27000

A milestone reached. My diurnal age today is 27000 which is the cube of the primorial 30 = 2 x 3 x 5, a sphenic number and thus representable as the sphenic brick shown in Figure 1.

Figure 1

This brick with its volume of 30 cubic units is symbolic of a month of 30 days. If these bricks are positioned to form a base of 15 blocks x 2 by 10 blocks x 3 and then stacked 6 high, a cube of side 30 units is formed. These 900 blocks can be seen as symbolic of 900 months of life.

Of course there being 12 months in a year, these 900 months are equivalent to 75 years and I'm yet to reach my 74th birthday. This is because 12 x 30 = 360 and there are on average 365.425 days in a year. There is an excess of 5.425 which, over the course of 73.92 years, amounts to 401 days. Thus the discrepancy which, in keeping with the notion of bricks, could be envisaged as the mortar between the bricks.

Suffice to say that these bricks would form an impressive structure, although they are not in the same proportions as the standard brick shown in Figure 2. The mortar between bricks normally has a width of 10 mm.

Figure 2

I asked ChatGPT the following question:

I'm looking for interesting things to say about the number 27000. The things of interest could be mathematical but not necessarily so. Anything of interest whether it be historical, scientific, numerological or whatever is acceptable.

This was its response:

Sure, here are some interesting things to say about the number 27000:

• 27000 is a perfect cube number. It can be written as 30³, which means that it is the product of three 30s multiplied together.
  
  
• In geometry, 27000 is the number of degrees in a 150-sided polygon (also called a hecatoicosagon).
  
  
• The number 27000 has a total of 27 factors, including 1 and itself. This means that it has a lot of divisors, which makes it a useful number in many mathematical contexts.
  
  
• In Hinduism, there is a sacred text called the Mahabharata, which is said to have around 27,000 verses.
  
  
• In astronomy, 27000 is roughly the number of light years that the Solar System travels around the center of the Milky Way galaxy in one orbit.
  
  
• The number 27000 is often used as a benchmark in statistical analysis, particularly in studies involving large datasets. This is because it represents a sufficiently large sample size to generate meaningful results.
  
  
• In the sport of sumo wrestling, 27,000 is the minimum number of spectators that must be in attendance for a tournament to be considered a "sell-out."
  
  
• The year 27000 AD is often cited as a possible time frame for the next Ice Age, based on current climate models.
  
  
• The binary representation of 27000 is 110100100101000, which consists of 15 digits. In binary code, this number can be used to represent various commands or functions.
  
  
• According to numerology, the number 27000 represents balance, harmony, and spiritual enlightenment. It is said to be a powerful number for meditation and personal growth.

Referring to the above points, 27000 does not have 27 factors. It has three distinct prime factors, nine prime factors with multiplicity and 64 divisors. Similarly the binary representation of 27000 is wrong. Correct is 110100101111000, although the stated number of binary digits is correct. One needs to remember to take anything ChatGPT says with a grain of salt when it comes to mathematics. 

I thought I'd pick ChatGPT's brain further and asked:

What examples are there from literature concerning the number 27000?

• There are several examples in literature that mention the number 27000. Here are a few:
  
  
• In the Hindu epic Mahabharata, there is a reference to an army of 27,000 soldiers that was sent by the Kauravas to capture the Pandavas.
  
  
• In Dante Alighieri's "Divine Comedy," the number 27 is significant as it represents the nine levels of heaven, each with three spheres, making a total of 27.
  
  
• In the book "The Bible Code," written by Michael Drosnin, the number 27000 is said to appear in a code in the Book of Exodus that predicts a future nuclear war.
  
  
• In J.R.R. Tolkien's "The Lord of the Rings," the number 27000 is mentioned in the context of the army of Rohirrim, which consists of 18,000 riders and 9,000 infantry.
  
  
• In the book "The Magic Mountain" by Thomas Mann, the number 27000 is used to describe the number of days that the main character, Hans Castorp, spends in a tuberculosis sanatorium.
  
  
• In the book "The Teachings of Don Juan" by Carlos Castaneda, the number 27000 is used to describe the number of days that the author spent learning from a Yaqui Indian shaman.
  
  
• Overall, the number 27000 appears to have different symbolic meanings in different works of literature, ranging from military might to spiritual significance to the passage of time.
  

Some of these statistics seem unlikely. The main character in Thomas Mann's book would have spent almost 74 years in the sanatorium and likewise for the Yaqui Indian apprenticeship. I'm not going to check which facts are correct and which are not.

I last celebrated a cubic number day when I turned 24389 (29 cubed) days old on Monday, the 11th January 2016. I'll next celebrate (if I'm around) on Saturday, 26th October 26th 2030 when I turn 29791 (31 cubed) days old.

Concatenating Prime Factors

I noticed that the number associated with my diurnal age today, 26989, is a semiprime that factors to 137 x 197. Concatenating these two factors, we get 137197 which is prime but if we concatenate 197 x 137 we get 197137 which is also prime. Semiprimes with this property constitute OEIS A330441:

A330441

Semiprimes \(p \times q\) such that the concatenations of \(p\) and \(q\) in both orders are prime.

The initial members of this sequence are:

21, 33, 51, 93, 111, 133, 177, 201, 219, 247, 253, 327, 411, 427, 573, 589, 679, 687, 763, 793, 813, 889, 993, 1077, 1081, 1119, 1243, 1339, 1347, 1401, 1411, 1497, 1501, 1603, 1623, 1651, 1671, 1821, 1839, 1843, 1851, 1981, 2019, 2047, 2059, 2103, 2157, 2199, 2217, 2469, 2479, 2629, 2761, 2787

Given that 26989 is an emirpimes (since its reverse is a distinct semiprime: 98962 = 2 x 49481), I thought I'd investigate how many emirpimes are in this sequence. It turns out that in the range up to one million there are the following:

3099, 9903, 10519, 11707, 13993, 16387, 18247, 19039, 30607, 32667, 36367, 38697, 39487, 39931, 70603, 70711, 72247, 73099, 74227, 74281, 74289, 76029, 76363, 76623, 78361, 78493, 78619, 79683, 91501, 91687, 92067, 93091, 98247, 99037, 100437, 101317, 101899, 104529, 108181, 108789, 120553, 126771, 133243, 134797, 137671, 144523, 147061, 149449, 159427, 160741, 168117, 176731, 176767, 177621, 181801, 184033, 197097, 199879, 312817, 322489, 325441, 328459, 330397, 330481, 331783, 337297, 337897, 338977, 342331, 345493, 350569, 355021, 357393, 365863, 368563, 386197, 387133, 393753, 394543, 711861, 713101, 716779, 717469, 718213, 724951, 734001, 767671, 779833, 790791, 791683, 792733, 793033, 797431, 798733, 925401, 944941, 951679, 954823, 964699, 964717, 965053, 976159, 977617, 978991, 984223, 987801, 996469, 998101

Let's check one of the entries, say the last. 998101 should have its reversal present in the list, 101899, which it does and both should have factors that can be concatenated in either way to produce primes:

998101 = 13 x 76777 with 1376777 and 7677713 both prime

101899 =  7 x 14557 with 714557 and 145577 both prime

This got me thinking about sphenic numbers, that is numbers of the form \(p \times q \times r\) where \(p\), \(q\) and \(r\) are distinct prime factors, with the property that any concatenation of these prime factors produces a prime. In other words:

• p | q | r is prime
• p | r | q is prime
• q | p | r is prime
• q | r | p is prime
• r | p | q is prime
• r | q | p is prime

In the range up to one million, there are 15 such numbers (permalink). They are:

3311, 27181, 32153, 41237, 53977, 86507, 110971, 125069, 208579, 256413, 500981, 543337, 853811, 901949, 964481

Here is the breakdown for each number:

3311 = 7 x 11 x 43 --> 71143 74311 11743 11437 43117 43711

27181 = 7 x 11 x 353 --> 711353 735311 117353 113537 353117 353711

32153 = 11 x 37 x 79 --> 113779 117937 371179 377911 793711 791137

41237 = 7 x 43 x 137 --> 743137 713743 437137 431377 137437 137743

53977 = 7 x 11 x 701 --> 711701 770111 117701 117017 701117 701711

86507 = 19 x 29 x 157 --> 1929157 1915729 2919157 2915719 1572919 1571929

110971 = 7 x 83 x 191 --> 783191 719183 837191 831917 191837 191783

125069 = 7 x 17 x 1051 --> 7171051 7105117 1771051 1710517 1051177 1051717

208579 = 7 x 83 x 359 --> 783359 735983 837359 833597 359837 359783

256413 = 3 x 127 x 673 --> 3127673 3673127 1273673 1276733 6731273 6733127

500981 = 13 x 89 x 433 --> 1389433 1343389 8913433 8943313 4338913 4331389

543337 = 17 x 31 x 1031 --> 17311031 17103131 31171031 31103117 10313117 10311731

853811 = 7 x 283 x 431 --> 7283431 7431283 2837431 2834317 4312837 4317283

901949 = 19 x 37 x 1283 --> 19371283 19128337 37191283 37128319 12833719 12831937

964481 = 7 x 211 x 653 --> 7211653 7653211 2117653 2116537 6532117 6537211

Note how 9 of the 15 numbers have 7 as a factor. Even though the reversals of some of the above numbers are also sphenic numbers (35123, 179011, 975802, 189005, 949109), none of them have the concatenating prime factors property. The first member of any sequence is special and so the takeaway here might be the following:

3099 is the smallest of a very special class of semiprimes because its distinct prime factors can be concatenated in either order to make primes and its reversal, 9903, is also a semiprime with the same property.

3311 is the smallest of a very special class of sphenic numbers because its prime factors can be concatenated in all six ways to make primes. 

While this topic falls squarely into the category of recreational mathematics, it's nonetheless fun to investigate and it helps to challenge my limited SageMath programming capabilities.