THE KORTEWEG-DE VRIES EQUATION AND A
DIOPHANTINE PROBLEM RELATED TO BERNOULLIPOLYNOMIALS
Á. PINTÉR AND SZ. TENGELY
Dedicated to Professor Hari Srivastava

Abstract. Some diophantine equations related to the soliton solutions of theKorteweg-de Vries equation are resolved. The main tools are the connection with Bernoulli polynomials and the application of certain computational
number-theoretical results.

1. introduction
In the paper [12] Fairlie and Veselov obtained a relation of the Bernoulli polynomials with the theory of the Korteweg-de Vries (KdV) equationut − 6uux + uxxx = 0.
This equation has infinitely many conservation laws of the formZ ∞Im [u] =Pm (u, ux , uxx , . . . , um )dx,−∞

where Pm are some polynomials of the function u and its x-derivatives up to orderm, see [18]. For example,
Z ∞
Z ∞
Z ∞
I−1 [u] =udx, I0 [u] =u2 dx, I1 [u] =(u2x + 2u3 )dx
−∞

and

−∞

Z ∞
I2 [u] =−∞(u2xx + 10uu2x + 5u4 )dx.

−∞

The KdV equation possesses a remarkable family of so-called n-soliton solutions
corresponding to the initial profile un (x, 0) = −2n(n + 1)sech2 x. For some recent
generalizations and applications of the Korteweg-de Vries equation we refer to [15],[14] and [22] and the references given therein.
Using the spectral theory of Schrödinger operators, see [30], Fairlie and Veselov[12] proved thatn(−1)k 4k+2 X 2k+3i
Ik [un ] =2k + 3 i=1
for k = −1, 0, 1, . . ..
Now let k 6= l be fixed integers with k, l ∈ {−1, 0, 1, 2, . . .} and suppose that
|Ik [un ]| = |Il [um ]|.
2010 Mathematics Subject Classification. Primary 11D41, 14H45; Secondary 11Y50.
Key words and phrases. Diophantine equations, curves of genus 2, Korteweg-de Vries equation.
Research was supported in part by the Hungarian Academy of Sciences, OTKA grants K75566,
K100339, NK101680, NK104208 (Á.P) and OTKA grants PD75264, NK104208, K100339 andJános Bolyai Research Scholarship of the Hungarian Academy of Sciences (Sz.T.).
1

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Á. PINTÉR AND SZ. TENGELY

One can ask that for given k and l, how often can these integrals be equal? In otherwords, what is the cardinality of the set of solutions m, n to the equationnm
4k X 2k+3
4l X 2l+3i=i,
2k + 3 i=1
2l + 3 i=1(1)

where k and l are fixed distinct integers?
Applying some recent results by Rakaczki, see [23] and [24], it is not too hardto give some ineffective and effective finiteness statements for the solutions m andn to equation (1). However, the purpose of this note is to resolve (1) for certain
values of m and n including an infinite family of the parameters.
Theorem 1. For k = −1 and l ∈ {0, 1, 2, 3}, equation (1) has only one solution,
namely (l, m, n) = (0, 24, 5).
Theorem 2. Assume that k = 0 and l is a positive integer such that 2l + 3 isprime. Then (1) has no solution in positive integers m and n.
2. auxiliary results
In our first lemma we summarize some classical properties of Bernoulli polynomials. For the proofs of these results we refer to [21].
Lemma 1. Let Bj (X) denote the jth Bernoulli polynomial and Bj = Bj (0), j =1, 2, . . .. Further, let Dj be the denominator of Bj . Then we havePj(A) Bj (X) = X n + i=1 ji Bi X j−i ,1(Bj+1 (x) − Bj+1 ),(B) Sj (x) = 1j + 2j + . . . + (x − 1)j = j+1
1(C) B1 = − 2 , B2j+1 = 0, j = 1, 2,Q...
(D) (von Staudt-Clausen) D2j = p−1|2j,p prime p(E) X 2 (X − 1)2 |B2j (X) − B2j (in Q[X]).
(F) Bj (X) = (−1)j Bj (1 − X).
Consider the hyperelliptic curve(2)C:

y 2 = F (x) := x5 + b4 x4 + b3 x3 + b2 x2 + b1 x + b0 ,where bi ∈ Z. Let α be a root of F and J(Q) be the Jacobian of the curve C. We
have thatx − α = κξ 2
where κ, ξ ∈ K = Q(α) and κ comes from a finite set. By knowing the Mordell-Weilgroup of the curve C it is possible
Pm to provide a method to compute such a finite set.
To each coset representative i=1 (Pi − ∞) of J(Q)/2J(Q) we associate
κ=

m
Y


γi − αd2i ,i=1where the set {P1 , . . . , Pm } is stable under the action of Galois, all y(Pi ) are nonzero and x(Pi ) = γi /d2i where γi is an algebraic integer and di ∈ Z≥1 . If Pi , Pj areconjugate then we may suppose that di = dj and so γi , γj are conjugate. We havethe following lemma (Lemma 3.1 in [8]).
Lemma 2. Let K be a set of κ values associated as above to a complete set of coset
representatives of J(Q)/2J(Q). Then K is a finite subset of OK and if (x, y) is an
integral point on the curve (2) then x − α = κξ 2 for some κ ∈ K and ξ ∈ K.

THE KORTEWEG-DE VRIES EQUATION AND A DIOPHANTINE PROBLEM

3

As an application of his theory of lower bounds for linear forms in logarithms,
Baker [1] gave an explicit upper bound for the size of integral solutions of hyperelliptic curves. This result has been improved by many authors (see e.g. [2], [3], [4],[9], [20], [26], [27] and [29]).
In [8] an improved completely explicit upper bound were proved combining ideas
from [9], [10], [11], [16], [17], [19], [29], [28]. Now we will state the theorem which
gives the improved bound. We introduce some notation. Let K be a number fieldof degree d and let r be its unit rank and R its regulator. For α ∈ K we denote byh(α) the logarithmic height of the element α. Let log 2
if d = 1, 2,
d∂K = 1  log log d 3

if d ≥ 3
4
log d
and
0
∂K
=


1+

π2
2
∂K

1/2
.

Define the constants(r !)2c1 (K) = r−1 r ,2 ddrc3 (K) = c1 (K)
,
∂K
rr+1
c5 (K) = r−1 .
2∂K
c2 (K) = c1 (K)

d
∂K

r−1
,

c4 (K) = rdc3 (K),

Let



0.16[K : Q]
0
∂L/K = max [L : Q] , [K : Q]∂K,,
∂K
where K ⊆ L are number fields. DefineC(K, n) := 3 · 30n+4 · (n + 1)5.5 d2 (1 + log d).
The following result will be used to get an upper bound for the size of the integral
solutions of our equations. It is Theorem 3 in [8].
Lemma 3. Let α be an algebraic integer of degree at least 3 and κ be an integer
belonging to K. Denote by α1 , α2 , α3 distinct conjugates of α and by κ1 , κ2 , κ3
the corresponding conjugates of κ. Let√√√
K1 = Q(α1 , α2 , κ1 κ2 ), K2 = Q(α1 , α3 , κ1 κ3 ), K3 = Q(α2 , α3 , κ2 κ3 ),and√√
L = Q(α1 , α2 , α3 , κ1 κ2 , κ1 κ3 ).
In what follows R stands for an upper bound for the regulators of K1 , K2 and K3
and r denotes the maximum of the unit ranks of K1 , K2 , K3 . Letc∗j = max cj (Ki )
1≤i≤3

and
N = max1≤i,j≤3

NormQ(αi ,αj )/Q (κi (αi − αj ))andH ∗ = c∗5 R +

2

log N
+ h(κ).
min1≤i≤3 [Ki : Q]

DefineA∗1 = 2H ∗ · C(L, 2r + 1) · (c∗1 )2 ∂L/L ·2rmax ∂L/Ki

1≤i≤3· R2 ,

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Á. PINTÉR AND SZ. TENGELY

andA∗2 = 2H ∗ + A∗1 + A∗1 log{(2r + 1) · max{c∗4 , 1}}.
If x ∈ Z\{0} satisfies x − α = κξ 2 for some ξ ∈ K thenlog|x| ≤ 8A∗1 log(4A∗1 ) + 8A∗2 + H ∗ + 20 log 2 + 13 h(κ) + 19 h(α).
To obtain a lower bound for the possible unknown integer solutions we are goingto use the so-called Mordell-Weil sieve. The Mordell-Weil sieve has been successfully
applied to prove the non-existence of rational points on curves (see e.g. [5], [7], [13]and [25]).
Let C/Q be a smooth projective curve (in our case a hyperelliptic curve) of
genus g ≥ 2. Let J be its Jacobian. We assume the knowledge of some rational
point on C, so let D be a fixed rational point on C and let  be the corresponding
Abel-Jacobi map:
 : C → J,
P 7→ [P − D].
Let W be the image in J of the known rational points on C and D1 , . . . , Dr generators for the free part of J(Q). By using the Mordell–Weil sieve we are going toobtain a very large and smooth integer B such that
(C(Q)) ⊆ W + BJ(Q).
Let
φ : Zr → J(Q),φ(a1 , . . . , ar ) =X

ai Di ,

so that the image of φ is the free part of J(Q). The variant of the Mordell-Weilsieve explained in [8] provides a method to obtain a very long decreasing sequenceof lattices in Zr
BZr = L0 ) L1 ) L2 ) · · · ) Lk
such that
(C(Q)) ⊂ W + φ(Lj )for j = 1, . . . , k.
The next lemma [8, Lemma 12.1] gives a lower bound for the size of rational
points whose image are not in the set W.
Lemma 4. Let W be a finite subset of J(Q) and L be a sublattice of Zr . Supposethat (C(Q)) ⊂ W + φ(L). Let µ1 be a lower bound for h − ĥ andqĥ(w) : w ∈ W .
µ2 = max
Denote by M the height-pairing matrix for the Mordell–Weil basis D1 , . . . , Dr andlet λ1 , . . . , λr be its eigenvalues. Letnpoλj : j = 1, . . . , rµ3 = minand m(L) the Euclidean norm of the shortest non-zero vector of L. Then, for any
P ∈ C(Q), either (P ) ∈ W or2h((P )) ≥ (µ3 m(L) − µ2 ) + µ1 .
The following lemma plays a crucial role in the proof of Theorem 1
Lemma 5. The integral solutions of the equation(3)

C : Y 2 = X(X + 20)2 (X 2 + 10X + 400) + 140625

are(X, Y ) ∈ {(0, ±375), (−20, ±375)}.

THE KORTEWEG-DE VRIES EQUATION AND A DIOPHANTINE PROBLEM

5Proof of Lemma 5. Let J(Q) be the Jacobian of the genus two curve (3). Using
MAGMA we determine a Mordell-Weil basis which is given by
D1 = (0, 375) − ∞,D2 = (−20, 375) − ∞.
Let f = x(x + 20)2 (x2 + 10x + 400) + 140625 and α be a root of f . We will chooseP2
for coset representatives of J(Q)/2J(Q) the linear combinations i=1 ni Di , whereni ∈ {0, 1}. Thenx − α = κξ 2 ,where κ ∈ K and K is constructed as described in Lemma 2. We have that K ={1, −α, −20 − α, α(α + 20)}. By local arguments it is possible to restrict the set K
further (see e.g. [5], [6]). In our case one can eliminateα(α + 20)
by local computations in Q3 . We apply Lemma 3 to get a large upper bound forlog |x| in the remaining cases. A MAGMA code were written to obtain the bounds
appeared in [8], it can be found at
http://www.warwick.ac.uk/∼maseap/progs/intpoint/bounds.m. We obtain thatthese bounds are as follows
κ1−α−20 − αbound for log |x|6.27 · 10307
4.48 · 10668
1.89 · 10612

The set of known rational points on the curve (3) is {∞, (0, ±375), (−20, ±375)}.
Let W be the image of this set in J(Q). Applying the Mordell-Weil implemented
by Bruin and Stoll and explained in [8] we obtain that (C(Q)) ⊆ W + BJ(Q),where
B = 28 · 53 · 72 · 112 · 132 · 172 · 19 · 31 · 37 · 41 · 43 · 53 · 59 · 71 · 79 · 83 · 89
that isB = 46128223306000188203435897312000.
Now we use an extension of the Mordell-Weil sieve due to Samir Siksek to obtaina very long decreasing sequence of lattices in Z2 . After that we apply Lemma 4 toobtain a lower bound for possible unknown rational points. We get that if (x, y) is
an unknown integral point, thenlog |x| ≥ 2.216448 × 10782 .
This contradicts the bound for log |x| we obtained by Baker’s method.



3. Proofs of the Theorems
Proof of Theorem 1. For k = −1 and l ∈ {0, 1, 2, 3} we have the diophantine equations
n(n + 1)m2 (m + 1)2
(4)
=
,
2
3
(5)

n(n + 1)1 2=z (2z − 1) with z = m(m + 1),
8
15

(6)

n(n + 12 2 2=z (3z − 4z + 2) with z = m(m + 1),
8
21

6

Á. PINTÉR AND SZ. TENGELY

and
(7)

n
m
1X
64 X 9
i=
i ,
4 i=1
9 i=1

respectively. One can see that the first three equations are elliptic diophatine
equations, thus using the program package MAGMA, subroutines IntegralPoints
or IntegralQuarticPoints is just a straightforward calculation to solve them. In
these cases the unique solution is (l, m, n) = (0, 24, 5). The forth equation can be
written as follows
128 2(m + m − 1)(m2 + m)2 (2m4 + 4m3 − m2 − 3m + 3) + 1.
(2n + 1)2 =45
So we easily obtain a hyperelliptic curve
Y 2 = X(X + 20)2 (X 2 + 10X + 400) + 140625,
where Y = 375(2n + 1) and X = 20m2 + 20m − 20. By Lemma 5 we have thatX = 0 or -20. Therefore we have that m ∈ {−1, 0}, a contradiction and there is no
solution in positive integers of (7).
Proof of Theorem 2. Now k = 0 and p = 2l + 3 ≥ 3 is a prime. From (1) we get
p · n2 (n + 1)2 = 3 · 4l+1 (1p + 2p + . . . + mp ).
Let m and n be an arbitrary but fixed solution. An elementary numbertheoreticalargument and Lemma 1 yield that p|m(m + 1) and p
1 + 2p + . . . + mp
Bp+1 (m + 1) − Bp+1
ordp
6= 0.
= ordpm2 (m + 1)2
m2 (m + 1)2
Suppose that p|m and let d the smallest positive integer such that Bp+1 (m + 1) −Bp+1 = d1 f (m)m2 (m + 1)2 , and f (X) ∈ Z[X]. Since p+1
is divisible by p forkk = 2, . . . , p − 1 and B1 = −1/2 we have
that
p
is
not
a
divisorof d. The constant
p+1
term of the polynomial f (X) is d p−1
Bp−1 and, by von Staudt-Clausen theorem,
it is not divisible by p. On the other hand, p is a divisor of m and f (m), we havea contradiction. If p|m + 1 the we can repeat the previous argument using the fact
f (X) = f (−X − 1), cf. Lemma 1.
Acknowledgement. The work is supported by the TÁMOP-4.2.2.C-11/1/KONV2012-0010 project. The project is implemented through the New Hungary Development Plan, co-financed by the European Social Fund and the European Regional
Development Fund.
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Institute of Mathematics
MTA-DE Research Group ”Equations, Functions and Curves”
Hungarian Academy of Sciences and University of Debrecen
P. O. Box 12, H-4010 Debrecen, Hungary
E-mail address: apinter@science.unideb.hu
Institute of Mathematics
University of Debrecen
P. O. Box 12, H-4010 Debrecen, Hungary
E-mail address: tengely@science.unideb.hu