More Upper Bounds on Taxicab and Cabtaxi Numbers

Po-Chi Su Hsing-Hua Senior High School Miaoli 35144 Taiwan

                    poky@webmail.mlc.edu.tw

Abstract

For positive integers a, b and integers x, y such that S = a^{3} + b^{3} = x^{3} + y^{3}, we prove that x + y ≡ a + b (mod 6); moreover, we give a parametric function r_{i} → (x(r_{i}), y(r_{i})) with (x(r_{i}))^{3} + (y(r_{i}))^{3} = a^{3} + b^{3} for chosen parameters r_{i}, and we conjecture that most such S are multiples of 18 if S is large enough. Accordingly, floating sieving is introduced and upper bounds on the Cabtaxi numbers Ca(n) with 43 ≤ n ≤ 57, and the Taxicab numbers Ta(n) with n = 23, 24 are given. Among them, Ta(n) with n = 23, 24, and Ca(n) with n = 43, 44, are included in the On-Line Encyclopedia of Integer Sequences.

1 Introduction

The n-th Taxicab number Ta(n) (respectively, the n-th Cabtaxi number Ca(n)) is the smallest that can be expressed as a sum of two cubes of positive integers (respectively, integers) in n ways, which are called n decompositions [1, 2]. For any positive n, Fermat proved the existence of Ta(n), as shown in the book by Hardy and Wright [4, Theorem 412]. Clearly, Ca(n) ≤ Ta(n) by definition. Specifically, Dardis found Ta(5) in 1994, but Ta(6) was not determined until 14 year later. Indeed, Wilson [5] found an upper bound on Ta(6) in 1997,

\begin{aligned}Ta(6) & ≤ 8, 230, 545, 258, 248, 091, 551, 205, 888 \\ & = 2^{9} · 3^{3} · 7 · 13 · 19^{3} · 31 · 67^{3} · 79 · 109^{3}. \end{aligned}

Rathbun improved this upper bound in 2002 to

\begin{aligned}Ta(6) & ≤ 24, 153, 319, 581, 254, 312, 065, 344 \\ & = 2^{6} · 3^{3} · 7^{4} · 13 · 19 · 43 · 73 · 79^{3} · 97 · 157. \end{aligned}

Calude et al. [3] showed that this upper bound is Ta(6) with a probability of greater than 0.99. Finally, in 2008, Hollerbach showed that it was exactly Ta(6) [1, 2].

Clearly, the determinations of Ta(n) or Ca(n) are not trivial. Indeed, these are problems \underline{A011541} and \underline{A047696} in the On-Line Encyclopedia of Integer Sequences, OEIS [6]. Up to now, those known Ta(n) and Ca(n) are given in Table 1. Straightforward relations, called magnifications, among these values; for example, Ca(3) = 2^{3}·Ca(2), Ca(9) = 5^{3}·67^{3}·Ca(7) = 2^{3} ·5^{3} ·67^{3} ·Ca(6) are given in Fig. 1. Magnifications among the best known upper bounds on Ta(n) and Ca(n) are described similarly by the diagrams in Figs. 13 and 14 in the Appendix.

Ta(n)  Ca(n)
Ta(1) = 2  Ca(1) = 1
Ta(2) = 7 · 13 · 19  Ca(2) = 7 · 13
Ta(3) = 3^{3} · 7 · 31 · 67 · 223  Ca(3) = 2^{3} · 7 · 13
Ta(4) = 2^{10} · 3^{3} · 7 · 13 · 19 · 31 · 37 · 127  Ca(4) = 2^{3} · 3^{3} · 7^{3} · 37
Ta(5) = 2^{6} · 3^{3} · 7^{4} · 13 · 19 · 43 · 73 · 97 · 157  Ca(5) = 3^{3} · 7 · 13 · 31 · 79
Ta(6) = 2^{6} · 3^{3} · 7^{4} · 13 · 19 · 43 · 73 · 79^{3} · 97 · 157 Ca(6) = 3^{3} · 7^{4} · 19 · 31 · 37
Ca(7) = 2^{3} · 3^{3} · 7^{4} · 19 · 31 · 37
Ca(8) = 2^{3} · 3^{3} · 7^{4} · 19 · 23^{3} · 31 · 37

Ca(9) = 2^{3} · 3^{3} · 5^{3} · 7^{4} · 19 · 31 · 37 · 67^{3} Ca(10) = 2^{3} · 3^{3} · 5^{3} · 7^{4} · 13^{3} · 19 · 31 · 37 · 67^{3}

Table 1: Known Ta(n) and Ca(n)

Figure 1: Magnifications among Ta(n) and Ca(n)

Following the discovery of Ta(n) with 1 ≤ n ≤ 6 and Ca(n) with 1 ≤ n ≤ 10, in 2008, Boyer found upper bounds on Ta(7), …, Ta(19) and Ca(11), …, Ca(30) respectively [1, 2]. As mentioned by Boyer later on his webpage, Moore improved the upper bounds on Ca(11), Ca(12), Ca(14), and Boyer and Wroblewski improved the upper bounds on Ta(11), …, Ta(19) and Ca(13), Ca(15), …, Ca(30). Boyer also gave the upper bounds on Ta(20), Ta(21), Ta(22) and Ca(31), …, Ca(42)[2]. In this paper, the known upper bounds on Ta(n) and Ca(n), given on Boyer’s webpage, are denoted by BTa(n) with 7 ≤ n ≤ 22, and BCa(n) with 11 ≤ n ≤ 42, respectively. Complete decompositions of BTa(n) with 7 ≤ n ≤ 12 and BCa(n) with 11 ≤ n ≤ 22 can be found in [2].

For given positive integers a and b, a condition for sieving integers x ≥ y to satisfy S = a^{3} + b^{3} = x^{3} + y^{3} is given in Lemma 4. The relation x + y ≡ a + b (mod 6) is a crucial condition in sieving the upper bounds on Ta(n) and Ca(n). We conjecture that Ta(n) with n ≥ 7 and Ca(n) with n ≥ 11 are multiples of 18 (as discussed in Section 3). If S = a^{3} +b^{3} is a multiple of 18, then the introduced sieving process can be utilized to find the upper bounds on Ta(n) and Ca(n). Based on the sieving conditions in Theorem 5, together with BTa(n) and BCa(n) that were provided by Boyer, the sieving process is modified herein by applying the concept of floating sieving to reduce the number of computations (Section 4). Applying the floating sieving process, the upper bounds on Ta(23), Ta(24) and Ca(43), …, Ca(57) together with their corresponding parameters are given in Section 5. The upper bounds on Ta(23), Ta(24), Ca(43) and Ca(44) were collected in OEIS, October 2014.

2 Magnifications

Magnification, introduced by Wilson [5] and Boyer [1], is an efficient and frequently used technique for finding the upper bounds on Ta(n) and Ca(n), that is finding a number with n + 1 decompositions starting from a number with n decompositions.

2.1 Magnifications among BTa(n) and among BCa(n)

If S can be described in n ways as a sum of two cubes, S = x_{i}^{3} + y_{i}^{3} with i = 1, 2, …, n, and k is an integer, then Sk^{3} can be described in at least n ways as a sum of two cubes: Sk^{3} = (kx_{i})^{3} + (ky_{i})^{3} with i = 1, …, n. If a value k is found such that there exists another sum of two cubes, Sk^{3} = x_{n+1}^{3} + y_{n+1}^{3} with (x_{n+1}, y_{n+1}) ≠ (x_{i}, y_{i}) for i = 1, 2, …, n, then Sk^{3} can be described as n + 1 sums of two cubes, yielding an upper bound on Ta(n + 1) or Ca(n + 1). Such a number k is called a splitting factor by Boyer [1].

For most values of n, the quotient BCa(n)/ BCa(n − 1) or BTa(n)/ BTa(n − 1) is k^{3} or the product k_{1}^{3} · k_{2}^{3} (where k, k_{1} and k_{2} are primes). Starting from BTa(7) and BCa(11) on, the quotients BTa(n)/ BTa(n−1) and BCa(n)/ BCa(n−1) are given in Table 2 respectively. Some of the quotients are complicated (see below for examples). However, most of them are in a simple form.

1. BTa(11)/ BTa(10) = (2^{3} · 5^{3} · 13^{2} · 17^{3} · 31 · 37 · 97^{2} · 109^{3}) / (19 · 29^{3} · 101^{3} · 127^{3})

2. BCa(11)/ BCa(10) = (2^{4} · 3^{3} · 37^{2} · 43 · 61^{3}) / (5^{3} · 7 · 13^{2} · 31 · 67^{2})

3. BCa(13)/ BCa(12) = (2^{2} · 3^{3} · 7 · 13^{3} · 109 · 193) / (19 · 43 · 61^{2} · 67)

4. BCa(14)/ BCa(13) = (2^{4} · 19 · 31^{3} · 43 · 61^{2} · 67) / (3^{3} · 7 · 13^{3} · 109 · 193)

5. BCa(15)/ BCa(14) = (3^{3} · 7 · 13^{3} · 73^{3} · 109 · 193) / (2^{4} · 19 · 31^{3} · 43 · 61^{2} · 67)

6. BCa(19)/ BCa(18) = (5^{3} · 11^{3} · 37 · 43 · 61^{2} · 67^{3} · 109^{2} · 157) / (2^{6} · 13 · 19^{6} · 31^{2} · 73^{2} · 193)

7. BCa(20)/ BCa(19) = (2^{6}·5^{3}·13·19^{3}·31^{2}·73^{2}·103^{3}·193) / (11^{3}·37·43·61^{2}·67^{3}·109^{2}·157)

8. BCa(21)/ BCa(20) = (11^{3}·43·61^{2}·67^{3}·79^{3}·109^{2}·157) / (2^{6}·5^{3}·13·31^{2}·37^{2}·73^{2}·103^{3}·193)

(a) BTa(n)

n BTa(n)/ BTa(n − 1)
7  101^{3}
8  127^{3}
9  139^{3}
10  13^{3} · 29^{3}
11  See 1.
12  3^{3} · 19^{3}
13  3^{3} · 61^{3}
14  397^{3}
15  503^{3}
16  2^{3} · 607^{3}
17  4261^{3}
18  37^{3} · 181^{3}
19 5^{6} · 457^{3} · 521^{3}/4261^{3}
20  4261^{3}
21  127^{3} · 197^{3}
22  11^{3} · 31^{3} · 103^{3}

(b) BCa(n)
n BCa(n)/ BCa(n − 1) n BCa(n)/ BCa(n − 1)
11  See 2.  27  5^{6}
12  19^{3}  28 7^{3} · 13^{3} · 97^{3}/5^{6} · 17^{3}
13  See 3.  29  17^{3}
14  See 4.  30  5^{6}
15  See 5.  31  29^{3}
16  19^{3}  32  43^{3}
17  2^{6} · 31^{3}/19^{3}  33  181^{3}
18  19^{3}  34  193^{3}
19  See 6.  35 397^{3} · 457^{3}/181^{3} · 193^{3}
20  See 7.  36  181^{3}
21  See 8.  37  101^{3} · 229^{3}/181^{3}
22  37^{3}/3^{3}  38  181^{3}
23  3^{3}  39  163^{3}
24  17^{3}  40  193^{3}
25  139^{3}/17^{3}  41  223^{3}
26  17^{3}  42  307^{3}

Table 2: Quotients of consecutive BTa(n) and BCa(n)

The magnifications among BTa(7), …, BT s(22) and BCa(11), …, BCa(42) can be found in Figs. 2, 3, 4, 5 and also Figs. 13 and 14 in the Appendix.

2.2 Consecutive magnifications

If k_{i} with 1 ≤ i ≤ h are splitting factors of S that can be described by n sums of two cubes, such that k_{i}^{3}S can be described by n + 1 sums of two cubes, then k_{1}^{3} · k_{2}^{3} ⋯ k_{h}^{3} · S may be described by n+h sums of two cubes, as in Example 1. The k-th known upper bounds, from small to large, are denoted by BTa(n, k) and BCa(n, k) respectively. Some upper bounds on Ca(n + h) may be improved by applying the magnification technique over some BCa(n, k), although these are not the best known upper bounds when k > 1, as in Examples 2 and 3.

Example 1. Ta(6) can be described as 6 sums of two cubes; 101^{3} · Ta(6) and 127^{3} · Ta(6) can be described as 7 sums of two cubes, and 101^{3} · 127^{3} · Ta(6) can be described as 8 sums of two cubes. This gives

Ta(8) ≤ BTa(8) = 127^{3} · BTa(7) = 101^{3} · 127^{3} · Ta(6).

When k = 23, 29, 38, 43, each k^{3} · Ca(10) can be described as 11 sums of two cubes, and 23^{3} · 29^{3} · 38^{3} · 43^{3} · Ca(10) can be described by 14 sums of two cubes. (see also Examples 6 and 7 in Section 4.1).

Figure 2: Magnifications among Ta(6), BTa(7) and BTa(8)

Example 2. The second and third smallest of the numbers with 9 decompositions are denoted by BCa(9, 2) and BCa(9, 3), respectively:

\begin{aligned}BCa(9, 2) & = 2^{6} · 3^{3} · 7^{7} · 19 · 31 · 73 · 97 · 139, \\ BCa(9, 3) & = 2^{7} · 3^{6} · 7^{3} · 13 · 19 · 37^{3} · 43 · 67. \end{aligned}

The upper bounds on Ca(11) that were derived by Boyer and Moore in 2008 are

Ca(11) ≤ 13^{3} · 17^{3} · BCa(9, 2), and
BCa(11) ≤ 61^{3} · BCa(9, 3)

respectively. Although BCa(9, 3) is larger than BCa(9, 2), we have

Ca(11) ≤ BCa(11) = 61^{3} · BCa(9, 3) < 13^{3} · 17^{3} · BCa(9, 2).

Fig. 3 presents the magnifications among the best known bounds.

Figure 3: BCa(11), BCa(12), BCa(14) associated with BCa(9, 3)

Example 3. The fifth smallest known with 10 decompositions is

BCa(10, 5) = 2^{9} · 3^{3} · 7^{4} · 13^{4} · 19^{3} · 61 · 109 · 193.

The upper bounds associated with BCa(10, 5) are given below:

BCa(13) = 3^{6} · 37^{3} · BCa(10, 5),
BCa(15) = 3^{6} · 37^{3} · 73^{3} · BCa(10, 5) = 73^{3} · BCa(13),
BCa(16) = 3^{6} · 19^{3} · 37^{3} · 73^{3} · BCa(10, 5) = 19^{3} · BCa(15),
BCa(17) = 2^{6} · 3^{6} · 31^{3} · 37^{3} · 73^{3} · BCa(10, 5) = 2^{6} · 31^{3} · BCa(15),
BCa(18) = 2^{6} · 3^{6} · 19^{3} · 31^{3} · 37^{3} · 73^{3} · BCa(10, 5) = 19^{3} · BCa(17),

BCa(20) = 2^{6} · 3^{6} · 5^{6} · 31^{3} · 37^{3} · 73^{3} · 103^{3} · BCa(10, 5) = 2^{6} · 5^{6} · 31^{3} · 103^{3} · BCa(15),

which are summarized in Fig. 4.

In Fig. 5, the magnifications among BCa(15), BCa(16), BCa(17) and BCa(18) are displayed in the shape of a parallelogram. Similar situations also hold in BCa(23), BCa(24), BCa(25) and BCa(26), as displayed in Fig. 7, and BCa(35), BCa(36), BCa(37), BCa(38), as displayed in Fig. 9, respectively.

Figure 4: BCa(n) associated with BCa(10, 5)

Figure 5: Magnifications among BCa(15), BCa(16), BCa(17), BCa(18)

3 Parameters of decompositions

If S = a^{3} + b^{3}, then a^{3} + b^{3} is called a decomposition of S as a sum of two cubes, abbreviated as a decomposition of S. Hence Ta(n) (resp., Ca(n)) is the smallest positive integer with n decompositions with nonnegative integral summands (or respectively integral summands). We will show that each decomposition of S can be expressed as a parametric function; see Theorem 5.

3.1 Decompositions as sums of two cubes

Consider decompositions S = x_{1}^{3} + y_{1}^{3} = x_{2}^{3} + y_{2}^{3} as sums of two cubes, and observe that (x_{1} + y_{1}) − (x_{2} + y_{2}) is a multiple of 6 in Table 3, and will be proved in Lemma 4.

\begin{matrix}S & x_{1} & y_{1} & x_{2} & y_{2} & (x_{1} + y_{1}) − (x_{2} + y_{2}) \\ 1,729 & 10 & 9 & 12 & 1 & 6 \\ 4,104 & 15 & 9 & 16 & 2 & 6 \\ 20,683 & 24 & 19 & 27 & 10 & 6 \\ 39,312 & 33 & 15 & 34 & 2 & 12 \\ 40,033 & 33 & 16 & 34 & 9 & 6 \\ 65,728 & 33 & 31 & 40 & 12 & 12 \\ 64,232 & 36 & 26 & 39 & 17 & 6 \\ 134,379 & 43 & 38 & 51 & 12 & 18 \\ 149,389 & 50 & 29 & 53 & 8 & 18 \\ 171,288 & 54 & 24 & 55 & 17 & 6 \end{matrix}

Table 3: Relations between pairs of decompositions

Lemma 4. For integers a, b, x, y with a < x and a^{3} + b^{3} = x^{3} + y^{3}, then

x + y ≡ a + b (mod 6).

Proof. We notice that a^{3} − a = (a − 1)a(a + 1) is a multiple of 6, and then a^{3} ≡ a (mod 6) holds for any integer a. Hence, x + y ≡ a + b (mod 6).

Based on observations on known Ta(n) with n = 4, 5, 6, BTa(n) with 7 ≤ n ≤ 12, Ca(n) with 7 ≤ n ≤ 10, and BCa(n) with 11 ≤ n ≤ 22, note that 6|(x + y) in each decomposition S = x^{3}+y^{3} and therefore x^{3}+y^{3} is a multiple of 18 because of x^{3}+y^{3} = (x+y)((x+y)^{2}−3xy). We conjecture that Ta(n) with n ≥ 7 and Ca(n) with n ≥ 11 are all multiples of 18. This is because when the magnification technique is used to find the upper bound on Ta(n + 1) and Ca(n + 1) from BTa(n) and BCa(n) of multiples of 18, the one derived will also be a multiple of 18. Section 4 introduces the sieving process that can be used to find the upper bounds on Ta(n) and Ca(n).

3.2 Parameters of decompositions

All sums of two cubes can be expressed as a parametric function (see Theorem 5), and this fact leads to a sieving process in Section 4.

For a given S = a^{3} + b^{3} with positive integers a ≥ b, to determine integers x ≥ y such that S = a^{3} +b^{3} = x^{3} +y^{3}, k = x+y is introduced and x, y must be obtained for appropriate k (or else may have no solution).

\begin{cases}x + y = k, \\ x^{3} + y^{3} = S. \end{cases}

Hence, x = x(k, S) and y = y(k, S) can be expressed as functions of k and S. Substituting y = k − x into x^{3} + (k − x)^{3} = S yields 3x^{2} − 3kx + (k^{3} − S)/k = 0 and therefore,

x = (3k + \sqrt{(3k)^{2} − 4 · 3 · (k^{3} − S)/k})/6,
y = (3k − \sqrt{(3k)^{2} − 4 · 3 · (k^{3} − S)/k})/6.

The necessary and sufficient condition for x and y to be positive integers is that the expression −3k^{2} + 12S/k ≥ 0 inside the square root is a perfect square. Since −3k^{2} + 12S/k ≥ 0 and k^{3} = (x + y)^{3} > x^{3} + y^{3} = S, we then have \sqrt[3]{S} < k ≤ \sqrt[3]{4S}.

Repeating the above for k = x + y = 6r yields x^{2} − 6rx + 12r^{2} − S/18r = 0. Substituting y = 6r − x yields

x = 3r + \sqrt{−3r^{2} + (S/18r)},
y = 3r − \sqrt{−3r^{2} + (S/18r)}.

The necessary and sufficient condition for x and y to be positive integers is that −3r^{2}+S/18r is a perfect square. Moreover, \sqrt[3]{S/216} < r ≤ \sqrt[3]{S/54} because −3r^{2} + S/18r ≥ 0 and (6r)^{3} = (x + y)^{3} > x^{3} + y^{3} = S. The above is summarized as follows:

Theorem 5. If S = x^{3} + y^{3}, x + y = k and x ≥ y, then

x = \frac{3k + \sqrt{−sk^{2} + 12S/k}}{6} , y = \frac{3k − \sqrt{−3k^{2} + 12S/k}}{6} .

Moreover,

(a) x = ^{\frac{3k+\sqrt{−3k^{2}+12S/k}}{6}} and y = ^{\frac{3k−\sqrt{−3k^{2}+12S/k}}{6}} are positive rationals if and only if k|S, \sqrt[3]{S} < k ≤ \sqrt[3]{4S}, and −3r^{2} + 12S/k are perfect squares.

(b) When k = 6r, x = 3r + \sqrt{−3r^{2} + (S/18r)} and y = 3r − \sqrt{−3r^{2} + (S/18r)} are positive integers if and only if 18|S, r is a divisor of S/18, \sqrt[3]{S/216} < r ≤ \sqrt[3]{S/54} and −3r^{2} + (S/18r) is a perfect square.

Among the sieving conditions in Theorem 5, the condition of perfect square is most crucial, as shown in Examples 6 and 7. The condition that r is a factor of S/18 is required in the Sieving Process and floating sieving process. When n parameters r = r_{1}, r_{2}, …, r_{n} are sieved, then the parametric functions

r_{i} → (x_{i}, y_{i}) = (x(r_{i}), y(r_{i}))

provides n decompositions of S = (x(r_{i}))^{3} + (y(r_{i}))^{3} with 1 ≤ i ≤ n.

4 Sieving and floating sieving

Based on the parametric expression given in Theorem 5, a sieving process for upper bounds on Ta(n) and Ca(n) is given in Section 4.1. Upper bounds on Ta(n) with n = 7, 8, 9 and on Ca(n) with n = 11, …, 16, derived by this process are given in Examples 6, 7 and 8 with illustrations. To reduce computational load, the floating sieving process is introduced in Section 4.2 with illustrations by applying the concept of floating sieving.

4.1 Sieving process

For given integers a and b, the conditions that govern the parameters given in Theorem 5 can be used to sieve for possible integers x, y that satisfy S = a^{3} + b^{3} = x^{3} + y^{3}, providing a means of exploring upper bounds on Ta(n) and Ca(n) respectively.

Sieving process

1. Input S = a^{3} + b^{3} and k, let counter = 0.

2. List all positive factors r_{1} < r_{2} < r_{3} ⋯ < r_{t} of Sk^{3}/18.

3. For i = 1, ⋯, t,

If \sqrt[3]{Sk^{3}/216} < r_{i} ≤ \sqrt[3]{Sk^{3}/54},

If −3r_{i}^{2} + Sk^{3}/18r_{i} is a perfect square, output r_{i}, x(r_{i}) = 3r_{i} + \sqrt{−3r_{i}^{2} + Sk^{3}/18r_{i}} and y(r_{i}) = 3r_{i} − \sqrt{−3r_{i}^{2} + Sk^{3}/18r_{i}}. counter ← counter + 1, i ← i + 1, return to Step 3.

otherwise i ← i + 1, return to Step 3. Otherwise, i ← i + 1, return to Step 3.

4. Output counter.

Let S = a^{3} + b^{3}. To find x, y with Sk^{3} = x^{3} + y^{3}, the function counter gives the number of decompositions, it is more likely to increase for prime k. If the condition \sqrt[3]{Sk^{3}/216} < r ≤ \sqrt[3]{Sk^{3}/54} above is replaced by 0 < r ≤ \sqrt[3]{Sk^{3}/54}, then all integral solutions of Sk^{3} = k^{3}(a^{3} + b^{3}) = x^{3} + y^{3} can be derived, yielding a sieving process for Cabtaxi numbers. In the following examples, the sieving process is illustrated in terms of Ta(6) with k = 101, 127 and Ca(10) with k = 23, 29, 38, 43, 127.

Example 6. For S = Ta(6) = 2^{6} · 3^{3} · 7^{4} · 13 · 19 · 43 · 73 · 79^{3} · 97 · 157 and k = 101, Sk^{3} has 143,360 positive factors, of the 61,440 positive factors of Sk^{3}/18, 629 lie between \sqrt[3]{Sk^{3}/216} and \sqrt[3]{Sk^{3}/54}, and finally −3r^{2} + Sk^{3}/18r is a perfect square for only 7 of them. Therefore, 7 decompositions are derived by using Theorem 5, yielding Ta(7) ≤ 101^{3} · Ta(6).

Example 7. For S = Ta(6) · 101^{3} = 2^{6} · 3^{3} · 7^{4} · 13 · 19 · 43 · 73 · 79^{3} · 97 · 101^{3} · 157 and k = 127, Sk^{3} has 573, 440 factors, of the 245,760 positive factors of Sk^{3}/18, 2,004 lie between \sqrt[3]{Sk^{3}/216} and \sqrt[3]{Sk^{3}/54}, and finally −3r^{2} + Sk^{3}/18r is a perfect square for only 8 of them. Therefore, 8 decompositions are derived by using Theorem 5, yielding

Ta(8) ≤ 101^{3} · 127^{3} · Ta(6).

Notably, the numbers 101 and 127 above are primes, and counter = 9 is derived for 101^{3} · 127^{3} · 139^{3} · Ta(6), which therefore has 9 decompositions, so

Ta(9) ≤ 101^{3} · 127^{3} · 139^{3} · Ta(6).

Example 8. For S = Ca(10), then counter = 11 is derived when k = 23, 29, 38, 43 and 46. Moreover, each of 23^{3} · Ca(10), 23^{3} · 29^{3} · Ca(10), 23^{3} · 29^{3} · 38^{3} · Ca(10), and 23^{3} · 29^{3} · 38^{3} · 43^{3} · Ca(10) have 11, 12, 13 and 14 parameters respectively. The above bounds on Ca(12), Ca(13), Ca(14) can be improved further. Consider the prime k = 127 ≤ 23 · 29, for which counter = 12, meaning that 127^{3} · Ca(10) also has 12 parameters, a better bound, so

Ca(12) ≤ 127^{3} · Ca(10).

Similar arguments show that

29^{3} · 127^{3} · Ca(10) has 13 parameters, an upper bound of Ca(13), 29^{3} · 43^{3} · 127^{3} · Ca(10) has 14 parameters, an upper bound of Ca(14), 23^{3} · 29^{3} · 38^{3} · 127^{3} · Ca(10) has 15 parameters, an upper bound of Ca(15), 23^{3} · 29^{3} · 38^{3} · 43^{3} · 127^{3} · Ca(10) has 16 parameters, an upper bound of Ca(16).

We let SCa(11), …, SCa(16) denote the upper bounds on Ca(11), …, Ca(16) obtained by sieving process based on Ca(10). These bounds are summarized below for reference, though they are not as good as bounds given by Boyer. The magnifications among them are given in Fig. 6. The same technique can also be used to derive upper bounds on Ca(43), …, Ca(57) based on BCa(42); see Theorem 11, and upper bounds on Ta(23), Ta(24) based on BTa(22) as well; see Theorem 12.

Ca(11) ≤ BCa(11) ≤ SCa(11) = 23^{3} · Ca(10)
= 2^{3} · 3^{3} · 5^{3} · 7^{4} · 13^{3} · 19 · 23^{3} · 31 · 37 · 67^{3},
Ca(12) ≤ BCa(12) ≤ SCa(12) = 127^{3} · Ca(10)
= 2^{3} · 3^{3} · 5^{3} · 7^{4} · 13^{3} · 19 · 31 · 37 · 67^{3} · 127^{3},
Ca(13) ≤ BCa(13) ≤ SCa(13) = 29^{3} · 127^{3} · Ca(10)
= 2^{3} · 3^{3} · 5^{3} · 7^{4} · 13^{3} · 19 · 29^{3} · 31 · 37 · 67^{3} · 127^{3},
Ca(14) ≤ BCa(14) ≤ SCa(14) = 29^{3} · 43^{3} · 127^{3} · Ca(10)
= 2^{3} · 3^{3} · 5^{3} · 7^{4} · 13^{3} · 19 · 29^{3} · 31 · 37 · 43^{3} · 67^{3} · 127^{3},
Ca(15) ≤ BCa(15) ≤ SCa(15) = 2^{3} · 19^{3} · 23^{3} · 29^{3} · 127^{3} · Ca(10)
= 2^{6} · 3^{3} · 5^{3} · 7^{4} · 13^{3} · 19^{4} · 23^{3} · 29^{3} · 31 · 37 · 67^{3} · 127^{3},
Ca(16) ≤ BCa(16) ≤ SCa(16) = 2^{3} · 19^{3} · 23^{3} · 29^{3} · 43^{3} · 127^{3} · Ca(10)
= 2^{6} · 3^{3} · 5^{3} · 7^{4} · 13^{3} · 19^{4} · 23^{3} · 29^{3} · 31 · 37 · 43^{3} · 67^{3} · 127^{3}.

4.2 Floating sieving process

Upper bounds for Ta(7), Ta(8), Ta(9), and for Ca(11), …, Ca(16) were derived by the above sieving process. However, the computing time that is needed for sieving increases as an

Figure 6: Magnifications among Ca(10), SCa(11), …, SCa(16)

exponential function of n in both Ca(n) and Ta(n). The process is therefore modified by considering the exponent sum of its standard factorization, rather than a value itself, and this modified process is the called floating sieving process.

Recall that each parameter r is a divisor of S/18 as shown in Theorem 5. Let r_{i} = ∏_{j=1}^{m} p_{j}^{β_{i,j}}

for primes p_{1} < p_{2} < ⋯ < p_{m} with 1 ≤ i ≤ n be the standard product of prime powers of the parameter of BCa(n), abbreviated as r_{i} = (β_{i,1}, β_{i,2}, …, β_{i,m}) with base (p_{1}, p_{2}, …, p_{m}). When a number S that can be described as n + 1 sums of two cubes, we may try S = k^{3} · BCa(n) first for a splitting factor k. Since S = k^{3}·BCa(n) itself already has n decompositions with parameters kr_{1}, kr_{2}, …, kr_{n}, the key lies in finding an additional parameter r_{n+1}. Let

a_{i} = ∑_{j=1}^{m} β_{i,j} with 1 ≤ i ≤ n be the exponent sum of r_{i}, and let [L, U] be an interval

that contains all a_{i}. Based on the assumption that k is a prime, the exponent sums of kr_{1}, kr_{2}, …, kr_{n} lie in the interval [L+1, U +1]. The exponent-sums of additional parameters are likely in the interval [L + 1, U + 1] as well, rather than in the range of the parameter [\sqrt[3]{S/216}, \sqrt[3]{S/54}] so many and huge numbers can be avoided.

More specifically, an additional parameter r_{n+1} = k^{β_{m+1}} · ∏_{i=1}^{m} p_{i}^{β_{i}}, associated with

k^{3} · BCa(n) = k^{3} · ∏_{i=1}^{m} p_{i}^{α_{i}}, may satisfy the conditions 0 ≤ β_{1} ≤ α_{1} − 1, 0 ≤ β_{2} ≤ α_{2} − 2 (since

18 = 2 · 3^{2} is a divisor of S), 0 ≤ β_{i} ≤ α_{i}/2 with 3 ≤ i ≤ m, and β_{m+1} = 0 or 3. Based on a comparison with original sieving over all (β_{1}, β_{2}, …, β_{m+1}) for 0 ≤ β_{i} ≤ α_{i} with i ≤ m, and 0 ≤ β_{m+1} ≤ 3, only restricted values of r are sieved for, efficiently reducing the time needed for sieving. Therefore, the number of searches will be reduced to about 1/2^{m+1} of the original number, where m is the number of prime factors in the standard factorization of BCa(n).

Floating sieving process

1. Input primes {p_{i}} and nonnegative integers {α_{i}} with 1 ≤ i ≤ m with (p_{1}, p_{2}) = (2, 3)

(for BCa(n) = ∏_{i=1}^{m} p_{i}^{α_{i}}), k (for magnification S = k^{3} · ∏_{i=1}^{m} p_{i}^{α_{i}}), and L, U (range for

scanning).

2. Input β_{i} with i = 1, …, m + 1, where 0 ≤ β_{1} ≤ α_{1} − 1, 0 ≤ β_{2} ≤ α_{2} − 2, 0 ≤ β_{i} ≤ α_{i}/2, i = 3, …, m, and β_{m+1} = 0, 3. (candidates for scanning)

3. For each sequence (β_{1}, β_{2}, …, β_{m+1}), let r = k^{β_{m+1}} · ∏_{i=1}^{m} p_{i}^{β_{i}}, and a = ∑_{i=1}^{m+1} β_{i}.

4. If L ≤ a ≤ U,

if −3r^{2} + S/18r is a perfect square, output r, x(r) = 3r + \sqrt{−3r^{2} + S/18r}, and y = 3r − \sqrt{−3r^{2} + S/18r}, return to Step 3.

otherwise, return to Step 3. Otherwise, return to Step 3.

The choices of L, U and β_{i} are crucial in floating sieving for upper bounds, as properly chosen values greatly reduce the number of computations. A risk of missing searching targets is taken; however, it is still worthwhile if performance efficiency is taken into consideration. Example 9. Based on the floating sieving process, sets of parameters for BCa(22), BCa(30), BCa(42) and BTa(22) are given in Tables 8–12 in the Appendix. The first rows of these tables present the bases (p_{1}, p_{2}, …, p_{m}). The magnifications among BCa(23), …, BCa(26) are summarized in Table 4 with the base (p_{1}, p_{2}, …, p_{18}) = (2, 3, 5, 7, 11, 13, 17, 19, 31, 37, 43, 61, 67, 73, 79, 109, 139, 157), and the parameters for BCa(23), …, BCa(26) are thus provided.

BCa(n)  additional parameters
\begin{aligned}& BCa(23) \\ = & 3^{3} · BCa(22) \end{aligned}  r_{23} = (6, 7, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0)
\begin{aligned}& BCa(24) \\ = & 17^{3} · BCa(23) \end{aligned}  r_{24} = (2, 2, 1, 2, 1, 1, 3, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0)
\begin{aligned}& BCa(25) \\ = & 139^{3} · BCa(23) \end{aligned}  \begin{matrix}r′_{24} = (2, 2, 3, 2, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0) \\ r_{25} = (4, 2, 3, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0) \end{matrix}

BCa(26)

= 139^{3} · BCa(24)
= 17^{3} · BCa(25)

\begin{matrix}139 · r_{24} \\ 17 · r′_{24} \\ 17 · r′_{25} \end{matrix}

Table 4: Parameters of BCa(23), BCa(24), BCa(25), BCa(26)

5 Upper bounds on Ca(43), …, Ca(57), and Ta(23), Ta(24)

To find an additional parameter of an upper bound on Ca(43) starting from BCa(42) by using sieving process, too much computations are required because BCa(42) has 29 prime

Figure 7: Magnifications among BCa(23), BCa(24), BCa(25), BCa(26)

factors. On the other hand, a value k can be a splitting factor of BCa(n) as well as of BCa(n + h) simultaneously, splitting factors of BCa(30), rather than of BCa(42), tend to be sought because BCa(30) has 19 prime factors.

First, the parameters for BCa(31), BCa(32), BCa(35), BCa(37), BCa(38), …, and finally BCa(42) are obtained by a sequence of consecutive magnifications (Table 5). Then 15 splitting factors of BCa(30), either primes or products of two primes, are provided along with their additional parameters, as in Table 6. Combining these 15 splitting factors of BCa(30) and the 42 parameters for BCa(42) enables an upper bound on Ca(43) to be derived. In addition to an upper bound on Ca(43), upper bounds on Ca(44), …, Ca(57) can be derived in terms of the magnifications

BCa(42) = Q^{3} · BCa(30) and k^{3} · BCa(42) = k^{3} · Q^{3} · BCa(30)

with respect to specific splitting factor k of BCa(30), as in Figs. 10 and 11. A similar technique can be used to the derive of upper bounds on Ta(23) and Ta(24) from BTa(12) as in Section 5.3.

5.1 The set of 42 parameters of BCa(42)

The set of 30 parameters of

BCa(30) = 2^{9} ·3^{9} ·5^{9} ·7^{7} ·11^{3} ·13^{6} ·17^{3} ·19^{3} ·31^{1} ·37^{4} ·43^{1} ·61^{3} ·67^{3} ·73^{1} ·79^{3} ·97^{3} ·109^{3} ·139^{3} ·157^{1}

is given in Table 9, which yields 30 decompositions. A sequence of magnifications of BCa(31), BCa(32), BCa(35), BCa(37), …, BCa(42), as shown in Fig. 8, follows, and Table 5 presents their corresponding additional parameters. Notably the base

(p_{1}, p_{2}, …, p_{29})
= (2, 3, 5, 7, 11, 13, 17, 19, 29, 31, 37, 43, 61, 67, 73, 79, 97,
101, 109, 139, 157, 163, 181, 193, 223, 229, 307, 397, 457).

The set of 42 parameters of BCa(42) is given in Tables 10 and 11 with bases (p_{1}, p_{2}, …, p_{29}) in the first rows in the Appendix.

Figure 8: Magnifications among BCa(30), BCa(31), BCa(32), BCa(35), BCa(37), …, BCa(42)

Figure 9: Magnifications among BCa(35), BCa(36), BCa(37), BCa(38)

5.2 Upper bounds on Ca(43), …, Ca(57)

Upper bounds on Ca(43) can be derived by a two-step strategy as follows:

1. Find a sequence of magnifications of BCa(30), …, BCa(42) (Table 5).

2. Find 15 splitting factors of BCa(30) by floating sieving (Table 6), starting from BCa(30), and then upper bounds on Ca(44), …, Ca(55) are derived in a sequence of magnifications.

Floating sieving process is used to find splitting factors k of BCa(30) with additional parameter R. Let BCa(30) = ∏_{i=1}^{19} p_{i}^{α_{i}}, and the corresponding 30 parameters of r_{i} be ∏_{j=1}^{19} p_{j}^{β_{i,j}}.

Additional parameter R = k^{β_{20}} · ∏_{i=1}^{19} p_{i}^{β_{i}}, whose possible β_{i}, i = 1, …, 19 are summarized in Table 7, with β_{20} = 0 or 3, can be found in the following.

BCa(n)  additional parameters
\begin{aligned}& BCa(31) \\ = & 29^{3} · BCa(30) \end{aligned}  r_{31} = (2, 2, 3, 2, 1, 2, 1, 1, 3, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0)
\begin{aligned}& BCa(32) \\ = & 43^{3} · BCa(31) \end{aligned}  r_{32} = (2, 3, 5, 2, 1, 2, 1, 1, 3, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0)
\begin{aligned}& BCa(35) \\ = & 397^{3} · 457^{3} · BCa(32) \end{aligned}  \begin{matrix}r_{33} = (2, 2, 9, 2, 1, 2, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0) \\ r_{34} = (2, 5, 3, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1) \\ r_{35} = (4, 2, 7, 3, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0) \end{matrix}
\begin{aligned}& BCa(37) \\ = & 101^{3} · 229^{3} · BCa(35) \end{aligned}  \begin{matrix}r_{36} = (4, 2, 3, 0, 1, 2, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 3, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1) \\ r_{37} = (6, 3, 3, 5, 1, 2, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1) \end{matrix}
\begin{aligned}& BCa(38) \\ = & 181^{3} · BCa(37) \end{aligned}  r_{38} = (2, 5, 5, 3, 1, 2, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1)
\begin{aligned}& BCa(39) \\ = & 163^{3} · BCa(38) \end{aligned}  r_{39} = (2, 2, 3, 5, 1, 2, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 3, 0, 0, 0, 1, 0, 0, 1)
\begin{aligned}& BCa(40) \\ = & 193^{3} · BCa(39) \end{aligned}  r_{40} = (2, 2, 3, 2, 1, 2, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1)
\begin{aligned}& BCa(41) \\ = & 223^{3} · BCa(40) \end{aligned}  r_{41} = (6, 3, 3, 2, 1, 2, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1)
\begin{aligned}& BCa(42) \\ = & 307^{3} · BCa(41) \end{aligned}  r_{42} = (2, 3, 3, 2, 1, 2, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 3, 1, 1)

Table 5: Magnifications of BCa(30) and their additional parameters

The exponent-sums ∑_{j=1}^{19} β_{i,j}, 1 ≤ i ≤ 30, of the set of 30 parameters

r_{i} = ∏_{j=1}^{19} p_{j}^{β_{i,j}} of BCa(30) lie in the interval [21, 30]. Set L = 22 and U = 31. Floating sieving

is performed using possible values of β_{1}, …, β_{19} that are shown in Table 7, and β_{20} = 0 or 3.

Calculate r = k^{β_{20}} · ∏_{i=1}^{19} p_{i}^{β_{i}} whenever 22 ≤ ∑_{i=1}^{20} β_{i} ≤ 31. Moreover, if −3r^{2} + k^{3} · BCa(30)/18r

is a perfect square, then this r is an additional parameter, denoted by R, of k^{3} · BCa(30). It follows that 15 splitting factors in the form of a prime or a product of two primes of BCa(30), together with their additional parameters are given in Table 6 with the base

\begin{aligned}& (p_{1}, p_{2}, …, p_{22}) \\ = & (2, 3, 5, 7, 11, 13, 17, 19, 23, 31, 37, 43, 61, 67, 73, 79, 97, 109, 139, 157, 503, 1307) \end{aligned}

We then show that upper bound on Ca(43) can be derived from a sequence of magnifications BCa(30), BCa(31), BCa(32), BCa(35), BCa(36), BCa(38), …, BCa(42). Let r_{i}, i = 1, …, 42, be the set of parameters of BCa(42) and let

\begin{aligned}Q & = & 29 · 43 · (397 · 457) · 181 · (101 · 229) · 163 · 193 · 223 · 307 \\ & = & 29 · 43 · 101 · 163 · 181 · 193 · 223 · 229 · 307 · 397 · 457, \end{aligned}

as presented in Fig. 10. Then the magnification BCa(42) = Q^{3} · BCa(30) holds. If k_{j} is a splitting factor of BCa(30), relative prime with Q, and k_{j}^{3} · BCa(30) has an additional

i splitting factors k_{i}  additional parameters R
1  487  (4,5,3,2,1,2,1,0,0,1,1,0,1,1,0,1,1,1,1,0,0,0)
2  503  (2,2,3,1,1,2,1,0,0,0,1,0,1,1,0,1,1,1,1,0,3,0)
3  2 · 607  (11,2,3,2,1,2,1,1,0,1,1,0,0,1,1,1,1,1,1,1,0,0)
4  1307  (2,2,3,2,1,2,1,1,0,0,1,0,1,0,0,1,0,1,1,0,0,3)
5  31 · 103  (2,3,3,3,3,2,1,1,0,0,1,1,1,1,1,1,1,1,1,0,0,0)
6  3559  (2,7,3,3,1,2,1,1,0,1,1,0,1,1,1,1,1,1,1,0,0,0)
7  4057  (2,2,5,2,1,2,1,0,0,0,2,0,1,1,1,1,1,1,1,1,0,0)
8  4261  (4,2,3,3,1,2,1,1,0,0,2,1,1,1,0,1,1,1,1,1,0,0)
9  4339  (4,2,3,2,1,2,1,1,0,1,1,0,1,1,1,1,1,1,1,1,0,0)
10  4957  (2,5,3,2,1,2,1,1,0,1,1,1,1,1,0,1,1,1,1,1,0,0)
11  23 · 283  (2,3,3,1,1,2,1,1,3,0,2,1,1,1,0,1,1,1,1,0,0,0)
12  6661  (2,2,3,2,1,2,1,1,0,1,2,1,1,1,1,1,1,1,1,0,0,0)
13  7489  (2,2,3,2,1,1,1,1,0,0,2,0,1,1,1,1,1,1,1,0,0,0)
14  8353  (4,3,3,3,1,2,1,1,0,0,2,0,1,1,1,1,1,1,1,0,0,0)
15  9043  (2,2,3,2,1,2,1,1,0,1,2,0,1,1,0,1,1,1,1,1,0,0)

Table 6: Splitting factors of BCa(30) and corresponding additional parameters

p_{i} 2 3 5 7 11 13 17 19 31 37 43 61 67 73 79 97 109 139 157
α_{i} 9 9 9 7 3 6 3 3 1 4 1 3 3 1 3 3 3 3 1
2 2 3 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
4 3 5 1 3 2 3 1 1 1 1 1 1 1 1 1 1 1 1
β_{i} 6 5 9 2 4 2 3 3
8 7  3

5 Table 7: Possible β_{i} for parameter r of BCa(30)

parameter R_{j}, then a set of 43 parameters of k_{j}^{3} · BCa(42) = Q^{3} · k_{j}^{3} · BCa(30) is given below: k_{j} · r_{i}, i = 1, …, 42, because of k_{j}^{3} · BCa(42) , and Q · R_{j}, because of Q^{3} · (k_{j}^{3} · BCa(30)).

Hence, the set of 43 parameters of k_{j}^{3} · BCa(42) and an upper bound of Ca(43) are obtained. Example 10. For k_{1} = 487, all parameters r_{i} with i = 1, …, 42 of BCa(42) are relative prime to 487. The magnification BCa(42) = Q^{3} · BCa(30) holds. Further let

S = 487^{3} · BCa(42), and R = 2^{4} · 3^{5} · 5^{3} · 7^{2} · 11 · 13^{2} · 17 · 31 · 37 · 61 · 67 · 79 · 97 · 109 · 139 · 487^{0}, then R is an additional parameter of 487^{3} · BCa(30). Clearly, the 42 parameters of S are 487 · r_{i}, i = 1, …, 42, because S = 487^{3} · BCa(42), and an additional parameter r_{43} = Q · R, because S = Q^{3} · (487^{3} · BCa(30)). Notice that r_{43} differs from each of 487 · r_{i}, i = 1, …, 42. Therefore, 487^{3} ·BCa(42) has 43 parameters, so 487^{3} ·BCa(42) is an upper bound on Ca(43), denoted by SCa(43).

Upper bounds on Ca(n) with 44 ≤ n ≤ 57 can be similarly derived, and are denoted by SCa(n) with 44 ≤ n ≤ 57, respectively.

Figure 10: Magnifications among BCa(30), 487^{3} × BCa(30), BCa(42), SCa(43)

Figure 11: SCa(n) with 43 ≤ n ≤ 57 derived from BCa(30) by magnifications

Figure 12: Magnifications among BCa(42), SCa(43), …, SCa(57)

The above mentioned upper bounds are summarized in Theorem 11. Among them, SCa(43) and SCa(44) were included in OEIS, August 2014.

Theorem 11.

Ca(43) ≤ SCa(43) = 487^{3} · BCa(42) Ca(44) ≤ SCa(44) = 503^{3} · SCa(43) Ca(45) ≤ SCa(45) = (2 · 607)^{3} · SCa(44) Ca(46) ≤ SCa(46) = 1307^{3} · SCa(45) Ca(47) ≤ SCa(47) = (31 · 103)^{3} · SCa(46) Ca(48) ≤ SCa(48) = 3559^{3} · SCa(47) Ca(49) ≤ SCa(49) = 4057^{3} · SCa(48) Ca(50) ≤ SCa(50) = 4261^{3} · SCa(49) Ca(51) ≤ SCa(51) = 4339^{3} · SCa(50) Ca(52) ≤ SCa(52) = 4957^{3} · SCa(51) Ca(53) ≤ SCa(53) = (23 · 283)^{3} · SCa(52) Ca(54) ≤ SCa(54) = 6661^{3} · SCa(53) Ca(55) ≤ SCa(55) = 7489^{3} · SCa(54) Ca(56) ≤ SCa(56) = 8353^{3} · SCa(55) Ca(57) ≤ SCa(57) = 9043^{3} · SCa(56)

Remark: As pointed by Boyer that 673 is a splitting factor for BCa(30) which is missing in Table 6, we confirm that

r = 2^{2} · 3^{2} · 5^{3} · 7^{3} · 11 · 13^{2} · 17 · 19 · 37 · 61 · 67^{3} · 79 · 97 · 109 · 139 · 673^{0}

is its additional parameter. As a consequence, SCa(45), …, SCa(57) in Theorem 11 can be improved easily by shifting the splitting factors properly.

5.3 Upper bounds on Ta(23) and Ta(24)

The strategy that was used in Section 5.2 can be applied to search for upper bounds on Ta(23) and Ta(24) in terms of two splitting factors of BTa(12) and a sequence of magnifications BTa(12), …, BTa(16), BTa(19), BTa(21) and BTa(22) in the order.

By floating sieving, 47, 627(= 97×491) and 91, 037(= 59×1543) are two splitting factors of BTa(12) with additional parameters

\begin{aligned}R_{1} & = & 2^{2} · 3 · 5 · 7 · 13^{2} · 17 · 19 · 73 · 79 · 97^{0} · 109 · 139 · 491^{3}, \\ R_{2} & = & 2^{2} · 3 · 5 · 7^{2} · 13^{2} · 17 · 19 · 59^{3} · 79 · 97 · 109 · 139 · 157 · 1543^{0} \end{aligned}

respectively. Let

\begin{aligned}Q′ & = & (3 · 61) · 397 · 503 · (2 · 607) · (5^{2} · 37 · 181 · 457 · 521) · 4261 · (127 · 197) · (11 · 31 · 103) \\ & = & 2 · 3 · 5^{2} · 11 · 31 · 37 · 61 · 103 · 127 · 181 · 197 · 397 · 457 · 503 · 521 · 607 · 4261. \end{aligned}

The magnification BTa(22) = Q^{′3} · BTa(12) holds. Then

(97 · 491)^{3} · BTa(22) = (97 · 491)^{3} · Q′^{3} · BTa(12)

has 22 parameters, which are 97 · 491 · r_{i}, i = 1, …, 22, where r_{1}, …, r_{22} are the parameters of BTa(22) as shown in Table 12, together with an additional parameter

\begin{aligned}Q′ · R_{1} & = & 2^{3} · 3^{2} · 5^{3} · 7 · 11 · 13^{2} · 17^{1} · 19^{1} · 31 · 37 · 61 · 73 · 79 · 103 · 109 · 127 · 139 · 181 · 197 \\ & & ·397 · 457 · 491^{3} · 503 · 521 · 607 · 4261, \end{aligned}

which differs from the previous 22 parameters, so (97 · 491)^{3} · BTa(22) has 23 parameters, and (97 · 491)^{3} · BTa(22) gives an upper bound on Ta(23), denoted by STa(23),

STa(23) = (97 · 491)^{3} · BTa(22).

Similarly,

STa(24) = (59 · 1543)^{3} · STa(23)

is an upper bound on Ta(24). Theorem 12 summarizes the above results. Both upper bounds STa(23) and STa(24) were included in OEIS, October 2014.

Theorem 12.

Ta(23) ≤ STa(23) = 97^{3} · 491^{3} · BTa(22),
Ta(24) ≤ STa(24) = 59^{3} · 1543^{3} · STa(23).

6 Acknowledgement

The author would like to thank Prof. C. Boyer for suggestive comments on the original version of this paper; in particular, for pointing out additional splitting factors for BCa(30).

7 Appendix

Figure 13: Magnifications among Ta(5), Ta(6), BTa(7), …, BTa(22), STa(23), STa(24) 20 Figure 14: Magnifications among BCa(19), BCa(21) …, BCa(42), SCa(43), …, SCa(57)

21

2 3 5 7 11 13 19 31 37 43 61 67 73 79 109 157 a_{i}
r_{1} 2 1 1 0 3 1 1 1 1 0 1 1 0 1 1 0 15
r_{2} 2 1 1 1 1 1 1 0 2 0 1 0 1 1 1 0 14
r_{3} 2 1 1 1 1 1 1 1 2 0 1 1 0 0 1 1 15
r_{4} 2 1 1 1 1 1 3 0 1 0 1 1 1 0 1 0 15
r_{5} 2 1 1 1 1 3 1 0 2 1 1 1 0 1 0 0 16
r_{6} 2 1 1 2 1 1 1 0 0 0 1 1 0 1 1 0 13
r_{7} 2 1 1 2 1 1 1 0 1 1 1 1 0 1 1 0 15
r_{8} 2 1 1 2 1 1 1 1 1 0 1 0 0 1 1 1 15
r_{9} 2 1 1 4 1 3 1 0 0 0 1 1 0 1 1 0 17
r_{10} 2 1 3 1 1 1 1 0 1 1 1 1 0 1 0 1 16
r_{11} 2 2 1 0 1 1 1 0 2 0 1 1 0 1 1 1 15
r_{12} 2 2 1 2 1 1 1 0 1 1 1 1 1 1 0 0 16
r_{13} 2 4 1 1 1 1 0 1 2 0 1 1 0 1 1 0 17
r_{14} 4 1 1 1 3 1 1 0 1 0 0 1 0 1 1 1 17
r_{15} 4 1 1 2 1 0 1 1 2 0 1 1 0 1 1 0 17
r_{16} 4 2 1 1 1 0 1 0 1 1 1 1 0 3 0 0 17
r_{17} 4 2 1 1 1 1 1 0 1 1 1 1 0 1 1 0 17
r_{18} 4 2 1 2 1 0 1 0 2 0 0 1 1 1 1 0 17
r_{19} 6 1 1 1 1 1 1 0 1 0 1 1 1 1 1 0 18
r_{20} 6 1 1 1 1 1 1 1 1 1 0 1 0 1 1 0 18
r_{21} 8 1 1 2 1 1 1 0 2 0 1 0 0 1 1 0 20
r_{22} 8 1 3 1 1 1 1 0 1 0 1 1 0 1 1 0 21

Table 8: 22 parameters for BCa(22)

2 3 5 7 11 13 17 19 31 37 43 61 67 73 79 97 109 139 157
r_{1} 2 2 3 1 3 2 1 1 1 1 0 1 1 0 1 1 1 1 0
r_{2} 2 2 3 2 1 2 1 1 0 2 0 1 0 1 1 1 1 1 0
r_{3} 2 2 3 2 1 2 1 1 1 2 0 1 1 0 0 1 1 1 1
r_{4} 2 2 3 2 1 2 1 3 0 1 0 1 1 1 0 1 1 1 0
r_{5} 2 2 3 2 1 4 1 1 0 2 1 1 1 0 1 1 0 1 0
r_{6} 2 2 3 3 1 2 1 1 0 0 0 1 1 0 1 1 1 1 0
r_{7} 2 2 3 3 1 2 1 1 0 1 1 1 1 0 1 1 1 1 0
r_{8} 2 2 3 3 1 2 1 1 1 1 0 1 0 0 1 1 1 1 1
r_{9} 2 2 3 3 1 2 3 1 0 1 0 1 1 0 1 1 1 1 0
r_{10} 2 2 3 5 1 4 1 1 0 0 0 1 1 0 1 1 1 1 0
r_{11} 2 2 5 2 1 2 1 1 0 1 1 1 1 0 1 1 0 1 1
r_{12} 2 2 5 3 1 2 1 1 1 1 1 1 1 0 1 1 1 0 0
r_{13} 2 3 3 1 1 2 1 1 0 2 0 1 1 0 1 1 1 1 1
r_{14} 2 3 3 3 1 2 1 1 0 1 0 1 1 1 1 0 1 1 1
r_{15} 2 3 3 3 1 2 1 1 0 1 1 1 1 1 1 1 0 1 0
r_{16} 2 3 9 1 1 1 1 1 0 2 1 1 0 0 1 1 1 0 0
r_{17} 2 5 3 2 1 2 1 0 1 2 0 1 1 0 1 1 1 1 0
r_{18} 2 2 3 2 1 0 1 1 0 2 0 1 1 0 1 3 1 1 0
r_{19} 4 2 3 2 3 2 1 1 0 1 0 0 1 0 1 1 1 1 1
r_{20} 4 2 3 3 1 1 1 1 1 2 0 1 1 0 1 1 1 1 0
r_{21} 4 2 5 2 1 2 1 1 0 1 1 1 1 1 1 1 1 0 0
r_{22} 4 3 3 2 1 1 1 1 0 1 1 1 1 0 3 1 0 1 0
r_{23} 4 3 3 2 1 2 1 1 0 1 1 1 1 0 1 1 1 1 0
r_{24} 4 3 3 3 1 1 1 1 0 2 0 0 1 1 1 1 1 1 0
r_{25} 6 2 3 2 1 2 1 1 0 1 0 1 1 1 1 1 1 1 0
r_{26} 6 2 3 2 1 2 1 1 1 1 1 0 1 0 1 1 1 1 0
r_{27} 6 7 3 2 1 2 1 1 1 1 0 0 1 0 1 1 1 1 0
r_{28} 8 2 3 0 1 4 1 1 0 1 0 1 1 1 1 0 1 1 0
r_{29} 8 2 3 3 1 2 1 1 0 2 0 1 0 0 1 1 1 1 0
r_{30} 8 2 5 2 1 2 1 1 0 1 0 1 1 0 1 1 1 1 0

Table 9: 30 parameters for BCa(30)

2 3 5 7 11 13 17 19 29 31 37 43 61 67 73 79 97 101 109 139 157 163 181 193 223 229 307 397 457
r_{1} 2 2 3 1 3 2 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1
r_{2} 2 2 3 2 1 2 1 1 1 0 2 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1
r_{3} 2 2 3 2 1 2 1 1 1 1 2 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1
r_{4} 2 2 3 2 1 2 1 3 1 0 1 1 1 1 1 0 1 1 1 1 0 1 1 1 1 1 1 1 1
r_{5} 2 2 3 2 1 4 1 1 1 0 2 2 1 1 0 1 1 1 0 1 0 1 1 1 1 1 1 1 1
r_{6} 2 2 3 3 1 2 1 1 1 0 0 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1
r_{7} 2 2 3 3 1 2 1 1 1 0 1 2 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1
r_{8} 2 2 3 3 1 2 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1
r_{9} 2 2 3 3 1 2 3 1 1 0 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1
r_{10} 2 2 3 5 1 4 1 1 1 0 0 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1
r_{11} 2 2 5 2 1 2 1 1 1 0 1 2 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1
r_{12} 2 2 5 3 1 2 1 1 1 1 1 2 1 1 0 1 1 1 1 0 0 1 1 1 1 1 1 1 1
r_{13} 2 3 3 1 1 2 1 1 1 0 2 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1
r_{14} 2 3 3 3 1 2 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1
r_{15} 2 3 3 3 1 2 1 1 1 0 1 2 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 1 1
r_{16} 2 3 9 1 1 1 1 1 1 0 2 2 1 0 0 1 1 1 1 0 0 1 1 1 1 1 1 1 1
r_{17} 2 5 3 2 1 2 1 0 1 1 2 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1
r_{18} 4 2 3 2 3 2 1 1 1 0 1 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1
r_{19} 4 2 3 3 1 1 1 1 1 1 2 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1
r_{20} 4 2 5 2 1 2 1 1 1 0 1 2 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1
r_{21} 4 3 3 2 1 1 1 1 1 0 1 2 1 1 0 3 1 1 0 1 0 1 1 1 1 1 1 1 1
r_{22} 4 3 3 2 1 2 1 1 1 0 1 2 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1

Table 10: 42 parameters for BCa(42) (part a)

2 3 5 7 11 13 17 19 29 31 37 43 61 67 73 79 97 101 109 139 157 163 181 193 223 229 307 397 457
r_{23} 4 3 3 3 1 1 1 1 1 0 2 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1
r_{24} 6 2 3 2 1 2 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1
r_{25} 6 2 3 2 1 2 1 1 1 1 1 2 0 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1
r_{26} 6 7 3 2 1 2 1 1 1 1 1 1 0 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1
r_{27} 8 2 3 0 1 4 1 1 1 0 1 1 1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1
r_{28} 8 2 3 3 1 2 1 1 1 0 2 1 1 0 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1
r_{29} 8 2 5 2 1 2 1 1 1 0 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1
r_{30} 2 2 3 2 1 0 1 1 1 0 2 1 1 1 0 1 3 1 1 1 0 1 1 1 1 1 1 1 1
r_{31} 2 2 3 2 1 2 1 1 3 0 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1
r_{32} 2 3 5 2 1 2 1 1 3 0 1 0 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1
r_{33} 2 2 9 2 1 2 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0
r_{34} 2 5 3 1 1 2 1 1 1 1 1 2 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1
r_{35} 4 2 7 3 1 2 1 1 1 1 2 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 0 0
r_{36} 4 2 3 0 1 2 1 1 1 0 1 1 1 1 0 1 1 3 1 1 1 1 1 1 1 0 1 1 1
r_{37} 6 3 3 5 1 2 1 1 1 0 2 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 1 0 1
r_{38} 2 5 5 3 1 2 1 1 1 1 0 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1
r_{39} 2 2 3 5 1 2 1 0 1 0 1 1 1 1 0 1 1 1 1 1 1 3 0 1 1 1 1 0 1
r_{40} 2 2 3 2 1 2 1 1 1 0 1 2 1 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1
r_{41} 6 3 3 2 1 2 1 1 1 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 0 1 1 1 1
r_{42} 2 3 3 2 1 2 1 1 1 0 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 0 3 1 1

Table 11: 42 parameters for BCa(42) (part b)

2 3 5 7 11 13 17 19 31 37 43 61 73 79 97 103 109 127 139 157 181 197 397 457 503 521 607 4261
r_{1} 3 2 9 1 1 2 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 1 0 1 1 1 1
r_{2} 3 2 3 1 1 4 1 1 1 2 1 1 0 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1
r_{3} 3 3 7 1 1 1 1 1 1 0 0 1 0 1 1 1 1 1 1 0 1 1 1 0 1 3 1 1
r_{4} 3 3 3 0 1 2 1 1 1 2 1 1 0 1 1 1 0 0 1 0 1 3 1 1 1 1 1 1
r_{5} 3 3 3 2 3 2 1 1 0 1 1 1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1
r_{6} 3 2 3 1 1 0 1 1 1 2 0 1 0 1 3 1 1 1 1 0 1 1 1 1 1 1 1 1
r_{7} 3 2 3 0 1 2 1 0 1 1 0 1 0 1 1 1 1 1 1 0 1 1 1 1 3 1 1 1
r_{8} 3 2 3 1 1 2 1 1 2 2 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1
r_{9} 3 2 5 2 1 2 1 1 2 1 1 1 0 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1
r_{10} 3 3 3 0 1 2 1 1 1 2 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
r_{11} 3 2 3 2 1 2 3 1 1 1 0 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1
r_{12} 3 3 3 2 1 2 1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1
r_{13} 3 5 3 0 1 2 1 1 2 1 1 1 0 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1
r_{14} 3 5 3 1 1 2 1 0 2 2 0 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1
r_{15} 3 5 5 2 1 2 1 1 2 0 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1
r_{16} 5 3 3 1 1 1 1 1 1 1 1 1 0 3 1 1 0 1 1 0 1 1 1 1 1 1 1 1
r_{17} 5 2 5 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1
r_{18} 5 2 3 2 1 1 1 1 2 2 0 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1
r_{19} 5 2 3 2 1 2 1 1 1 2 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0
r_{20} 7 7 3 1 1 2 1 1 2 1 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1
r_{21} 9 2 5 1 1 2 1 1 1 1 0 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1
r_{22} 11 2 3 1 1 2 1 1 2 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1

Table 12: 22 parameters for BTa(22)

References

[1] C. Boyer, New upper bounds for Taxicab and Cabtaxi numbers. J. Integer Sequences, 11 (2008), Article 08.1.6.

[2] C. Boyer, New upper bounds for Taxicab and Cabtaxi numbers. Available at http://www.christianboyer.com/taxicab/.

[3] C. S. Calude, E. Calude and M. J. Dinneen, What is the value of Taxicab(6)?, J. Univ. Comput. Sci. 9 (2003), 1196–1203.

[4] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth edition, Oxford University Press, 1980.

[5] D. W. Wilson, The fifth Taxicab number is 48988659276962496, J. Integer Sequences, 2 (1999), Article 99.1.9.

[6] N. J. A. Sloane, On-Line Encyclopedia of Integer Sequences, http://oeis.org.

2010 Mathematics Subject Classification: Primary 11D25.

Keywords: Taxicab number, Cabtaxi number, magnification, splitting factor, sieving, floating sieving.

(Concerned with sequences \underline{A011541} and \underline{A047696}.)

Received April 29 2015; revised versions received November 17 2015; January 20 2016; March 28 2016. Published in Journal of Integer Sequences, April 21 2016.

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