A Special Class of Semiprimes

A well-known factorisation involves the sum of two cubes:$$ a^3 + b^3=(a+b)(a^2-ab+b^2)$$Assuming that both \(a\) and \(b\) are positive integers, then this sum of two cubes is a semiprime if \(a+b\) and \(a^2-ab+b^2\) are both prime. It's interesting to explore when this happens.

Let's begin with \(b=1\) and restrict ourselves to semiprimes between 4 and 4000. There are only seven that qualify and they are:$$9, 65, 217, 4097, 5833, 10649, 21953 $$These numbers form the initial terms of OEIS A237040. Their factorisations are as follows:$$ \begin{align} 9 &= 3^2 = 2^3 + 1^3\\65 &= 5 \times 13 = 4^3 + 1^3\\217 &= 7 \times 31 = 6^3 + 1^3\\4097 &= 17 \times 241 = 16^3 + 1^3\\5833 &= 19 \times 307 = 18^3 + 1^3\\10649 &= 23 \times 463 = 22^3 + 1^3\\21953 &= 29 \times 757 = 28^3 + 1^3 \end{align}$$The next is \(b=8\) which yields six numbers:$$35, 133, 737, 1339, 3383, 24397$$These numbers factorise as follows:$$ \begin{align} 35 &= 5 \times 7 = 3^3 + 2^3\\133 &= 7 \times 19 = 5^3 + 2^3\\737 &= 11 \times 67 = 9^3 + 2^3\\1339 &= 13 \times 103 = 11 ^3 + 2^3\\3383 &= 17 \times 199 = 15^3 + 2^3\\24397 &= 31 \times 787 = 29^3 + 2^3 \end{align} $$These terms are not listed in the OEIS database. Next we'll let \(b=27\). This yields eight numbers and we see that 35 makes a reappearance as the 2 and 3 swap places. The numbers are:$$35, 91, 1027, 2771, 8027, 17603, 21979, 39331 $$These numbers are not listed in the OEIS database and their factorisations are:$$ \begin{align} 35 &= 5 \times 7 = 2^3 + 3^3\\91 &= 7 \times 13 = 4^3 + 3^3\\1027 &= 13 \times 79 = 10^3 + 3^3\\2771 &= 17 \times 163 = 14^3 + 3^3\\8027 &= 23 \times 349 = 20^3 + 3^3\\17603 &= 29 \times 607 = 26^3 + 3^3\\21979 &= 31 \times 709 = 28^3 + 3^3\\39331 &= 37 \times 1063 = 34^3 + 3^3 \end{align} $$Here is a permalink to an algorithm that allows further exploration using additional values of \(b\). A further avenue for investigation would be to consider semiprimes that are a difference of two cubes since we know that:$$ a^3 - b^3=(a-b)(a^2+ab+b^2)$$Using \(b=-1\), we find that the following semiprimes qualify:$$26, 215, 511, 1727, 2743, 7999, 13823$$These numbers form the initial members of OEIS A242262. Their factorisations are as follows:$$ \begin{align} 26 &= 2 \times 13 = 3^3 -1^3\\215 &= 5 \times 43 = 6^3 - 1^3\\511 &= 7 \times 73 = 8^3 - 1^3\\1727 &= 11 \times 157 = 12^3 - 1^3\\2743 &= 13 \times 211 = 14^3 - 1^3\\7999 &= 19 \times 421 = 20^3-1^3\\13823 &= 23 \times 601 = 24^3-1^3 \end{align} $$

Special Numbers, Special Dates

On the 26th January 2023, I posted about Turning Dates Into Numbers and made the observation that the 10th February 2023 was a special day because the number associated with this date (20230210) had the interesting property that it was the only date in the year that produced a number that had seven distinct prime factors:

20230210 = 2 x 5 x 7 x 11 x 13 x 43 x 47

It turns out that this number has another interesting property in that the sum of its digits equals the last two digits (representing the day of the month). In fact, it begins a run of ten consecutive numbers that end with 20230219 (the 19th February 2023) ... permalink:

20230210, 20230211, 20230212, 20230213, 20230214, 20230215, 20230216, 20230217, 20230218, 20230219

Another run of ten similar numbers occurs in November, starting on the 10th and ending on the 19th.

20231110, 20231111, 20231112, 20231113, 20231114, 20231115, 20231116, 20231117, 20231118, 20231119

For many, many years to come, the sum of the digits of the numbers associated with the dates will always contain two digits. Today is the 15th February 2023 and so we are into the second half of the first-mentioned run.

This number can be expressed as a sum of three cubes
See Figure 2

Similar runs will occur every year but the dates will differ naturally. For example, in 2024, the first run of numbers will begin on 11th January 2024 with 20240111 and end on 19th January 2024 with 20240119. 

In my previous post Turning Dates Into Numbers I looked at what numbers were prime, semiprime and so on. In this post I'll turn my attention to cubes and see what numbers can be expressed as the sum of two cubes and as a sum of three cubes. 

It turns out that there is only one number in the entire year that can be expressed as a sum of two cubes and that is \( 20231020 = 69^3 + 271^3\). This corresponds to the 20th October 2023.

The sums of three cubes are more common. The complete list is shown in Figure 2, arranged in chronological order with 20th January being the first date and 26th December the last.

Figure 2: permalink

It can be noted that some dates produce numbers that can be expressed as sums of three cubes in more than one way. Specifically, 20230426 (two ways), 20230523 (two ways), 20230722 (two ways), 20230904 (two ways), 20230911 (three ways), 20231028 (two ways) and 20231226 (two ways). Thus the 11th September (the famous 911 in United States date format) produces the only number (20230911) that can be expressed as a sum of three cubes in three ways.

The Original Taxi Cab Number in a New Light

Today I turned 25829 days old and, amongst the number's many different properties, one in particular caught my eye. The property was that it is a member of OEIS A262054: Euler pseudoprimes to base 7: composite integers such that:$$ |7^{(n-1)/2}| \equiv 1 \pmod {n}$$Now there's no sign of 1729, the original taxi cab number, but we'll get there. Firstly however, how did 1729 earn its sobriquet? Here an excerpt from Wikipedia:

The name is derived from a conversation in about 1919 involving mathematicians G. H. Hardy and Srinivasa Ramanujan. As told by Hardy: 

"I remember once going to see him [Ramanujan] when he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to be rather a dull one, and that I hoped it was not an unfavourable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two [positive] cubes in two different ways."

Figure 1

The two different ways are: \(1^3 + 12^3\) and \(9^3 + 10^3\).

I won't go further into taxicab numbers here as the Wikipedia article explains things well enough. What I want to do is cast a new light on 1729, the number that Hardy originally thought was a rather dull number. The light I'm casting comes from the Euler pseudoprimes. 

I've already discussed pseudoprimes in two earlier posts: Fermat Pseudoprimes and Carmichael Numbers. I did make make passing mention of Euler pseudoprimes in the former post but didn't go into the matter further. Figure 1 shows a screenshot of part of what Wikipedia has to say about Euler pseudoprimes. 

The excerpt in Figure 1 concludes with the observation that:

The absolute Euler pseudoprimes are a subset of the absolute Fermat pseudoprimes, or Carmichael numbers, and the smallest absolute Euler pseudoprime is 1729 = 7×13×19.

It's surprising then that a mathematician of Hardy's calibre should not have recognised 1729 as having quite some claim to fame. In fact, the Online Encyclopaedia of Integer Sequences (OEIS) has 794 entries for the number so it is far from dull. Let's consider some of the other entries for 1729 in the OEIS. 

One entry is not surprising when the factorisation of 1729 is considered and the factors are arranged in descending order: 19 x 13 x 7. Let's add a 1 to give 19 x 13 x 7 x 1. The numbers 19, 13, 7 and 1 form an arithmetic sequence and this shows 1729 to be a so-called sextuple factorial. This can be written as 19!!!!!! or 19!6.

1729 counts the ways that a 2 x 2 matrix can be populated with integers from -7 to +7 in such a way that every matrix is singular (that is has a determinant of zero). It thus forms part of OEIS A209981. 

Figure 2: 35 points in a body-centered cubic lattice, 
forming two cubical layers around a central point

In the realm of figurate numbers, 1729 is a centred cube number. These are numbers of the form:\((n+1)^3+n^3\) and of course 1729 can be written as \(10^3+9^3\). Figure 2 shows the example of the centred cube number 35 and Wikipedia explains:

A centred cube number is a centred figurate number that counts the number of points in a three-dimensional pattern formed by a point surrounded by concentric cubical layers of points, with \(i^2\) points on the square faces of the \(i\)-th layer. Equivalently, it is the number of points in a body-centred cubic pattern within a cube that has \(n + 1\) points along each of its edges. 

The first few centred cube numbers are:

1, 9, 35, 91, 189, 341, 559, 855, 1241, 1729, 2331, 3059, 3925, 4941, 6119, 7471, 9009, ... (sequence A005898 in the OEIS).

1729 is also an heptagonal number, a 12-gonal or dodecagonal number, a 24-gonal or icosotetragonal number but that's probably enough for the moment.