Beyond 23

The algorithm that I got Gemini to create and that I used in the post 23 Revisited was the result of the following prompt:

Write a program in SageMath that determines all the composite, positive integers in a given range that contain the digits 2 and 3 in ascending order from left to right (but not necessarily consecutively) and whose prime factors also all contain the digits 2 and 3 in ascending order but not necessarily consecutively. The default range can be from 1 to 40000. The output should a table showing number and factors followed by a comma-separated list of the qualifying numbers.

Instead of the prime 23, other prime numbers can be investigated. Composite numbers could be investigated too but this would lead to rather large numbers and so I'll restrict this investigation to prime numbers but I'll increase the range from 1 to one million.

Here are the results for 11 (permalink):

Number       | Prime Factors

------------------------------

110231       | 11, 911

111221       | 11, 10111

112321       | 11, 10211

117821       | 11, 10711

128711       | 11, 11701

129811       | 11, 11801

131153       | 11, 11923

131197       | 11, 11927

171121       | 211, 811

211321       | 11, 19211

231121       | 11, 21011

232111       | 11, 21101

261143       | 113, 2311

351119       | 311, 1129

451121       | 11, 41011

511621       | 11, 46511

781121       | 11, 71011

892111       | 11, 81101

Here are the results for 13 (permalink):

Number       | Prime Factors

------------------------------

13169         | 13, 1013

16913         | 13, 1301

22139         | 13, 131

113269       | 13, 8713

131729       | 13, 10133

131807       | 13, 10139

132703       | 131, 1013

134069       | 13, 10313

136669       | 13, 10513

137069       | 113, 1213

137969       | 13, 10613

138781       | 137, 1013

147013       | 113, 1301

161369       | 13, 12413

169013       | 13, 13001

171301       | 13, 13177

171379       | 13, 13183

172133       | 13, 13241

180713       | 13, 13901

190013       | 139, 1367

211303       | 131, 1613

313469       | 13, 24113

313729       | 13, 24133

313781       | 13, 24137

360139       | 13, 2131

407213       | 313, 1301

413473       | 313, 1321

421369       | 13, 32413

661369       | 313, 2113

797513       | 613, 1301

813397       | 13, 4813

813451       | 613, 1327

Here are the results for 17 (permalink):

Number       | Prime Factors

------------------------------

51731         | 17, 179

117589       | 17, 6917

173009       | 17, 10177

178313       | 17, 617

198917       | 17, 11701

217843       | 179, 1217

291703       | 17, 17159

295817       | 17, 17401

301733       | 17, 17749

308417       | 179, 1723

351713       | 17, 1217

368917       | 17, 21701

506617       | 17, 1753

517021       | 17, 1789

541753       | 317, 1709

717043       | 17, 42179

Here, and finally, are the results for 19 (permalink):

Number       | Prime Factors

------------------------------

19361         | 19, 1019

36119         | 19, 1901

119467       | 193, 619

121943       | 197, 619

193667       | 19, 10193

194629       | 191, 1019

196667       | 193, 1019

251929       | 191, 1319

361019       | 19, 19001

361969       | 19, 19051

366719       | 19, 19301

370519       | 19, 19501

371963       | 19, 19577

376219       | 19, 19801

381947       | 193, 1979

519329       | 191, 2719

601967       | 193, 3119

714419       | 19, 1979

719473       | 19, 1993

752419       | 19, 199

796519       | 419, 1901

I won't continue with primes larger than 23 because really what I want is an algorithm that will identify a sequence of digits that form a prime within a number and then check that the same digit sequence occurs with each of its prime factors. The number being checked is thus composite. Here is a permalink to the program that I've now included in my daily number analysis.

Details of the Einstein and Spectre Monotiles

Having read an article in Quanta about the Einstein Tile, I asked Gemini to do some deep research into the topic. It produced a detailed report without a single diagram which was ridiculous and highlights the deficiencies of this sort of report. Lesson learned. Even the Quanta article is sorely lacking in visual support. 

I opted instead to ask Gemini for a detailed diagram of the original Einstein tile. Figure 1 shows what it came up with.

Figure 1

The tile is made up of eight "deltoidal trihexagonal kites", one of which is shown on far right of Figure 1. However, to tile the plane the reflection of the hat is required as shown on the bottom right of Figure 1. The "Spectre" or "Vampire Einstein" monotile fixes this problem (see Figure 2).

Figure 2

I made the following observation:

The Spectre tile is clearly not a polygon. Wasn't the aim to fill 2-D space with a suitable polygon. Does this mean that there is no vampire monotile that is a polygon?

Here was Gemini's response:

Does a truly "polygonal" vampire monotile exist?

Actually, yes! Although curves are the easiest way to visualize the "Spectre's" chiral enforcement, you can replace those curves with polygonal (jagged) features.

If instead of smooth curves you use jagged, puzzle-like zig-zagged sub-edges (effectively turning the 14-sided tile into a vastly more complex polygon with many tiny, straight-line notches), you can strictly forbid reflection without using a single curve. However, doing so requires creating a heavily complex, jagged polygon with dozens of vertices, which makes it much harder to print, analyze, or illustrate clearly on a grid.

For simplicity, mathematicians and illustrators typically show the smooth-curved Spectre as the definitive "vampire monotile" because it naturally forces its own chirality without the visual clutter of dozens of tiny polygonal zig-zags.

23 Revisited

The number 23 never ceases to be of interest. Back in November of 2020 I made a post titled Twenty Three in which I looked at some of the properties of this prime number. These included the following:

• 23 and 239 are the only integers requiring nine positive cubes for their representation
• 23 has a reciprocal with a period of 22
• 23 features in and is central to the birthday paradox
• 23 is a Sophie Germain and safe prime
• 23 forms part of the first Cunningham chain 2, 5, 11, 23, 47
• 23 is the smallest odd prime that is not a twin prime
• 23 is the second Woodell prime (the first is 7)
• 23 is a factorial prime equal to 4! -1
• 23 is the second Smarandache–Wellin prime (the first is 2)
• 23 divides 874, the sum of the first 23 primes
• 23 is the length of the repunit prime 11111111111111111111111
• 23 is the only prime \(p\) such that \(p!\) is \(p\) digits long
• 23 is conjectured to be the only pointer prime that does not contain a 1

The number associated with my diurnal age today, 28313, caught my attention because the number 23 stood out:$$ \textcolor{red}{2}8\textcolor{red}{3}13 = \textcolor{red}{23} \times 1\textcolor{red}{23}1 $$I wondered how common this sort of pattern was so I asked Gemini to write a program to find out. This was the prompt:

Write a program in SageMath that determines all the composite, positive integers in a given range that contain the digits 2 and 3 in ascending order from left to right (but not necessarily consecutively) and whose prime factors also all contain the digits 2 and 3 in ascending order but not necessarily consecutively. The default range can be from 1 to 40000. The output should a table showing number and factors followed by a comma-separated list of the qualifying numbers.

Here is a permalink to the program that returned only one result: 28313. Extending the range to 100,000 gives us the following numbers:

Number     | Prime Factors
--------------------------------------------------
28313      | 23, 1231
52739      | 23, 2293
58213      | 23, 2531
62399      | 23, 2713
62813      | 23, 2731

It is only when we extend the range to one million that we find numbers such that the 23's  are all formed from consecutive digits. I modified the prompt and got Gemini to create a new program:

Write a program in SageMath that determines all the composite, positive integers in a given range that contain the digits 2 and 3 consecutively in ascending order from left to right and whose prime factors also all contain the digits 2 and 3 consecutively in ascending order. The default range can be from 1 to one million. The output should be a table showing number and factors followed by a comma-separated list of the qualifying numbers (permalink).

Here are the results when the program is run: 

123257, 235129, 282923, 286823, 442313, 523381, 533623, 542363, 547423, 552329, 555239, 563323, 672313, 902359, 985823

Number       | Prime Factors

--------------------------------------------------

123257       | 23, 233

235129       | 23, 10223

282923       | 23, 12301

286823       | 233, 1231

442313       | 23, 19231

523381       | 223, 2347

533623       | 23, 23201

542363       | 23, 23581

547423       | 23, 23801

552329       | 239, 2311

555239       | 233, 2383

563323       | 239, 2357

672313       | 23, 29231

902359       | 23, 39233

985823       | 233, 4231

These numbers of course are far beyond what my diurnal age will ever reach but we can identify 29231 as a special number because it contains the digit sequence 23 and when multiplied by 23 produces a new number, 672313, that also contains the digit sequence 23. I may reach this number but it's doubtful whether I'll reach the next (39233) that has the same property. 

When Dates and Days Align

I put this little problem to Gemini:

A person is born on April 3rd 1949. The day that he is born is reckoned as Day 0 and for every subsequent day 1 is added and this count becomes his diurnal age. Let the number associated with any date be determined as DDMYY where DD is the number associated with the month(1 to a maximum of 31), M is the month (from 1 to 9) and YY is the last two digits of the current year. Thus a date of 14th September 2026 would become 14926. Is there any date, using this system, where the diurnal age equals the date?

Here was its response:

September 23, 2014, is the sole date where the diurnal age mathematically equates to its DDMYY numerical representation. 

That is interesting and so, on the 23rd of September 2014, I was 23914 days old. I got Gemini to write a program that would accept any birthdate as input using this prompt:

Can you write a program in SageMath that will accept any birthdate as input and from there calculate if there is a date where the diurnal age equals the date number using DDMYY format. Let’s realistically limit the person’s possible age to less than 100.  

Here is a permalink to the program and the result for the following birthdate:

• 23rd August 1964:
  Match found: 05 February 1979 | Diurnal Age / Date Number: 5279

However, I realised that I could be less restrictive with the date format and modified the prompt as follows:

I’d like to modify the program so the acceptable date format can be DMMYY, DDMYY or DMYY. This ensures that the resultant number derived from the date is always less than six digits. Everything else remains the same.

Here is a permalink to the modified program which produces the same results for the two dates examined above. Here are some additional dates:

• 23rd November 1980:
  Match found: 06 July 1999 | Diurnal Age: 6799 | Date Format: 6799
  Match found: 16 January 2025 | Diurnal Age: 16125 | Date Format: 16125
  
  
• 6th October 2002:
  Match found: 10 September 2032 | Diurnal Age: 10932 | Date Format: 10932

So just a little curiosity. Not all birthdates yield a result within the span of 100 years. You could ask the person born on the 6th October 2002: how many days old will you be on the 10th September 2032? Don't know? Here's a way to find out:

• 10th \( \rightarrow\) 10
• September \( \rightarrow\) 9
• 2032 \( \rightarrow\)  32
• Concatenation \( \rightarrow\) 10932 \( \rightarrow\) your diurnal age!

The Wolfram Alpha Logo

 I used Wolfram Alpha today to determine when I would be 77.77777 years old. 

Yep, 9:21pm on January 11th 2027, midway between my father's birthday on January 9th and my brother Tony's birthday on January 13th. More about that later but, perhaps because of my recent foray into sketching, I happened to notice the logo shown below that appears on the web page. Oddly, I failed to notice the red stellated polyhedra that appears in the image above, maybe because it's quite small.

I asked Gemini the following:

What can you tell me about the Wolfram Alpha logo. It’s an interesting shape. Does it have any history or special significance? Why was this shape shown? Does it have name?

This was the response:

The shape shown is a 2D stylized projection of a \( \textbf{rhombic hexecontahedron}\).

Mathematical History and Properties

• Origins: The shape was formally described mathematically in 1940 by the German mathematician Helmut Unkelbach.

• Geometric Structure: It is a \( \textbf{non-convex zonohedron} \) composed entirely of 60 intersecting \( \textbf{golden rhombi} \) (rhombuses whose diagonals align exactly with the golden ratio).

• Vertices and Edges: The structure features 62 vertices and 120 edges, all organized with perfect \( \textbf{icosahedral symmetry} \).

• Stellation: It is formed as a stellation of the \( \textbf{rhombic triacontahedron} \), meaning it is created by extending the face planes of a core polyhedron until they intersect again to form a new, more complex outer boundary.

• Internal Geometry: The internal volume of the shape can be cleanly dissected into 20 acute \( \textbf{golden rhombohedra} \) that meet exactly at the center point.

Significance and Selection as a Logo

• The "Spikey" Tradition: The rhombic hexecontahedron continues a decades-long corporate tradition initiated by Stephen Wolfram. Since the release of Mathematica 1.0 in 1988, Wolfram products have been represented by complex, mathematically generated 3D polyhedra, affectionately known internally as "Spikeys." While early Mathematica logos featured shapes like the stellated icosahedron and hyperbolic dodecahedron, the rhombic hexecontahedron was designated specifically for the launch of the Wolfram|Alpha engine in 2009.

• Visualizing Computation: The logo was selected to reflect the computational complexity, algorithmic precision, and mathematical rigor underpinning Wolfram|Alpha. The shape’s composition—made of identical, mathematically strict intersecting faces that combine to form a highly intricate and globally symmetric structure—serves as a visual metaphor for the engine's ability to process massive amounts of disparate raw data and compute unified, structured answers.

Here is what Gemini did with a sketch that I made of the shape using the following prompt:

This image is a 2D stylized projection of a rhombic hexecontahedron, used by Wolfram Alpha as its logo. I need you impose a strict symmetry on the shape and make it look as solid and 3D-looking as possible with shading or whatever. The faces I’ve represented as windows and these should be rendered as translucent with a yellow light emanating, seemingly from within. The overall shape should then look like a polyhedral house with a warm and inviting glow.

Instagram Link

The Golden Rhombus and Golden Rhombohedron referred to earlier look this:

A rhombohedron shown above is an acute golden rhombohedron and it is a cube that has been pushed out of shape so that all of its faces are rhombi. A golden rhombus has a ratio of long diagonal to short diagonal of \( \phi \; : \; 1\). A golden rhombohedron is simply a rhombus with all faces golden rhombi. 

• The acute angle in a golden rhombus is given by \( 2 \text{arctan} (1/ \phi) \approx 63.43^{\circ} \)
• The obtuse angle is given by \( 2 \text{arctan} ( \phi) \approx 116.57^{\circ} \) 

Twenty acute golden rhombohedra can be combined to form a solid rhombic hexecontahedron. There is also an obtuse golden rhombohedron but I'll deal with that later.

Geometric Interpretation of Tetraprimes

Following on from my previous post titled Tetraprimes, I asked Gemini the following:

I’d like you to create a program in SageMath that will accept any positive integer with up to four not necessarily distinct prime factors as INPUT and if the integer has FOUR prime factors it should return 4D Hypervolume, 3D Surface Volume, 2D Total Face Area and 1D Total Edge Length. If it has THREE factors then 3D Volume, 2D Surface Area and 1D Edge Length are returned. If it has TWO factors then 2D Surface Area and 1D Edge Length are returned. If the input is prime or has more than four factors then this is stated and no output is returned. The factorisation of the numbers shown in the output should also be displayed.

Here is a permalink to the program that it created using 28302 as input. The output looks of the program is shown below:

Input: 28302
4D Hypervolume: 28302 = 2 * 3 * 53 * 89
3D Surface Volume: 48874 = 2 * 7 * 3491
2D Total Face Area: 21732 = 2^2 * 3 * 1811
1D Total Edge Length: 1176 = 2^3 * 3 * 7^2

The program also deals with triprimes and biprimes. Here is the output using 28303 and 28299 as inputs.

Input: 28303
3D Volume: 28303 = 11 * 31 * 83
2D Surface Area: 7654 = 2 * 43 * 89
1D Edge Length: 500 = 2^2 * 5^3

Input: 28299
2D Surface Area: 28299 = 3 * 9433
1D Edge Length: 18872 = 2^3 * 7 * 337

I've incorporated this algorithm into my daily number analysis. 

In the example shown above, we can see easily that 28303 can be interpreted as the volume of a rectangular prism with a surface area that is also a sphenic number and thus interpretable as the volume of another rectangular prism. This additional information for my daily number analysis will prove useful in other ways I'm sure.

Tetraprimes

While I have looked at numbers with four prime factors counted with multiplicity whose reversals also have this property, I've not actually looked at reversible tetraprimes. Numbers with four distinct prime factors are called tetraprimes. The first example of such a number is 1518 where we have:

• \(1518 = 2 \times 3 \times 11 \times 23 \)
• \( 8151 = 3 \times 11 \times 13 \times 19 \)

There are 273 such numbers in the range up to 40000 (permalink):

1518, 2046, 2226, 2262, 2418, 2478, 2618, 2622, 2814, 2838, 2886, 3135, 3927, 4170, 4182, 4386, 4389, 4746, 4785, 4935, 5313, 5394, 5406, 5478, 5565, 5655, 5838, 5874, 6018, 6045, 6222, 6402, 6438, 6474, 6486, 6690, 6699, 6834, 6846, 6882, 7293, 7458, 8106, 8142, 8151, 8162, 8346, 8382, 8385, 8547, 8742, 8745, 9834, 9966, 10434, 10506, 11022, 11346, 11814, 11946, 12243, 12441, 12738, 12765, 13026, 13299, 13542, 13629, 13695, 14105, 14118, 14421, 14469, 14574, 15114, 15873, 16005, 16107, 16359, 16665, 16786, 16962, 16995, 17017, 17358, 17589, 17655, 17754, 17922, 18183, 18258, 18447, 18462, 18546, 18615, 18879, 19434, 19437, 19446, 19578, 19662, 20022, 20055, 20085, 20118, 20145, 20163, 20190, 20262, 20310, 20355, 20382, 20405, 20526, 20553, 20559, 20562, 20658, 20746, 20769, 20774, 20878, 20922, 20958, 22002, 22011, 22074, 22098, 22154, 22242, 22290, 22458, 22515, 22533, 22578, 22695, 22710, 22737, 22755, 22854, 22902, 22946, 22962, 22971, 24115, 24123, 24178, 24186, 24198, 24222, 24270, 24297, 24339, 24465, 24530, 24531, 24582, 24585, 24618, 24645, 24662, 24717, 24766, 24846, 24890, 24906, 24915, 24969, 26013, 26103, 26106, 26187, 26202, 26286, 26358, 26394, 26445, 26481, 26490, 26502, 26535, 26554, 26634, 26642, 26691, 26733, 26895, 26922, 26961, 26978, 26985, 26994, 28083, 28086, 28194, 28245, 28302, 28329, 28410, 28446, 28455, 28497, 28509, 28542, 28545, 28578, 28623, 28626, 28635, 28785, 28866, 28886, 28905, 28974, 28985, 28990, 30162, 30426, 31038, 31062, 31146, 31494, 31515, 31614, 31955, 32142, 32574, 32637, 32682, 32745, 32835, 32865, 33198, 33285, 33522, 33594, 33654, 33666, 33762, 34017, 34221, 34386, 34419, 34518, 34554, 34626, 34854, 35502, 35673, 35706, 35826, 36102, 36195, 36218, 36645, 36966, 36993, 37037, 37074, 37158, 37365, 37605, 37653, 37785, 37851, 38055, 38082, 38181, 38445, 39045, 39666, 39738, 39963

I've also considered biprimes as rectangles and triprimes as rectangular prisms but what about tetraprimes. I asked Gemini the following:

A triprime with prime factors of \(a\), \(b\) and \(c\) can be conceived of as a rectangular prism with sides \(a\), \(b\), \(c\), volume \(a \, b \, c\) and surface area equal to \(2 \times (ab + bc + ac) \). Are there equivalent formulae for volume and surface area of a four dimensional object derived from a tetraprime with four prime factors.

Here was Gemini's response:

The direct geometrical equivalent of a tetraprime with four prime factors \(a\), \(b\), \(c\), and \(d\) is a four-dimensional rectangular orthotope (commonly called a hyperrectangle).

Because a 4D object possesses boundaries in multiple lower dimensions (3D, 2D, and 1D), the equivalent to your 3D formulas expands into a hierarchy of measurements. Just as a 3D prism is bounded by 2D faces, a 4D orthotope is bounded by 3D cells, 2D faces, and 1D edges.

Here are the equivalent geometric formulas derived from a tetraprime:

• 4D Hypervolume (The equivalent of 3D Volume): The total 4D spatial content of the object is simply the product of all four prime factors, which is exactly equal to the tetraprime itself.$$V_4 = a \cdot b \cdot c \cdot d$$

• 3D Surface Volume (The structural equivalent of 3D Surface Area): In 3D geometry, "surface area" measures the (n-1)-dimensional boundary of the object. For a 4D object, its outer boundary is three-dimensional. A hyperrectangle is enclosed by 8 solid 3D rectangular prisms (called "cells"). There are two cells for each possible triplet of the dimensions.$$S_3 = 2(abc + abd + acd + bcd)$$

• 2D Total Face Area (The literal equivalent of 2D Area): If you are looking specifically for flat, two-dimensional surface area, a hyperrectangle has 24 flat 2D rectangular faces. For every pair of dimensions, there are 4 identical faces throughout the 4D structure.$$A_2 = 4(ab + ac + ad + bc + bd + cd)$$

• 1D Total Edge Length: For complete geometric breakdown, a hyperrectangle has 32 one-dimensional edges. There are 8 parallel edges for each of the four prime factors.$$L_1 = 8(a + b + c + d)$$

By substituting your four prime factors into \(V_4\) and \(S_3\), you achieve the exact mathematical analogies for volume and surface area as you step up from three dimensions into four.

I'll investigate this further in a subsequent blog.