Abstract
Let \( \{P_{n}\}_{n\ge 0} \) be the sequence of Padovan numbers defined by \( P_0=0 \), \( P_1 =1=P_2\), and \( P_{n+3}= P_{n+1} +P_n\) for all \( n\ge 0 \). In this paper, we find all repdigits in base 10 which can be written as a sum of three Padovan numbers.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Let \( \{P_{n}\}_{n\ge 0} \) be the sequence of Padovan numbers given by
This is sequence A000931 on the On-Line Encyclopedia of Integer Sequences (OEIS) [12]. The first few terms of this sequence are
A repdigit is a positive integer N that has only one distinct digit when written in base 10. That is, N is of the form
for some positive integers d and \( \ell \) with \( 1\le d\le 9 \) and \( \ell \ge 2 \). The sequence of repdigits is sequence A010785 on the OEIS.
2 Main Result
In this paper, we study the problem of writing repdigits as sums of three Padovan numbers. More precisely, we completely solve the Diophantine equation:
in non-negative integers \( (N, n_1, n_2, n_3, d, \ell ) \) with \( n_1\ge n_2\ge n_3 \ge 0 \), \( \ell \ge 2 \), and \( 1\le d\le 9 \).
We discard the situations when \( n_1=1 \) and \( n_1=2 \) and just count the solutions for \( n_1=3 \) since \( P_1=P_2=P_3 =1 \). For the same reasons, we discard the situation when \( n_1=4 \) and just count the solutions for \( n_1=5 \) since \( P_4=P_5 =2 \). Thus, we always assume that \( n_1,n_2, n_3 \notin \{1,2,4\}\). Our main result is the following.
Theorem 1
All non-negative integer solutions \( (N, n_1, n_2, n_3, d, \ell ) \) with \( n_1\ge n_2\ge n_3 \ge 0 \), \( \ell \ge 2 \), and \( 1\le d\le 9 \) to the Diophantine equation (2) arise from
This paper serves as a continuation of the results in [3, 6,7,8,9, 11]. The method of proof involves the application of Baker’s theory for linear forms in logarithms of algebraic numbers, and the Baker-Davenport reduction procedure. Computations are done with the help of a simple computer program in Mathematica.
3 Preliminary results
3.1 The Padovan sequence
Here, we recall some important properties of the Padovan sequence \( \{P_n\}_{n\ge 0} \). The characteristic equation
has roots \( \alpha , \beta ,\) and \( \gamma = \bar{\beta } \), where
and
Furthermore, the Binet formula is given by
where
Numerically, the following estimates hold:
From (3), (4), and (7), it is easy to see that the contribution the complex conjugate roots \( \beta \) and \( \gamma \), to the right-hand side of (5), is very small. In particular, setting
holds for all \( n\ge 1 \). Furthermore, by induction, it can be proved that
3.2 Linear forms in logarithms
Let \( \eta \) be an algebraic number of degree d with minimal primitive polynomial over the integers:
where the leading coefficient \( a_{0} \) is positive and the \( \eta ^{(i)} \)’s are the conjugates of \( \eta \). Then the logarithmic height of \( \eta \) is given by
In particular, if \( \eta = p/q \) is a rational number with \( \gcd (p,q) = 1 \) and \( q>0 \), then \( h(\eta ) = \log \max \{|p|, q\} \). The following are some of the properties of the logarithmic height function \( h(\cdot ) \), which will be used in the next sections of this paper without reference:
For the proofs of (10) and further details, we refer the reader to the book of Bombieri and Gubler [1].
We recall the result of Bugeaud, Mignotte, and Siksek [2, Theorem 9.4, p. 989], which is a modified version of the result of Matveev [10], which is one of our main tools in this paper.
Theorem 2
Let \(\eta _1,\ldots ,\eta _t\) be positive real algebraic numbers in a real algebraic number field \(\mathbb {K} \subset \mathbb {R}\) of degree \(D_\mathbb {K}\), \(b_1,\ldots ,b_t\) be nonzero integers, and assume that
is nonzero. Then
where
and
3.3 Reduction procedure
During the calculations, we get upper bounds on our variables which are too large, thus we need to reduce them. To do so, we use some results from the theory of continued fractions.
For the treatment of linear forms homogeneous in two integer variables, we use the well-known classical result in the theory of Diophantine approximation. The following lemma is the criterion of Legendre.
Lemma 1
Let \(\tau \) be an irrational number, \( \frac{p_0}{q_0}, \frac{p_1}{q_1}, \frac{p_2}{q_2}, \ldots \) be all the convergents of the continued fraction expansion of \( \tau \) and M be a positive integer. Let N be a nonnegative integer such that \( q_N> M \). Then, putting \( a(M):=\max \{a_{i}: i=0, 1, 2, \ldots , N\} \), the inequality:
holds for all pairs (r, s) of positive integers with \( 0<s<M \).
For a nonhomogeneous linear form in two integer variables, we use a slight variation of a result due to Dujella and Pethő (see [4, Lemma 5a]). For a real number X, we write \(\Vert X \Vert := \min \{|X-n|: n\in \mathbb {Z}\}\) for the distance from X to the nearest integer.
Lemma 2
Let M be a positive integer, \(\frac{p}{q}\) be a convergent of the continued fraction expansion of the irrational number \(\tau \) such that \(q>6M\), and \(A,B,\mu \) be some real numbers with \(A>0\) and \(B>1\). Furthermore, let \(\varepsilon : = \Vert \mu q \Vert -M\Vert \tau q\Vert \). If \( \varepsilon > 0 \), then there is no solution to the inequality:
in positive integers u, v, and w with
Finally, the following Lemma is also useful. It is Lemma 7 in [5].
Lemma 3
If \(r\geqslant 1\), \(H>(4r^2)^r\), and \(H>L/(\log L)^r\), then
4 Bounding the variables
We assume that \( n_1\ge n_2\ge n_3 \). From (2) and (9), we have
and
where we use \( \alpha ^4 >3 \). Thus
Since \( \log \alpha /\log 10 = 0.122123\ldots <1/5 \), we can conclude from the above that
Running a Mathematica program in the range \( 0\le n_3 \le n_2 \le n_1 \le 500 \), \( 1 \le d \le 9 \), and \( 1 \le \ell \le 100 \) we obtain only the solutions listed in Theorem 1. From now on, we assume that \( n_1> 500 \).
By using (8), equation (2) can be written as
We then consider (13) in three different cases as follows.
4.1 Case 1
We have that
This is equivalent to
This shows that
and so
We divide through (14) by \( a\alpha ^{n_1}\) to get
Thus, we have
We put
To apply Theorem 2, we need to check that \( \varLambda _1\ne 0 \). Suppose that \( \varLambda _1=0 \), then we have
To see that this is not true, we consider the \( \mathbb {Q} \)-automorphism \( \sigma \) of the Galois extension \( \mathbb {Q}(\alpha , \beta ) \) over \( \mathbb {Q} \) given by \( \sigma (\alpha ):=\beta \) and \( \sigma (\beta ):=\alpha \). Now, since \( \varLambda _1=0 \), we get \( \sigma (\varLambda _1)=0 \). Thus, conjugating the relation (16) under \( \sigma \), and taking absolute values on both sides, we get
which is false for \( \ell \ge 2 \) and \( d \ge 1 \). Therefore, \( \varLambda _1\ne 0 \).
Therefore, we apply Theorem 2 with the data
It is a well–known fact that
the minimal polynomial of a is \( 23x^3-23x^2+6x-1 \) and has roots a, b, c. Since \( |b|=|c|< |a|=a <1 \) (by (7)), we get
Since \( \eta _1, \eta _2, \eta _2\in \mathbb {Q}(\alpha ) \), we take the field \( \mathbb {K}:= \mathbb {Q}(\alpha ) \) with degree \( D_{\mathbb {K}}:=3 \). Since \( \max \{1, \ell , n_1\} \le n_1\), we take \( B:=n_1 \). Further, the minimal polynomial of \( \alpha \) over \( \mathbb {Z} \) is \( x^3-x-1 \) has roots \( \alpha , ~ \beta , ~ \gamma \) with \( 1.32< \alpha < 1.33 \) and \( |\beta |=|\gamma | <1 \). Thus, we can take \( h(\alpha ) = \frac{1}{3} \log \alpha \). Similarly, \( h(10) = \log 10 \) . Also,
Thus, we can take \( A_1:= 3\log 10 \), \( A_2:=\log \alpha \) and \( A_3:=15\log 3 \). So, Theorem 2 tells us that the left-hand side of (15) is bounded below by
By comparing the above inequality with the right-hand side of (15) we get that
4.2 Case 2
We have that
This is equivalent to
Thus, it follows that
and so
We divide through (14) by \( a(\alpha ^{n_1}+\alpha ^{n_2})\) to get
This means that
We put
As before, to apply Theorem 2, we need to check that \( \varLambda _2\ne 0 \). Suppose that \( \varLambda _2=0 \), then we have
To see that this is not true, we again consider the \( \mathbb {Q} \)-automorphism \( \sigma \) of the Galois extension \( \mathbb {Q}(\alpha , \beta ) \) over \( \mathbb {Q} \) given by \( \sigma (\alpha ):=\beta \) and \( \sigma (\beta ):=\alpha \). Now, since \( \varLambda _2=0 \), we get \( \sigma (\varLambda _2)=0 \). Thus, conjugating the relation (20) under \( \sigma \), and taking absolute values on both sides, we get
which is false for \( \ell \ge 2 \) and \( d \ge 1 \). Therefore, \( \varLambda _2\ne 0 \).
Therefore, we apply Theorem 2 with the data
Since \( \eta _1, \eta _2, \eta _2\in \mathbb {Q}(\alpha ) \), we take the field \( \mathbb {K}:= \mathbb {Q}(\alpha ) \) with degree \( D_{\mathbb {K}}:=3 \). Since \( \max \{1, \ell , n_2\} \le n_1\), we take \( B:=n_1 \). Further,
Thus, we can take \( A_1:= 3\log 10 \), \( A_2:=\log \alpha \) and \( A_3:=5.31\times 10^{14}\log n_1 \). So, Theorem 2 tells us that the left-hand side of (19) is bounded below by
By comparing the above inequality with the right-hand side of (19), we get that
4.3 Case 3
We have that
This is equivalent to
Thus, we have
and so
We divide through (14) by \( a(\alpha ^{n_1} +\alpha ^{n_2}+\alpha ^{n_3})\) to get
Thus, it follows that
We put
As before, in order to apply Theorem 2 we need to check that \( \varLambda _3\ne 0 \). Suppose that \( \varLambda _3=0 \), then we have
To see that this is not true, we again consider the \( \mathbb {Q} \)-automorphism \( \sigma \) of the Galois extension \( \mathbb {Q}(\alpha , \beta ) \) over \( \mathbb {Q} \) given by \( \sigma (\alpha ):=\beta \) and \( \sigma (\beta ):=\alpha \). Now, since \( \varLambda _3=0 \), we get \( \sigma (\varLambda _3)=0 \). Thus, conjugating the relation (24) under \( \sigma \), and taking absolute values on both sides, we get
which is false for \( \ell \ge 2 \) and \( d \ge 1 \). Therefore, \( \varLambda _2\ne 0 \).
Therefore, we apply Theorem 2 with the data:
Since \( \eta _1, \eta _2, \eta _2\in \mathbb {Q}(\alpha ) \), we take the field \( \mathbb {K}:= \mathbb {Q}(\alpha ) \) with degree \( D_{\mathbb {K}}:=3 \). Since \( \max \{1, \ell , n_3\} \le n_1\), we take \( B:=n_1 \). Furthermore
Thus, we can take \( A_1:= 3\log 10 \), \( A_2:=\log \alpha \), and \( A_3:=5.16\times 10^{28}(\log n_1)^2 \). So, Theorem 2 tells us that the left-hand side of (23) is bounded below by
By comparing the above inequality with the right-hand side of (19), we get that
Now, we apply Lemma 3 on the above inequality (25) with the data: \( r:=3, ~ H:=1.94\times 10^{42}\), and \( L:=n_1 \). We obtain that \( n_1 < 2.7\times 10^{48} \). We now record what we have proved.
Lemma 4
Let \((N, n_1, n_2, n_3, d, \ell ) \) be the nonnegative integer solutions to the Diophantine equation (2) with \( n_1\ge n_2 \ge n_3 \ge 0 \), \( 1\le d\le 9 \), and \( \ell \ge 2 \). Then, we have
5 Reducing the bounds
The bounds obtained in Lemma 4 are too large to carry out meaningful computations on the computer. Thus, we need to reduce these bounds. To do so, we return to (15), (19), and (23) and apply Lemma 2 via the following procedure.
First, we put
For technical reasons, we assume that \( n_1-n_2 \ge 20 \) and go to (15). Note that \( e^{\varGamma _1}-1= \varLambda _1 \ne 0 \). Thus, \( \varGamma _1\ne 0 \). If \( \varGamma _1< 0 \), then
If \( \varGamma _1> 0 \), then we have that \( |e^{\varGamma _1}-1| <1/2 \). Hence \( e^{\varGamma _1}<2 \). Thus, we get that
Therefore, in both cases, we have that
Dividing through the above inequality by \( \log \alpha \), we get
If we put
we can rewrite (26) as
We now apply Lemma 2 on (27). We put \( M:=3\times 10^{48} \). A quick computer search in Mathematica reveals that the convergent
of \( \tau \) is such that \( q_{106}> 6M \) and \( \varepsilon (d) \ge 0.0129487>0 \). Therefore, with \( A:=36 \) and \( B:=\alpha \), we calculated each value of \( \log (36q_{106}/\varepsilon (d))/\log \alpha \) and found that all of them are at most 432. Thus, we have that \( n_1-n_2 \le 432. \) In the case \( n_1-n_2 <20 \), we have \( n_1-n_2<20<432 \). Thus, \( n_1-n_2 \le 432 \) holds in both cases.
Next, we put
For technical reasons, as before we assume that \( n_2-n_3 \ge 20 \) and go to (19). Note that \( e^{\varGamma _2}-1= \varLambda _2 \ne 0 \). Thus, \( \varGamma _2\ne 0 \). If \( \varGamma _2< 0 \), then
If \( \varGamma _2> 0 \), then we have that \( |e^{\varGamma _2}-1| <1/2 \). Hence \( e^{\varGamma _2}<2 \). Thus, we get that
Therefore, in both cases, we have that
Dividing through the above inequality by \( \log \alpha \), we get
We put
where \( k:=n_1-n_2 \). We can rewrite (28) as
We now apply Lemma 2 on (29). We put \( M:=3\times 10^{48} \). A quick computer search in Mathematica reveals that the 106-th convergent of \( \tau \) is such that \( q_{106}> 6M \) and \( \varepsilon (d,k) \ge 0.000134829>0 \) for all \( 1\le d\le 9 \) and \( 1\le k \le 432 \) except for the case \( \varepsilon (9,11) \), which is always negative. Therefore, with \( A:=22 \) and \( B:=\alpha \) we calculated each value of \( \log (22q_{106}/\varepsilon (d,k))/\log \alpha \) and found that all of them are at most 446. Thus, we have that \( n_2-n_3 \le 446. \)
The problem in the case of \( \varepsilon (9,11) \) is due to the fact that
Thus, if we consider the identity (30), the inequality (28) becomes
In this case, we apply the classical result from Diophantine approximation given in Lemma 1. We assume that \( n_2-n_3 \) is so large that the right-hand side of the inequality in (31) is smaller than \( 1/(2\ell ^2) \). This certainly holds if
Since \( \ell< n_1< 3\times 10^{48} \), it follows that the last inequality (32) holds provided that \( n_2-n_3\ge 415 \), which we now assume. In this case \( r/s:=(n_2+9)/\ell \) is a convergent of the continued fraction of \( \tau : = \log 10 / \log \alpha \) and \( \ell < 3\times 10^{48} \). We are now set to apply Lemma 1.
We write \( \tau : =[a_0; a_1, a_2, a_3, \ldots ] = [8; 5, 3, 3, 1, 5, 1, 8, 4, 6, 1, 4, 1, 1, 1, 9, 1, 4, 4, 9, \ldots ] \) for the continued fraction expansion of \( \tau \) and \( p_k/q_k \) for the \( k- \)th convergent. We get that \( r/s=p_j/q_j \) for some \( j\le 106 \). Furthermore, putting \( a(M):=\max \{a_j: j=0,1, \ldots , 106\} \), we get \( a(M):=564 \). By Lemma 1, we get
which gives
This implies that \( n_2-n_3\le 435 \). Thus, in both cases we have that \( n_2-n_3\le 446 \). In the case \( n_2-n_3 < 20 \), we get that \( n_2-n_3<20< 446 \). Thus, \( n_2-n_3\le 446 \) holds in all cases.
Finally, we put
We use the assumption that \( n_1>500 \) and go to (23). Note that \( e^{\varGamma _3}-1= \varLambda _3 \ne 0 \). Thus, \( \varGamma _3\ne 0 \). If \( \varGamma _3< 0 \), then
If \( \varGamma _3> 0 \), then we have that \( |e^{\varGamma _3}-1| <1/2 \). Hence \( e^{\varGamma _3}<2 \). Thus, we get that
Therefore, in both cases, we have that
Dividing through the above inequality by \( \log \alpha \), we get
We put
where \( 1\le k:=n_1-n_3 =(n_1-n_2)+(n_2-n_3) \le 878 \) and \( 1\le s:=n_2-n_3 \le 446 \). We can rewrite (33) as
We now apply Lemma 2 on (34). We put \( M:=3\times 10^{48} \). A quick computer search in Mathematica reveals that the 106-th convergent of \( \tau \) is such that \( q_{106}> 6M \) and \( \varepsilon (d,k,s) \ge 0.000125>0 \). Therefore, with \( A:=36 \) and \( B:=\alpha \) we calculated each value of \( \log (36q_{106}/\varepsilon (d,k,s))/\log \alpha \) and found that all of them are at most 485. Thus, \( n_1\le 485 \). This contradicts our assumption that \( n_1> 500 \). Hence, Theorem 1 holds. \(\square \)
References
Bombieri, E., Gubler, W.: Heights in Diophantine Geometry. Cambridge University Press, Cambridge (2006)
Bugeaud, Y., Mignotte, M., Siksek, S.: Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers. Ann. Math. 163(2), 969–1018 (2006)
Ddamulira, M.: Repdigits as sums of three balancing numbers (preprint) (2019)
Dujella, A., Pethő, A.: A generalization of a theorem of Baker and Davenport. Q. J. Math. 49(195), 291–306 (1998)
Gúzman Sánchez, S., Luca, F.: Linear combinations of factorials and \(s\)-units in a binary recurrence sequence. Ann. Math. du Qué. 38(2), 169–188 (2014)
Lomelí, A.C., García., Hernández, S.H.: Repdigits as sums of two Padovan numbers. J. Integer Seq.22(2), Art. 19.2.3 (2019)
Luca, F.: Repdigits as sums of three Fibonacci numbers. Math. Commun. 17, 1–11 (2012)
Luca, F., Normenyo, B.V., Togbé, A.: Repdigits as sums of three Lucas numbers. Colloq. Math. 156(2), 255–265 (2019)
Luca, F., Normenyo, B.V., Togbé, A.: Repdigits as sums of four Pell numbers. Bol. Soc. Mat. Mex. 25(2), 249–266 (2019)
Matveev, E.M.: An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers II. Izv. Ross. Akad. Nauk Ser. Mat. 64(6), 125–180 (2000) [in Russian: English translation in Izv. Math., 64(6), 1217–1269 (2000)]
Normenyo, B.V., Luca, F., Togbé, A.: Repdigits as sums of three Pell numbers. Period. Math. Hungar. 77(2), 318–328 (2018)
OEIS Foundation Inc.: The On-Line Encyclopedia of Integer Sequences. https://oeis.org (2019)
Acknowledgements
Open access funding provided by Austrian Science Fund (FWF). The author thanks the referee for the careful reading of the manuscript and the useful comments and suggestions that improved on the quality of the presentation of this paper. The author was supported by the Austrian Science Fund (FWF) projects: F5510-N26—Part of the special research program (SFB), “Quasi-Monte Carlo Methods: Theory and Applications” and W1230—‘Doctoral Program Discrete Mathematics”. Part of the work was done during the research stay of the author at the Max Planck Institute for Mathematics in Bonn, from September to November 2019. He thanks this institution for the hospitality and the fruitful working environment.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Ddamulira, M. Repdigits as sums of three Padovan numbers. Bol. Soc. Mat. Mex. 26, 247–261 (2020). https://doi.org/10.1007/s40590-019-00269-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40590-019-00269-9