithmetica
|
2015
|
tom 171
|
nr 2
173-182
EN
Let a,b,c be fixed coprime positive integers with min{a,b,c} > 1, and let m = max{a,b,c}. Using the Gel'fond-Baker method, we prove that all positive integer solutions (x,y,z) of the equation $a^x+b^y = c^z$ satisfy max{x,y,z} < 155000(log m)³. Moreover, using that result, we prove that if a,b,c satisfy certain divisibility conditions and m is large enough, then the equation has at most one solution (x,y,z) with min{x,y,z} > 1.
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