(PDF) On a problem of Diophantus for higher powers
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On a problem of Diophantus for higher powers
Andrej Dujella
2003, Mathematical proceedings of the Cambridge Philosophical Society
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Abstract
Let k ≥ 3 be an integer. We study the possible existence of finite sets of positive integers such that the product of any two of them increased by 1 is a k-th power. The Greek mathematician Diophantus observed that the rational numbers 1 16 , 33 16 , 17 4 and 105 16 have the following property: the product of any two of them increased by 1 is a square of a rational number. Later, Fermat found a set of four positive integers with the above property, namely the set {1, 3, 8, 120}. We call a Diophantine m-tuple any set of m positive integers a 1 , . . . , a m such that a i a j + 1 is a perfect square whenever 1 ≤ i < j ≤ m. It was known already to Euler that there are infinitely many Diophantine quadruples (see for instance [5, pp. 513-520]). Among the broad literature on that topic, let us mention that Baker & Davenport [3] proved that {1, 3, 8} cannot be extended to a Diophantine quintuple, a result improved by Dujella & Pethő [10], who showed that even {1, 3} cannot be extended to a Diophantine quintuple. The first absolute upper bound for the size of Diophantine m-tuples was given by the second author in , where it was proved that Diophantine 9-tuples do not exist. Very recently, he was able to considerably improve upon his result, by showing [9] that there exist no Diophantine sextuple and only finitely many Diophantine quintuples. However, the question of the existence of a Diophantine quintuple remains a challenging open problem. We refer to for further references on this topic. In the present work, we are interested in an analogous problem, namely the existence of sets {a, b, c} of positive integers such that the three numbers ab + 1, ac + 1 and bc + 1 are perfect k-th powers, for an integer k ≥ 3. Examples of such triples for k=3 and k = 4 are given, respectively, by {2, 171, 25326} and {1352, 9539880, 9768370}. To our knowledge, no example of such triple is known for k ≥ 5. In order to investigate this question, we study a slightly more general problem, recently considered by Gyarmati [12]. Let N ≥ 1 and k ≥ 3 be integers. Let A and B be subsets of {1, . . . , N } such that ab + 1 is a perfect k-th power whenever a ∈ A and b ∈ B. What can be said about the cardinalities of the sets A and B ? Let |S| denote the cardinality of a finite set S. Using elementary arguments, Gyarmati proved that min{|A|, |B|} ≤ 1 + (log log N )/ log(k -1). As a corollary of our main result, we show that, except for small values of k, we have the considerably better 2000 Mathematics Subject Classification : 11D61.
Key takeaways
AI
The existence of finite sets where products increased by 1 yield k-th powers is rigorously analyzed.
Theorem 1 establishes that for k ≥ 3, k must be ≤ 176 for certain integer sets.
Corollary 1 asserts no quadruples exist for k ≥ 177 under the specified conditions.
The paper improves bounds on the cardinalities of sets A and B, establishing min{|A|, |B|} ≤ 2.
New results extend classical Diophantine problem solutions, particularly regarding k ≥ 6 and k = 3 cases.
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On a problem of Diophantus for higher powers
YANN BUGEAUD (Strasbourg) & ANDREJ DUJELLA (Zagreb)
Abstract. Let k ≥ 3 be an integer. We study the possible existence of
finite sets of positive integers such that the product of any two of them
increased by 1 is a k-th power.
1. Introduction
1 33 17
, 16 , 4
The Greek mathematician Diophantus observed that the rational numbers 16
have
the
following
property:
the
product
of
any
two
of
them
increased
by
1 is a
and 105
16
square of a rational number. Later, Fermat found a set of four positive integers with the
above property, namely the set {1, 3, 8, 120}. We call a Diophantine m-tuple any set of m
positive integers a1 , . . . , am such that ai aj + 1 is a perfect square whenever 1 ≤ i < j ≤ m.
It was known already to Euler that there are infinitely many Diophantine quadruples (see
for instance [5, pp. 513–520]). Among the broad literature on that topic, let us mention
that Baker & Davenport [3] proved that {1, 3, 8} cannot be extended to a Diophantine
quintuple, a result improved by Dujella & Pethő [10], who showed that even {1, 3} cannot
be extended to a Diophantine quintuple. The first absolute upper bound for the size of
Diophantine m-tuples was given by the second author in [7], where it was proved that
Diophantine 9-tuples do not exist. Very recently, he was able to considerably improve
upon his result, by showing [9] that there exist no Diophantine sextuple and only finitely
many Diophantine quintuples. However, the question of the existence of a Diophantine
quintuple remains a challenging open problem. We refer to [6] for further references on
this topic.
In the present work, we are interested in an analogous problem, namely the existence
of sets {a, b, c} of positive integers such that the three numbers ab + 1, ac + 1 and bc + 1 are
perfect k-th powers, for an integer k ≥ 3. Examples of such triples for k=3 and k = 4 are
given, respectively, by {2, 171, 25326} and {1352, 9539880, 9768370}. To our knowledge, no
example of such triple is known for k ≥ 5. In order to investigate this question, we study
a slightly more general problem, recently considered by Gyarmati [12]. Let N ≥ 1 and
k ≥ 3 be integers. Let A and B be subsets of {1, . . . , N } such that ab + 1 is a perfect k-th
power whenever a ∈ A and b ∈ B. What can be said about the cardinalities of the sets
A and B ? Let |S| denote the cardinality of a finite set S. Using elementary arguments,
Gyarmati [12] proved that min{|A|, |B|} ≤ 1 + (log log N )/ log(k − 1). As a corollary of our
main result, we show that, except for small values of k, we have the considerably better
2000 Mathematics Subject Classification : 11D61.
estimate min{|A|, |B|} ≤ 2. We also provide an absolute (i.e. independent of N ) upper
bound for min{|A|, |B|} for the other values of k.
Our proofs rest on classical tools of Diophantine approximation, namely the theory
of linear forms in logarithms and sharp irrationality measures for certain k-th roots of
rational numbers.
2. Statement of the results
Theorem 1. Let k ≥ 3 and 0 < a < b < c < d be integers such that the four numbers
ac + 1, ad + 1, bc + 1
and bd + 1
are perfect k-th powers. Then we have k ≤ 176.
Remark : The proof of Theorem 1 rests on the theory of linear forms in two logarithms
of algebraic numbers, and heavily depends on a refinement obtained by Shorey [17], who
was first to notice that one gets the best possible estimates when the algebraic numbers
involved are close to 1. Shorey’s trick has numerous applications (see [19] for a survey),
−1
= y q and
for instance to the exponential Diophantine equations axn − by n = c, xx−1
y n −1
xm −1
x−1 = y−1 , considered, respectively, in [15], [13] and [4]. The numerical value we get in
Theorem 1 is remarkably small. This is due to the use of the sharp estimate of Mignotte
[16] (see Lemma 2 below), and to the fact that our problem allows us to take a very large
ray ρ in the application of Lemma 2.
As an immediate corollary, we derive from Theorem 1 new results on the generalization
of the problem of Diophantus mentioned in the Introduction.
Corollary 1. For any integer k ≥ 177, there exist no set of four positive integers such
that the product of any two of them increased by 1 is a perfect k-th power.
Corollary 2 below considerably improved Theorem 1 of Gyarmati [12] when the integer
k is not too small.
Corollary 2. Let k ≥ 177 be an integer and A and B be sets of positive integers such
that ab + 1 is a perfect k-th power for any a ∈ A and b ∈ B. Then we have
min{|A|, |B|} ≤ 2.
Corollary 2 follows easily from Theorem 1. Indeed, if a1 < a2 < a3 (resp. b1 < b2 < b3 )
belong to A (resp. to B), then we have either a1 < a2 < b2 < b3 or b1 < b2 < a2 < a3 , and
we may apply Theorem 1.
Theorem 2. Let 4 ≤ k ≤ 176 be an integer. Assume that the integers 0 < a < b < c1 <
. . . < cm are such that aci + 1 and bci + 1 are perfect k-th powers for any 1 ≤ i ≤ m.
Then there exists an effectively computable constant C1 (k), depending only on k, such
that m ≤ C1 (k). More precisely, we may take C1 (4) = 3 and C1 (k) = 2 for k ≥ 5.
Remark : The proof of Theorem 2 depends on a result of Evertse [11] on Thue equations
aX n +bY n = c, whose proof uses hypergeometric methods. For k ≥ 6, we could also derive
Theorem 2 from Theorem 1 of Baker [1].
Unfortunately, the proof of Theorem 2 gives nothing for k = 3. In that case, we need
a stronger assumption.
Theorem 3. Assume that the integers 0 < a < b < c < d1 < . . . < dm are such that
adi + 1, bdi + 1 and cdi + 1 are perfect cubes for any 1 ≤ i ≤ m. Then m ≤ 6.
New results on the problem considered by Gyarmati and on the generalization of the
problem of Diophantus follow from Theorems 2 and 3.
Corollary 3. Let 3 ≤ k ≤ 176 be an integer and A and B be sets of positive integers
such that ab + 1 is a perfect k-th power for any a ∈ A and b ∈ B. Then there exists an
effectively computable constant C2 (k), depending only on k, such that
min{|A|, |B|} ≤ C2 (k).
More precisely, we may take C2 (3) = 8, C2 (4) = 4 and C2 (k) = 3 for k ≥ 5.
The statement of Corollary 3 follows directly from Theorems 2 and 3, as Corollary 1
follows from Theorem 1.
Corollary 4. Let k ≥ 2 be an integer. Assume that the integers 0 < a1 < a2 < . . . < am
are such that ai aj + 1 are perfect k-th powers whenever 1 ≤ i < j ≤ m. Then there exists
an effectively computable constant C3 (k), depending only on k, such that m ≤ C3 (k).
More precisely, we may take C3 (2) = 5, C3 (3) = 7, C3 (4) = 5, C3 (k) = 4 for 5 ≤ k ≤ 176
and C3 (k) = 3 for k ≥ 177.
The statement of Corollary 4 for k ≥ 4 follows directly from Corollaries 2 and 3. The
statement for k = 2 is just the main result from [9], while the statement for k = 3 will be
proved in Section 4 using a special gap principle.
One can obtain weeker results than in Theorem 1 by using a result of Shorey &
Nesterenko [20] on irrationality measures of k-th roots of certain rational numbers, derived
from a theorem of Baker [2]. Already in a few papers (see for instance [18], [13], [4] and
the survey [19]), the authors have successfully combined this method with the theory of
linear forms in logarithms. Here, we are able to complement Theorem 2 in the range
11 ≤ k ≤ 176.
Theorem 4. Let 11 ≤ k ≤ 176. Then there are only finitely many quadruples of integers
0 < a < b < c < d such that the four numbers
ac + 1, ad + 1, bc + 1
and bd + 1
are perfect k-th powers.
Remark : Let us mention that for k = 3, 4 and 6, there are triples a < b < c of positive
integers such that ac + 1 and bc + 1 are perfect k-powers. E.g. for k = 6 the triple
(a, b, c) = (8, 45, 91) has the above property. Moreover, for k = 3 and k = 4 there exist
infinite families of such triples.
For k = 3, let (xn , yn ) denote the sequence of the positive integer solutions of Pell
equation x2 − 7y 2 = 1 and let n ≡ 2 mod 7. Then we may take a = (xn + 5yn − 3)/14,
b = (5xn + 7yn − 3)/2 and c = ((5xn + 7yn )2 + 3)/4.
For k = 4, we may take a = (Fn2 − 1)/5, b = L2n − 1 and c = L2n + 1, where n ≡ 2 or 8
mod 10, while Fn , Ln denote, respectively, n-th Fibonacci and Lucas number.
Remark : The methods used to prove Theorems 1 and 2 can also be applied to investigate
similar questions, like the existence of quadruples of positive integers 0 < a < b < c < d
such that the product of any two of them increased by N is a k-th power, where N is a fixed
non-zero integer. For instance, we can explicitely compute an integer k0 (N ), depending
only on N , such that such quadruples do not exist whenever k > k0 (N ). The case k = 2
has been studied by the second author [8].
3. Auxiliary lemmas
Lemma 1. Let k ≥ 3 be an integer. Let a < b < c1 < c2 be positive integers such that
aci + 1 and bci + 1 are k-th powers for i ∈ {1, 2}. Then we have bc2 > k k ck−1
ak−1 and
k k−2 k−1
c2 > k c1 a
. Further, if a1 < a2 < · · · < a7 are positive integers such that ai aj + 1 is
a perfect cube for all 1 ≤ i < j ≤ 7, then a7 > 345 a93 a22
1 . Finally, if a < b < c < d1 < d2
are positive integers
such
that
ad
1,
bd
and
cd
1 are perfect cubes for i ∈ {1, 2},
3− 2
then d2 > 27d1
Proof : The first statement follows from the proof of [12, Theorem 1] applied to the sets
{a, b} and {c1 , c2 }.
Assume now that k = 3. Then, by the same result of Gyarmati, we have a4 a2 > 33 a23 a21
and a25 a23 > 36 a44 a42 . Multiplying these two inequalities we obtain
a25 > 39 a34 a32 a21 > 39 (33 a23 a21 )3 a21 = 318 a63 a81 .
Therefore, we get
a5 > 39 a33 a41 .
Now we have
a6 > 33 a25 a22 /a3 > 321 a63 a81 a22 /a3 > 321 a53 a10
and
a7 > 33 a26 a22 /a3 > 345 a93 a22
1 .
For the last statement of the lemma, first note that the Gyarmati’s gap principle gives
bd2 > 27a2 d21
(1)
cd2 > 27b2 d21 .
(2)
and
Set ϕ = 1 + 2. If b < aϕ or c > bϕ , the result follows from (1), resp. (2). Otherwise, we
have c > bϕ > aϕ , which, combined with (1), yields the result.
We need the following refinement, due to Mignotte [16], of a theorem of Laurent,
Mignotte & Nesterenko [14] on linear forms in two logarithms. For any non-zero algebraic
number α, we denote by h(α) its logarithmic absolute height. For instance, for any non-zero
rational number p/q, written under its irreducible form, we have h(p/q) = log max{|p|, |q|}.
Lemma 2. Consider the linear form
Λ = b2 log α2 − b1 log α1 ,
where b1 and b2 are positive integers. Suppose that α1 and α2 are multiplicatively independent. Put
D = [Q(α1 , α2 ) : Q] / [R(α1 , α2 ) : R].
Let a1 , a2 , h, k be real positive numbers, and ρ a real number > 1. Put λ = log ρ, χ = h/λ
and suppose that χ ≥ χ0 for some number χ0 ≥ 0 and that
b1
b2
h ≥ D log
+ log λ + f (dK0 e) + 0.023,
a2
a1
ai ≥ max 1, ρ | log αi | − log |αi | + 2Dh(αi ) , (i = 1, 2),
a1 a2 ≥ λ2
where
f (x) = log
1+
x−1
x−1
√
log x
3 log x−1
+ + log +
6x(x − 1) 2
x−1
and
K0 =
2 + 2χ0
!2
2(1 + χ0 ) 2λ 1
1 4λ 2 + χ0
a1 a2 .
3 a1
a2
3 a1 a2
Put
v = 4χ + 4 + 1/χ and m = max 25/2 (1 + χ)3/2 , (1 + 2χ)5/2 /χ .
Then we have the lower bound
!2
4λv 1
v 1 v2
1
8λm
+ √
a1 a2
6 2 9
3 a1
a2
3 a1 a2
√
3/2
− max λ(1.5 + 2χ) + log (2 + 2χ) + (2 + 2χ) k A + (2 + 2χ) , D log 2
log |Λ| ≥ −
where
A = max{a1 , a2 } and k = 2
1 + 2χ
3χ
2
2 (1 + 2χ)1/2
3χ 3
Proof : This is Theorem 2 of [16].
The proof of Theorems 2 and 3 depends on the following result of Evertse [11].
Lemma 3. If a, b and n are positive integers with n ≥ 3 and c is a positive real number,
then there is at most one positive integral solution (x, y) to the inequality
|axn − by n | ≤ c
with
max{|axn |, |by n |} > βn cαn ,
where αn and βn are effectively computable positive constants satisfying
α3 = 9,
3n − 2 2(n − 1)
αn = max
2(n − 3) n − 2
for n ≥ 4
and
β3 = 1152.2,
β4 = 98.53,
βn < n2
for n ≥ 5.
Proof : This is Theorem 2.1 of [11].
The proof of Theorem 4 uses an irrationality measure [20] of certain algebraic numbers
derived from a Theorem of Baker [1], using some improvements from [2].
Lemma 4. Let A, B, K and n be positive integers such that A > B, K < n, n ≥ 3 and
ω = (B/A)1/n is not a rational number. For 0 < φ < 1, put
δ =1+
2−φ
s=
1−φ
and
K+s+1
u−1
40n(K+1) .
2 = K2
u1 = 40n(K+1)(s+1)/(Ks−1) ,
Assume that
A(A − B)−δ u−1
1 > 1.
(3)
Then
ω−
u2
K(s+1)
Aq
for all integers p and q with q > 0.
Proof : This is Lemma 1 of Shorey & Nesterenko [20]. We notice that this has been
refined by Hirata-Kohno in [13] but the statement of [20] is sufficient for our purpose.
4. Proofs
Proof of Theorem 1 :
Let 0 < a < b < c < d be integers such that there exist positive integers r, s, t, u and
k ≥ 2 with
ac + 1 = rk , ad + 1 = sk , bc + 1 = tk and bd + 1 = uk .
Our aim is to prove that k is bounded by an absolute constant. Hence, we may assume
that k ≥ 160 and that, since
c2 > bc + 1 ≥ 3k ,
(4)
we have
d > c > 380 .
(5)
We also observe that, by Lemma 1, we have
log d > (k − 2) log c.
We set
α1 =
ur
st
α2 =
(6)
b ac + 1
a bc + 1
and we consider the linear form in logarithms
b ac + 1
ur
Λ = | log α2 − k log α1 | = log
− k log
a bc + 1
st
Before applying Lemma 2 with b2 = 1 and b1 = k in order to bound Λ, we need some
estimates.
Firstly, we have
b−a
|α2 − 1| = α2 − 1 =
< .
(7)
abc + a
Secondly, from (5) and the upper bound
b ac + 1
a bc + 1
ur
st
k
k
b−a
(b − a)(ac + 1)
a(ad + 1)
a(ad + 1)(bc + 1)
ad
we deduce that
ad
Let now define the quantities a1 , a2 , h, k, ρ appearing in Lemma 2.
We set
ρ = c (thus λ = log c),
Λ≤
(8)
and, by (5) and (7), we may take
a1 = 3 +
2(k + 1)
log d
k(k − 2)
and a2 = 3 + 6 log c.
Indeed, we easily see that kh(α1 ) = h((bd + 1)(ac + 1)) ≤ log(c3 d), whence by (6) we get
kh(α1 ) ≤ (1 + 3/(k − 2)) log d.
Further, we see that one can take h = λ/2, since c ≥ 3k/2 by (4). We should also
check that α1 and α2 are multiplicatively independent. However, a look at the proof of
Theorem 1.5 of [16] shows that this is not needed. Indeed, we apply it with the choice
L = 3, hence it is sufficient to check that the three numbers 1, α1 and α2 are distinct,
which is clearly the case.
It follows from our choice of h that χ0 = 1/2, whence v = 8 and m = 8 2. Using (5)
and (6), we get the lower bound
4 1
log Λ ≥ −
log c 3 2
√ 2
64 64 32 2
a1 a2 − 2.5 log c − log(20.8a1 ).
+ √
3 3
Combined with (8), after a few calculations, we obtain
log d ≤ 167
k+1
log d + 254.9 + 2.5 log c + log (log d)/k .
k(k − 2)
(9)
Using (4), (5) and (6), we infer from (9) that
k+1
254.9
2.5
1 ≤ 167
k(k − 2)
log d
k−2 k
log d
log d
log
(10)
Since we have assumed k ≥ 160, it follows from (4), (5), (6) and (10) that the integer k
satisfies
k ≤ 176,
as claimed.
Proof of Theorem 2 :
Let k ≥ 5 and assume that m ≥ 3. Let acm−1 + 1 = xk and bcm−1 + 1 = y k . Then
bxk − ay k = b − a
and Lemma 3 implies
abcm−1 < k 2 b3.25 .
Hence, cm−1 < k 2 b2.25 . On the other hand, Lemma 1 implies that
cm−1 ≥ c2 > k k ck−2
> k 5 b3 ,
a contradiction.
Let k = 4. Then, as above, we obtain cm−1 < 99b4 . By Lemma 1, we have c2 > 256b2
and c3 > 256c22 > 2563 b4 . Therefore m − 1 ≤ 2 and m ≤ 3.
Proof of Theorem 3 :
Let adm−1 + 1 = x3 and bdm−1 + 1 = y 3 . As in the proof of Theorem 2, an application
of Lemma 3 gives abdm−1 < 1153b9 and
dm−1 < 1153 b8 .
On the other hand, successive applications of Lemma 1 give
d2 > 27d13−
d3 >
d4 >
d5 >
d6 >
> 27 b3−
27d23− 2 > 4930 b2.51 ,
27d23 b−1 > 6 · 108 b4.02 ,
27d24 b−1 > 9 · 1018 b7.04 ,
27d25 b−1 > 2 · 1039 b13.08 .
Therefore, m − 1 ≤ 5 and m ≤ 6.
Proof of Corollary 4 :
It suffices to prove the corollary for k = 3. Let a1 < a2 < · · · < a8 be positive integers
such that the product of any two of them increased by 1 is a perfect cube. As in the proof
of Theorem 3, Lemma 3 implies a7 < 1153a82 . From Lemma 1, we have a7 > 345 a92 , a
contradiction.
Proof of Theorem 4 :
Let 11 ≤ k ≤ 176 be an integer. We denote by κ1 (k), . . . , κ6 (k) effectively computable
positive constants which depend only on k. Assume that the integers 0 < a < b < c < d
are such that there exist integers r, s, t and u with
ac + 1 = rk , ad + 1 = sk , bc + 1 = tk
and bd + 1 = uk .
We will apply Lemma 4 with K = 2 to the algebraic number
ω=
a(bc + 1)
b(ac + 1)
1/k
i.e. with A = b(ac + 1) and A − B = b − a. Firstly, we observe that
ω−
st
ru
ad
(11)
Let φ < 1/4 be a (very) small positive number, and, with the notation of Lemma 4,
set δ = 2 − φ/2 and s = δ/(1 − φ).
The assumption (3) in the statement of Lemma 4 is fulfilled if
b(ac + 1) > 403k(s+1)(2s−1) (b − a)2−φ/2 ,
thus, since c > b, it is fulfilled as soon as c > κ1 (k). Under this assumption, we infer from
Lemma 4 and (11) that
st
κ2 (k)
(12)
≥ ω−
ad
ru
bac(ur)6+7φ
Recalling that ur = (ac + 1)1/k (bd + 1)1/k , it follows from (12) that
ad < κ3 (k)abc(ac)(6+7φ)/k (bd)(6+7φ)/k .
(13)
By Lemma 1, we have
bd > k k ck−1 ak−1 .
(14)
Using b < c and combining (13) and (14), we get
dk−6−7φ < κ4 (k)(bc)k+6+7φ a6+7φ < κ4 (k)(ac)2k+12+14φ < κ5 (k)d(2k+12+14φ)/(k−2) ,
whence we deduce that d < κ6 (k), since k ≥ 11, as claimed.
Acknowledgements. This work was done when the first named author was visiting
the Technische Universität in Graz. He would like to thank the mathematical department
for its warm hospitality. The authors are also indebted to the referee for his very careful
reading of this text.
10
References
[1] A. Baker, Rational approximations to 3 2 and other algebraic numbers, Quart. J.
Math. Oxford (2) 15 (1964), 375–383.
[2] A. Baker, Simultaneous rational approximations to certain algebraic numbers, Proc.
Cambridge Phil. Soc. 63 (1967), 693–702.
[3] A. Baker and H. Davenport, The equations 3x2 − 2 = y 2 and 8x2 − 7 = z 2 , Quart. J.
Math. Oxford 20 (1969), 129–137.
[4] Y. Bugeaud and T. N. Shorey, On the Diophantine equation (xm − 1)/(x − 1) =
(y n − 1)/(y − 1), Pacific J. Math. (to appear).
[5] L. E. Dickson, History of the Theory of Numbers, Vol. 2, Chelsea, New York, 1966.
[6] A. Dujella, The problem of Diophantus and Davenport, Math. Commun. 2 (1997),
153–160.
[7] A. Dujella, An absolute bound for the size of Diophantine m-tuples, J. Number Theory
89 (2001), 126–150.
[8] A. Dujella, On the size of Diophantine m-tuples, Math. Proc. Cambridge Phil. Soc.
(to appear).
[9] A. Dujella, There are only finitely many Diophantine quintuples (submitted).
[10] A. Dujella and A. Pethő, A generalization of a theorem of Baker and Davenport,
Quart. J. Math. Oxford (2) 49 (1998), 291–306.
[11] J. H. Evertse, Upper Bounds for the Numbers of Solutions of Diophantine Equations,
MCT 168, Mathematisch Centrum, Amsterdam, 1983.
[12] K. Gyarmati, On a problem of Diophantus, Acta Arith. 97 (2001), 53–65.
[13] N. Hirata-Kohno and T. N. Shorey, On the equation (xm − 1)/(x − 1) = y q with
x power, in Analytic Number Theory (Kyoto, 1996), 119–125, London Math. Soc.
Lecture Note Ser. 247, Cambridge Univ. Press, Cambridge, 1997.
[14] M. Laurent, Yu. Nesterenko et M. Mignotte, Formes linéaires en deux logarithmes et
déterminants d’interpolation, J. Number Th. 55 (1995), 285–321.
[15] M. Mignotte, A note on the equation axn − by n = c, Acta Arith. 75 (1996), 287–295.
[16] M. Mignotte, A corollary to a theorem of Laurent–Mignotte–Nesterenko, Acta Arith.
86 (1998), 215–225.
[17] T. N. Shorey, Linear forms in the logarithms of algebraic numbers with small coefficients I, II, J. Indian Math. Soc. 38 (1974), 271–284; ibid. 38 (1974), 285–292.
[18] T. N. Shorey, Perfect powers in values of certain polynomials at integer points, Math.
Proc. Cambridge Phil. Soc. 99 (1986), 195–207.
[19] T. N. Shorey, Exponential Diophantine equations involving products of consecutive
integers and related equations, in Number Theory, edited by R. P. Bambah, V. C.
Dumir and R. J. Hans-Gill, Hindustan Book Agency (1999), 463–495.
[20] T. N. Shorey and Yu. Nesterenko, Perfect powers in product of integers from a block
of consecutive integers (II), Acta Arith. 76 (1996), 191–198.
11
Yann Bugeaud
Université Louis Pasteur
U. F. R. de mathématiques
7, rue René Descartes
67084 STRASBOURG
FRANCE
e-mail :
[email protected]
Andrej Dujella
University of Zagreb
Department of Mathematics
Bijenička cesta 30
10000 ZAGREB
CROATIA
e-mail :
[email protected]
12
References (20)
A. Baker, Rational approximations to 3 √ 2 and other algebraic numbers, Quart. J. Math. Oxford (2) 15 (1964), 375-383.
A. Baker, Simultaneous rational approximations to certain algebraic numbers, Proc. Cambridge Phil. Soc. 63 (1967), 693-702.
A. Baker and H. Davenport, The equations 3x 2 -2 = y 2 and 8x 2 -7 = z 2 , Quart. J. Math. Oxford 20 (1969), 129-137.
Y. Bugeaud and T. N. Shorey, On the Diophantine equation (x m -1)/(x -1) = (y n -1)/(y -1), Pacific J. Math. (to appear).
L. E. Dickson, History of the Theory of Numbers, Vol. 2, Chelsea, New York, 1966.
A. Dujella, The problem of Diophantus and Davenport, Math. Commun. 2 (1997), 153-160.
A. Dujella, An absolute bound for the size of Diophantine m-tuples, J. Number Theory 89 (2001), 126-150.
A. Dujella, On the size of Diophantine m-tuples, Math. Proc. Cambridge Phil. Soc. (to appear).
A. Dujella, There are only finitely many Diophantine quintuples (submitted).
A. Dujella and A. Pethő, A generalization of a theorem of Baker and Davenport, Quart. J. Math. Oxford (2) 49 (1998), 291-306.
J. H. Evertse, Upper Bounds for the Numbers of Solutions of Diophantine Equations, MCT 168, Mathematisch Centrum, Amsterdam, 1983.
K. Gyarmati, On a problem of Diophantus, Acta Arith. 97 (2001), 53-65.
N. Hirata-Kohno and T. N. Shorey, On the equation (x m -1)/(x -1) = y q with x power, in Analytic Number Theory (Kyoto, 1996), 119-125, London Math. Soc. Lecture Note Ser. 247, Cambridge Univ. Press, Cambridge, 1997.
M. Laurent, Yu. Nesterenko et M. Mignotte, Formes linéaires en deux logarithmes et déterminants d'interpolation, J. Number Th. 55 (1995), 285-321.
M. Mignotte, A note on the equation ax n -by n = c, Acta Arith. 75 (1996), 287-295.
M. Mignotte, A corollary to a theorem of Laurent-Mignotte-Nesterenko, Acta Arith. 86 (1998), 215-225.
T. N. Shorey, Linear forms in the logarithms of algebraic numbers with small coeffi- cients I, II, J. Indian Math. Soc. 38 (1974), 271-284; ibid. 38 (1974), 285-292.
T. N. Shorey, Perfect powers in values of certain polynomials at integer points, Math. Proc. Cambridge Phil. Soc. 99 (1986), 195-207.
T. N. Shorey, Exponential Diophantine equations involving products of consecutive integers and related equations, in Number Theory, edited by R. P. Bambah, V. C. Dumir and R. J. Hans-Gill, Hindustan Book Agency (1999), 463-495.
T. N. Shorey and Yu. Nesterenko, Perfect powers in product of integers from a block of consecutive integers (II), Acta Arith. 76 (1996), 191-198.
FAQs
AI
What is the upper bound for k when certain integers form perfect k-th powers?
add
The study determines that k is bounded by 176 for integers a, b, c, and d when ac+1, ad+1, bc+1, and bd+1 are perfect k-th powers.
How does the methodology utilize linear forms in logarithms in the proofs?
add
The proofs leverage linear forms in logarithms as key tools, relying on classic results, notably from Shorey, to derive optimal bounds in the context of Diophantine equations.
What are the specific results derived from Theorem 1 regarding integers k ≥ 177?
add
The theorem confirms that for k ≥ 177, no quadruple of positive integers exists such that the products increased by one yield perfect k-th powers.
What improvements does Corollary 2 offer compared to Gyarmati's results?
add
Corollary 2 enhances Gyarmati's initial findings for integers k not too small, yielding improved bounds on the sets of integers where defined perfect k-th powers exist.
How does the research handle cases for k = 3 specifically?
add
For k = 3, the study identifies that a stronger assumption is necessary, leading to a maximum of six integers in the respective set constructions.
Andrej Dujella
University of Zagreb, Faculty Member
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Generalized Fibonacci Numbers and the Problem of Diophantus
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On the number of Diophantine m-tuples
Andrej Dujella
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A Polynomial Variant of a Problem of Diophantus for Pure Powers
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