A Search for High Rank Congruent Number Elliptic Curves

Andrej Dujella Department of Mathematics University of Zagreb Bijenička cesta 30 10000 Zagreb Croatia

                     duje@math.hr

Ali S. Janfada^{1} and Sajad Salami Department of Mathematics University of Urmia P.O. Box 165 Urmia Iran

                a.sjanfada@urmia.ac.ir
                  asjanfada@gmail.com
                salami.sajad@gmail.com

Abstract

In this article, we describe a method for finding congruent number elliptic curves with high ranks. The method involves an algorithm based on the Monsky’s formula for computing 2-Selmer rank of congruent number elliptic curves, and Mestre-Nagao’s sum which is used in sieving curves with potentially large ranks. We apply this method for positive squarefree integers in two families of congruent numbers and find some new congruent number elliptic curves with rank 6.

1The second author was partially financed by a grant No. 009/4/87 from Urmia University.

1 Introduction

One of the major topics connected with elliptic curves is construction of elliptic curves with high ranks. Several authors considered this problem for elliptic curves with prescribed properties and relatively high ranks. For instance, we cite [6, 14] for the curves with given torsion groups, [2, 9] for the curves y^{2} = x^{3} + dx, [10, 19] for the curves x^{3} + y^{3} = k related to the so-called taxicab problem, [8] for the curves y^{2} = (ax + 1)(bx + 1)(cx + 1)(dx + 1) induced by Diophantine quadruples {a, b, c, d}, etc. Dujella [6] collected a list of known high rank elliptic curves with prescribed torsion groups. The largest known rank of elliptic curves, found by N. D. Elkies in 2006, is 28.

In this work we deal with a family of elliptic curves which are closely related to the classical Congruent Number problem. A positive squarefree integer n is called a congruent number if it is the area of a right triangle with rational sides \underline{A006991}, \underline{A003273}. The problem of determining congruent numbers is closely related to the curves E_{n} : y^{2} = x^{3} − n^{2}x, which are called congruent number elliptic curves or CN-elliptic curves. In fact, the positive squarefree integer n is a congruent number if and only if the Mordell-Weil rank r(n) of E_{n} is a positive integer [13, Chap. 1, Prop. 18]. In this case, we refer to n itself as a CN-elliptic curve, which corresponds to E_{n}. In 1972, Alter, Curtz, and Kubota [1] conjectured that n ≡ 5, 6, 7 (mod 8) are congruent numbers. In 1975, appealing to the Birch and Swinnerton-Dyer conjecture and Shafarevich-Tate conjecture, Lagrange [23] deduced a conjecture on the parity of the r(n) as follows:

r(n) ≡ \{ \begin{matrix}0 (mod 2), & if n ≡ 1, 2, 3 (mod 8); \\ 1 (mod 2), & if n ≡ 5, 6, 7 (mod 8). \end{matrix}

The problem of constructing high rank CN-elliptic curves was considered by several authors. In 1640, Fermat proved that r(1) = 0, so n = 1 is not a congruent number. Billing [3] proved that r(5) = 1. Wiman [26] proved that r(34) = 2, r(1254) = 3 and r(29274) = 4 \underline{A062693}, \underline{A062694}, \underline{A062695} . In 2000, Rogers [18], based on an idea of Rubin and Silverberg [22], found the first integers n = 4132814070, 61471349610 such that r(n) = 5, 6, respectively. Later, in his PhD thesis [19], Rogers gave other integers with r(n) = 5, 6 smaller than those presented in [18]. Also he found [19] the first integer n = 797507543735 with r(n) = 7. During the preparation of this paper, Rogers informed us that the smallest n with r(n) = 5 which he was aware is 48272239, while the smallest n with r(n) = 6 is 6611719866. This rank 6 curve is known to be minimal [27]. Here we give the complete list on n’s with r(n) = 6 communicated to us by Rogers [20], other than those curves which are noted above: 66637403074, 94823967361, 129448648329, 179483163699, 208645752554, 213691672290, 226713842409, 248767798521, 344731563386, 670495125874, 797804045274, 898811499201.

In Section 2, we briefly describe the complete 2-descents and 2-Selmer rank of CNelliptic curves, denoted by s(n), which is an upper bound for r(n). In Section 3, we describe Monsky’s formula for computing the value of s(n). In Section 4, we study Mestre-Nagao’s sum method [15, 16, 7] which is used as a sieving tool in our algorithm. In Section 5, we design an algorithm to find high rank CN-elliptic curves, based on the Monsky’s formula for 2-Selmer rank CN-elliptic curves s(n), and Mestre-Nagao’s sum S(N, n). We applied our algorithm for positive squarefree integers arisen from two specific families of congruent numbers. We found a large number of curves with rank 5 and twenty-four new curves with rank 6. We have not found any new curve with r(n) ≥ 7, although with some variants of our method we have rediscovered Rogers’ example with r(n) = 7 (and some of his examples with r(n) = 5 and 6). We have also found several curves with 5 ≤ r(n) ≤ 7, where the upper bound is obtained by MWRANK program (option -s). It might be a challenging problem to decide whether these curves have ranks equal to 5 or 7.

In our computations we used the PARI/GP software (version 2.4.0) [17] and Cremona’s MWRANK program [5] for computing the Mordell-Weil rank of the CN-elliptic curves (using the method of descent via 2-isogeny).

2 Complete 2-descent and 2-Selmer rank

In this section, we briefly describe an upper bound for Mordell-Weil rank of CN-elliptic curves r(n), which is based on the cardinality of 2-Selmer group S^{(2)}(E_{n}/ℚ). We denote this group by S^{(2)}. For more details on the (2-)Selmer groups and related topics, please see [24, Chap. X]. In the following we will describe 2-descents over ℚ for the CN-elliptic curves. The number of 2-descents is the order of S^{(2)}. This is a power of 2, and will be a multiple of 4, on account of the rational points of order 2 on the curve E_{n}. We shall therefore write #S^{(2)} = 2^{s(n)+2}. The exponent s(n) is called 2-Selmer rank of the curve E_{n}. Next we describe the 2-descent process on the curve E_{n}. For a similar argument of complete 2-descent, please see [24, Chap. X, §1], [23, Sec. 3] and [11, Sec. 2].

Let p_{1}, …, p_{t} be the odd prime factors of the squarefree integer n, and let M_{ℚ} be the set of all places of ℚ. Define the sets S and ℚ(S, 2) as follows.

S = {∞, 2, p_{1}, …, p_{t}},
ℚ(S, 2) = {a ∈ ℚ^{∗}/ℚ^{∗}^{2}|υ_{p}(a) ≡ 0 (mod 2) ∀p ∈ M_{ℚ}\S} .

Theorem 1. Let E_{n} be the elliptic curve y^{2} = x^{3} − n^{2}x and let O be the identity element of the group E_{n}(ℚ). With the above notation, we have:

(i) There is an injective homomorphism

θ : E_{n}(ℚ)/2E_{n}(ℚ) ⟶ ℚ(S, 2) × ℚ(S, 2)

P = (x, y) ↦ \{ \begin{matrix}(x, x − n), & if P ≠ O, (0, 0), (n, 0); \\ (−1, −n), & if P = (0, 0); \\ (n, 2), & if P = (n, 0); \\ (1, 1), & if P = O. \end{matrix}

(ii) Let (a, b) ∈ ℚ(S, 2) × ℚ(S, 2)\{(1, 1), (−1, −n), (n, 2)}. Then (a, b) is the image of a point P = (x, y) ∈ E_{n}(ℚ)/2E_{n}(ℚ) if and only if the following system of equations have a common solution (X, Y, Z) ∈ ℚ^{∗} × ℚ^{∗} × ℚ^{∗}.

(∗) aX^{2} − bY^{2} = n, aX^{2} − abZ^{2} = −n.

If such a solution exist then one can take P = (aX^{2}, abXY Z) = (bY^{2} + n, abXY Z).

For a proof of this theorem see [24, Chap. X, §1] or [23, Sec. 3] .

Note that the Mordell-Weil rank of the curve E_{n} can be found by

r(n) = log_{2} (\frac{Image(θ)}{4});

Also, the cardinality of S^{(2)} is equal to the number of the pairs (a, b) such that the system (∗) is everywhere locally solvable. If one take the set R = {±2^{α}p_{1}^{α_{1}} ⋯ p_{t}^{α_{t}}|α, α_{1}, …, α_{t} ∈ {0, 1}} as representatives for ℚ(S, 2), then it is immediate that #ℚ(S, 2) = 2^{t+2} and so

r(n) ≤ s(n) ≤ 2w(n).

3 Monsky’s formula for 2-Selmer rank

In 1994, P. Monsky [12] proved a theorem on the parity of the 2-Selmer rank of CN-elliptic curves. He gave a formula for computation of the s(n) through his proof of this theorem.

Theorem 2. Let n be a positive squarefree integer. Then

s(n) ≡ \{ \begin{matrix}0 (mod 2), & if n ≡ 1, 2, 3 (mod 8); \\ 1 (mod 2), & if n ≡ 5, 6, 7 (mod 8). \end{matrix}

For a proof of this theorem see Appendix of [12].

Let n be a positive squarefree integer with odd prime factors p_{1}, …, p_{t}. Define the diagonal t × t matrix D_{l} = (d_{i}), for l ∈ {−1, −2, 2} , and the square t × t matrix A = (a_{ij}) as follows:

d_{i} = \{ \begin{matrix}0, & if (\frac{l}{p_{i}}) = 1; \\ 1, & if (\frac{l}{p_{i}}) = −1, \end{matrix} a_{ij} = \{ \begin{matrix}0, & if (\frac{p_{j}}{p_{i}}) = 1, j ≠ i; \\ 1, & if (\frac{p_{j}}{p_{i}}) = −1, j ≠ i, \end{matrix} a_{ii} = ∑_{j: j≠i} a_{ij}.

Monsky showed that s(n) can be computed as

s(n) = \{ \begin{matrix}2t − rank_{𝔽_{2}}(M_{o}), & if n = p_{1}p_{2} ⋯ p_{t}; \\ 2t − rank_{𝔽_{2}}(M_{e}), & if n = 2p_{1}p_{2} ⋯ p_{t}, \end{matrix}

where M_{o} and M_{e} are the following 2t × 2t matrices:

M_{o} = [ \begin{matrix}A + D_{2} & D_{2} \\ D_{2} & A + D_{−2} \end{matrix} ] , M_{e} = [ \begin{matrix}D_{2} & A + D_{2} \\ A^{T} + D_{2} & D_{−1} \end{matrix} ] .

4 Mestre-Nagao’s sum

Now we describe a sieving method for finding the best candidates for high rank CN-elliptic curves. For any elliptic curve E : y^{2} = x^{3} + ax + b over ℚ, and every prime number p not dividing the discriminant Δ = −16(4a^{3} + 27b^{2}) of E, we can reduce a and b modulo p and view E as an elliptic curve over the finite field 𝔽_{p}. Let #E(𝔽_{p}) be the number of points on the reduced curve:

#E(𝔽_{p}) = 1 + #{0 ≤ x, y ≤ p − 1 : y^{2} ≡ x^{3} + ax + b (mod p)}.

There is both theoretical and experimental evidence which suggests that elliptic curves of high ranks have the property that #E(𝔽_{p}) is large for many primes p.

Definition 3. Let N be a positive integer and P_{N} be the set of all primes less than N. Mestre-Nagao’s sum is defined by

S(N, E) = ∑_{p∈P_{N}}(1 − \frac{p − 1}{#E(𝔽_{p})}) log p = ∑_{p∈P_{N}} \frac{−a_{p} + 2}{#E(𝔽_{p})} log p.

Note that S(N, E) can be computed efficiently with PARI/GP software [17], provided N is not too large. It is experimentally known [7, 15, 16] that we may expect that high rank curves have large S(N, E). See [4] for a heuristic argument which connects this assertion with the famous Birch and Swinnerton-Dyer conjecture. For a positive squarefree integer n, we denote S(N, E_{n}) by S(N, n).

5 An algorithm for finding high rank

Now we are ready to exhibit our algorithm for finding high rank CN-elliptic curves, based on Monsky’s formula for 2-Selmer rank of CN-elliptic curves s(n) and Mestre-Nagao’s sum S(N, n). In this algorithm, first of all, a list of different positive squarefree congruent number is considered. Next, for any integer n in this list, the value of s(n) is computed by the Monsky’s formula which is described in the section 3. Selecting those n’s with s(n) ≥ s for a given positive number s, a new list of integers n is scored by Mestre-Nagao sum S(N, n) using finitely many successive primes. Finally, the Mordell-Weil rank r(n) is computed by MWRANK for integers n with s(n) ≥ s and large values of Mestre-Nagao sums. To be more precise, we write our algorithm step by step as follows.

Step 1. Let s be a positive integer. Choose a non-empty set T of some squarefree congruent numbers. For any n ∈ T compute s(n) by the Monsky’s formula. Define the subset T_{s} of T containing all n ∈ T with s(n) = s. If T_{s} is empty choose another set T.

Step 2. Let k be a positive integer. Choose the set M_{s} as follows:

M_{s} = {(N_{i}, M_{i}) : 0 < N_{1} < ⋯ < N_{k}, 0 < M_{i}, 1 ≤ i ≤ k} .

Put T_{s}^{0} = T_{s}, and for any i with 1 ≤ i ≤ k, define the recursive sets

T_{s}^{i} = {n ∈ T_{s}^{i−1} : S(N_{i}, n) ≥ M_{i}} .

Step 3. Take j, 1 ≤ j ≤ k, such that for any i with j < i ≤ k, the sets T_{s}^{i} are empty. Now for any n ∈ T_{s}^{j}, compute r(n) using Cremona’s MWRANK [5].

Remark 4. For a given positive integer s in Step 1, choice of starting set T is very important. To save the time, we should avoid any repeated elements in T. By applying Theorem 2 and Lagrange’s conjecture about the parity of r(n), one can expect to find an integer n in the set T_{s} such that r(n) is less than s and has the same parity as s.

Remark 5. The most sensitive part of our algorithm is choosing the sets M_{s} in Step 2. For a prescribed value of s, we must choose the elements of M_{s} and its cardinality in such a way that the total time of available computations is as small as possible. Note that the elements of the sets T_{s}^{j}, in Step 3, are the best candidates for high rank CN-elliptic curves.

Remark 6. In Step 3, we try to compute r(n) for any n ∈ T_{s}^{j}. This is done by Cremona’s program MWRANK efficiently for small values of n. However, for large n’s the computation can be much slower, and MWRANK often gives only lower and upper bounds for r(n).

Given any positive integer s, our algorithm can be implemented in some different ways depending on the choice of the starting set T in Step 1. To explain our strategy, we need the next result which gives two specific families of congruent numbers. For a proof of the cases (I) and (II) see [21] and [23], respectively. Note that the construction of congruent numbers via case (I) is the same as that in [22] (originally due to Gouvéa and Mazur), applied to the

curves E_{1} : y^{2} = x^{3} − x and E1^{′} : y^{2} = x^{3} + 4x.

Theorem 7. Let u and v be arbitrary positive integers such that u < v, gcd(u, v) = 1 and u + v is odd. Then the squarefree parts of the following families of integers are congruent numbers:

(I) uv(v − u)(v + u), (II) uv(u^{2} + v^{2})/2.

In this paper, we focused on the integers s ≥ 5 and all different positive squarefree integers n of the forms (I) and (II) with u < v ≤ 10^{5} and ω(n) ≥ 5, where ω(n) denotes the number of distinct prime factors of n.

After choosing two sets T_{I} and T_{II} related to the integers of the form (I) and (II), we then took the starting set of the our algorithm as T = T_{I} ∪ T_{II} and got different sets T_{s} for each s ≥ 5. Then for each s ≥ 5, we considered the related sets M_{s} as follows:

{N_{i}}_{i=1}^{7} = {500, 1000, 5000, 10000, 15000, 20000, 50000} ,
M_{5} = {(N_{1}, 10), (N_{2}, 12), (N_{3}, 15), (N_{4}, 20), (N_{5}, 25), (N_{6}, 28), (N_{7}, 30)} ,
M_{6} = {(N_{1}, 10), (N_{2}, 14), (N_{3}, 18), (N_{4}, 22), (N_{5}, 25), (N_{6}, 30), (N_{7}, 35)} ,
M_{7} = {(N_{1}, 10), (N_{2}, 15), (N_{3}, 20), (N_{4}, 25), (N_{5}, 30), (N_{6}, 35), (N_{7}, 40)} ,
M_{8} = {(N_{1}, 10), (N_{2}, 14), (N_{3}, 16), (N_{4}, 20), (N_{5}, 25), (N_{6}, 30), (N_{7}, 35)} ,
M_{9} = {(N_{1}, 10), (N_{2}, 15), (N_{3}, 20), (N_{4}, 25), (N_{5}, 28), (N_{6}, 30), (N_{7}, 35)} ,
M_{≥10} = {(N_{1}, 10), (N_{2}, 12), (N_{3}, 15), (N_{4}, 18), (N_{5}, 22), (N_{6}, 25), (N_{7}, 30)} .

For each s ≥ 5 and each i, 1 ≤ i ≤ 7, by choosing (N, M) = (N_{i}, M_{i}) ∈ M_{s} and computing S(N_{i}, n) for all n ∈ T_{s}^{i−1}, gets the sets T_{s}^{i} of n’s that satisfy S(N_{i}, n) ≥ M_{i}. The elements of the sets T_{s}^{j} are best candidates to give high rank CN-elliptic curves. Finally, we used MWRANK to compute Mordell-Weil rank r(n), for n’s in each of the sets T_{s}^{j}. This stage of our algorithm was very time consuming. By the implementation of our algorithm, we have rediscovered some of the Rogers’ examples with r(n) = 5, 6, and 7. Also, we were able to find some new CN-elliptic curves with r(n) = 6 and some curves with 5 ≤ r(n) ≤ 7. We give these curves in the Tables 1 and 2, respectively.

We give also generators of the Mordell-Weil group for two smallest new examples with r(n) = 6. By using MWRANK we find 6 indepenent points on E_{n}, which are moreover generators of the Mordell-Weils group, while LLL-algorithm is used for finding the generators with smaller heights, which are listed below.

For n = 531670544130 we have the curve

y^{2} = x^{3} − 282673567495490277456900x

with the generators

P1 = [-317205078080, 240309412570889200],
P2 = [1110744023070, 1027815645288207600],
P3 = [-8842721250, 49989119984694000],
P4 = [2350922039070, 3511212519485048400],
P5 = [7424745951989070/361, 639554031769152257946000/6859],
P6 = [-165395800834700271/51351556, 11103259191546833925683935833/367985250296]

For n = 602730488666 we have the curve

y^{2} = x^{3} − 363284041967555154459556x

with the generators

P1 = [25844642800106/25, 106746067884077780496/125],
P2 = [-89776938384, 178580334935648520],
P3 = [3666632085466, 6925523273366507040],
P4 = [26198594092166458/10609, 4112253205326835858960032/1092727],
P5 = [2097707297289652801/1012036, 2906919721960250194451760705/1018108216],
P6 = [5187004732864967512122/8543489761,
     44888914750852091711316911386224/789683302098991]

6 Acknowledgements

The authors would like to express their gratitude to N. Rogers for giving the list of his unpublished results. They also thank the referee for his/her helpful suggestions. The third author would like to thank J. Cremona for his helpful guides for using MWRANK and his suggestions to resolve certain computational issues in computing ranks of CN-elliptic curves for large positive integers.

n  factorization  n mod 8 s(n)
531670544130  2·3·5·11·17·107·463·1913  2  6
602730488666  2·29·41·97·137·19073  2  6
1079812755065  5·11·23·41·89·449·521  1  6
1351528542210  2·3·5·7·11·29·31·47·61·227  2  6
1440993982946  2·7·17·23·41·73·281·313  2  8
1544991154746  2·3·13·19·83·163·251·307  2  6
1663586838899  17·103·137·756·9161  3  8
2280190889130  2·3·5·7·11·23·41·257·4073  2  6
4611082954146  2·3·19·41·113·953·9161  2  8
8231905771386  2·3·11·17·19·23·41·43·89·107  2  6
9033322597530  2·3·5·7·11·43·53·59·127·229  2  6
17434310103210  2·3·5·7·11·13·17·19·67·139·193  2  6
46485304142530  2·5·11·19·23·43·67·107·3137  2  6
90181020280890  2·3·5·7·11·251·397·401·977  2  6
165130972136130  2·3·5·7·11·13·29·103·233·7901  2  6
179009302343970  2·3·5·7·17·19·23·47·53·73·631  2  6
181025271456226  2·17·103·127·151·1259·2141  2  6
243339180933145  5·11·401·1049·3169·3319  1  8
339507119347242  2·3·7·17·19·23·37·59·113·401  2  6
444724421083665  3·5·17·31·71·103·137·233·241  1  8
846249312638730  2·3·5·7·11·13·31·37·41·101·349  2  6
1056710141801930 2·5·7·11·41·43·53·71·269·769  2  6
4601440550332626 2·3·7·11·13·17·19·37·41·101·113·137  2  6
13897395819317010 2·3·5·7·11·13·23·29·31·61·113·191  2  6

Table 1: Some new CN-elliptic curves with r(n) = 6

n  factorization  n mod 8 s(n)
1024801887174  2·3·13·37·409·769·1129  6  7
1025774078934  2·3·11·17·41·43·641·809  6  7
1649085975174  2·3·11·47·73·97·193·389  6  7
2093383150230  2·3·5·29·73·97·419·811  6  7
2392760979654  2·3·17·41·43·83·160313  6  7
2473595024934  2·3·11·17·41·83·347·1867  6  7
5080701332454  2·3·11·17·41·59·521·3593  6  7
5449406258406  2·3·11·17·41·251·683·691  6  7
7322494848870  2·3·5·17·19·137·151·36529  6  7
7391341307526  2·3·11·19·59·67·523·2851  6  7
7697325362694  2·3·11·137·401·547·3881  6  7
7836495180886  2·17·281·353·971·2393  6  9
7889458857566  2·11·19·881·1049·1571  6  7
8549294440966  2·17·19·37·137·353·5857  6  7
10571147972390  2·5·17·89·277·587·4297  6  7
11050024116846  2·3·11·13·17·29·31·569·1481  6  7
12651761296614  2·3·11·17·19·43·59·449·521  6  7
14020765617254  2·11·17·23·71·241·95257  6  7
19843964725254  2·3·17·19·937·2683·4073  6  7
25161173711039  19·23·29·103·1657·11633  7  7
25837148295902  2·31·97·593·1217·5953  6  9
26755379766174  2·3·23·59·233·353·39953  6  7
29130582949206  2·3·19·113·283·1913·4177  6  7
32334652741974  2·3·11·43·89·113·883·1283  6  7
34243576397574  2·3·73·89·457·953·2017  6  7
35876712238310  2·5·31·41·1289·1361·1609  6  7
44066140293846  2·3·11·17·41·43·59·491·769  6  9
56858065281654  2·3·7·13·19·73·89·769·1097  6  7
57705905931141  3·13·17·131·521·937·1361  5  7
57939619068870  2·3·5·7·11·37·53·89·137·1049  6  7
61639096639029  3·7·13·29·241·2113·15289  5  7
109995988504269  3·17·41·65809·114193  5  7
114490690064454  2·3·11·19·577·1873·84481  6  9
117205364344206  2·3·7·17·73·97·233·293·2377  6  7
119231629856526  2·3·11·17·29·41·59·83·18251  6  7
121466637600990  2·3·5·11·17·31·89·107·1033  6  7
130629627999390  2·3·5·13·17·37·41·97·257·521  6  7
146421396607926  2·3·11·17·19·449·2417·6329  6  7
175656508365734  2·11·97·113·10169·71633  6  9
180196195115046  2·3·11·17·43·83·179·251393  6  7
191519081464326  2·3·7·11·31·41·59·89·89·179·347  6  7
242515586992326  2·3·19·41·73·587·641·1889  6  9
433182183087126  2·3·11·17·41·251·2707·13859  6  7
459848288031405  3·5·7·13·17·41·61·389·20369  5  7
1687029282320910  2·3·5·11·1049·1729·2027  6  7
2053424339679966  2·3·11·17·19·31·43·179·499·809  6  7
2059195525185430  2·5·89·641·823·929·4721  6  9
3167344617712806  2·3·19·73·89·283·3137·4817  6  9
8797235243700486  2·3·11·19·313·577·5147·7547  6  9
342916139097905191 3·13·17·37·53·61·157·1753·6733  7  7

Table 2: Some CN-elliptic curves with 5 ≤ r(n) ≤ 7

References

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2000 Mathematics Subject Classification: Primary 11G05; Secondary 14H52. Keywords: CN-elliptic curve, Mordell-Weil rank, 2-Selmer rank, Mestre-Nagao sum.

(Concerned with sequences \underline{A003273}, \underline{A006991}, \underline{A062693}, \underline{A062694}, and \underline{A062695}.)

Received April 5 2009; revised version received July 14 2009. Published in Journal of Integer Sequences, July 16 2009.

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