Lonely Prime Pairs

Today I celebrate my first prime in 30 days. The previous prime was 28319 and the number associated with my diurnal age today is 28349. It's a twin prime and so 28351 is also prime. To my suprise however, the next prime after that is 36 days away. It is the number 28387 that falls on December 22nd 2026. 

Usually after a long drought of primes, there follows a flurry of them but not in this case. I was right to suspect this is an unusual occurrence. The total span of days before 28349 and after 28351 is 30 + 36 = 66. I decided to find all the record breaking twin prime pairs between 1 and one million. Here is the result where \(p\) and \(q\) are the twin primes (permalink).

  gap   p        q        gap   total 

  1     3        5        2     3

  2     5        7        4     6

  4     11       13       4     8

  6     29       31       6     12

  6     137      139      10    16

  10    419      421      10    20

  12    521      523      18    30

  22    1151     1153     10    32

  18    1931     1933     16    34

  16    1949     1951     22    38

  22    6449     6451     18    40

  34    10007    10009    28    62

  30    28349    28351    36    66

  52    35729    35731    16    68

  52    48731    48733    18    70

  22    85931    85933    58    80

  34    107507   107509   54    88

  70    173429   173431   42    112

  40    838247   838249   100   140

 If we use the smaller of the twin primes as the marker then the sequence becomes:

3, 5, 11, 29, 137, 419, 521, 1151, 1931, 1949, 6449, 10007, 28349, 35729, 48731, 85931, 107507, 173429, 838247

The sequence is not in the OEIS and I've no intention of putting it there.

Convolution of Fibonacci with Composites

 I've written about convolutions before in two posts:

• A Practical Example of a Convolution on the 12th of October 2025
• Convolutions Revisited on the 1st May 2026
• Cartesian Product versus Self-Convolution on the 1st of May 2026

Today, having turned 28347 days old, I noticed that this number is a member of OEIS A023609:

 A023609: convolution of Fibonacci numbers and composite numbers.

Here an explanation from Gemini about how this sequence is generated (link) as well as code and visualisations. In future, it's probably better to just post Gemini links rather than copying and pasting because this seems to always incur formatting problems.

Just a reminder that in integer sequences, a convolution combines two sequences, $A$ and $B$, to produce a third sequence. The $n$-th term of the new sequence is calculated by taking the first $n$ terms of $A$ and the first $n$ terms of $B$, reversing the order of $B$, multiplying the aligned pairs, and summing the results.

Here we have A = 1, 1, 2, 3, 5, 8, 13, 21... and B = 4, 6, 8, 9, 10, 12, 14, 15... and thus we get: 

• first term is A(1) x B(1) = 1 x 4 = 4 
• second term is A(1) x B(2) + A(2) x B(1) = 1 x 6 + 4 x 1 = 10
• third term is A(1) x B(3) + A(3) x B(1) + A(2) x B(2) = 1 x 8 + 2 x 4 + 1 x 6 = 22

The sequence begins:

 4, 10, 22, 41, 73, 126, 213, 354, 583, 955, 1558, 2534, 4114, 6672, 10811, 17509, 28347, 45884, 74261, 120177, 194471, 314682, 509188, 823906, 1333132, 2157077, 3490249, 5647368, 9137661

Biprimes With 2 As A Factor

I noticed that the number 28346 associated with my diurnal age today had an interesting property when viewed as a very elongated rectangle:

• 2D Surface Area: \(28346 = 2 \times 14173\)
  
  
• 1D Edge Length: \(28350 = 2 \times 3^4 \times 5^2 \times 7\)
  

This is a difference of 4 and a percentage difference of \( \dfrac{4}{28346} \% \approx 0.01411133846  \%\).

Little did I know that the difference between the area and the perimeter is always 4 when 2 is one of the factors of a biprime. Starting with the first two biprimes 4 and 6 we notice that:

• 2D Surface Area: \(4 = 2 \times 2\)
  
  
• 1D Edge Length: \(8 = 2 \times 4\)
  
  
• 2D Surface Area: \(6 = 2 \times 3\)
  
  
• 1D Edge Length: \(10 = 2 \times 5\)
  
  

In general terms, we have every even biprime \(n = 2q\) with a constant difference of \( \vert{}2q - (2q + 4)\vert{} = 4\) between its area and perimeter. The percentage difference is:$$\frac{4}{2q} \times 100\% = \frac{200}{q}\%$$As \(q\) gets larger the percentage difference shrinks. Once  26678 is reached, the percentage difference is \(0.01499 \% \) which rounds to \( 0.01 \% \) (permalink).

Product of a Number and its Reversal

In July of 2026, I posted about Pandigital Numbers Formed From the Product of a Number and its Reversal and my diurnal age today, 28344, is one such number:$$28344 \times 44382 = 1257963408$$The topic was treated adequately in that blog post so today I'd like to consider numbers that when multiplied by their reversals generate products with no digits in common with the originating number. While this is common enough for numbers with four or fewer digits, it should not be all that common for five digit numbers. In fact, in the range between 10000 and 40000, there are only 160 such numbers but NONE in the range between 10000 and 20000. The 160 numbers are (permalink):

20072, 20077, 20122, 20123, 20222, 20267, 20272, 20607, 20667, 20777, 21113, 21114, 21122, 21124, 21133, 21212, 21323, 21414, 21722, 21772, 21812, 22022, 22102, 22103, 22104, 22112, 22144, 22202, 22207, 22223, 22254, 22313, 22335, 22444, 22544, 22573, 22622, 22712, 22782, 22823, 22882, 22885, 22923, 23224, 23292, 23334, 23733, 23752, 23872, 23944, 24004, 24214, 24223, 24474, 24545, 24555, 25444, 25525, 25555, 25732, 25882, 26624, 26992, 27002, 27023, 27044, 27202, 27323, 27494, 27712, 27832, 28722, 28755, 28822, 28833, 28852, 29092, 29103, 29204, 29232, 29962, 30122, 30133, 30192, 30206, 30306, 30338, 30606, 31038, 31112, 31322, 31413, 32072, 32102, 32222, 32233, 32242, 32312, 32343, 32372, 32822, 32833, 32922, 33006, 33008, 33018, 33103, 33112, 33116, 33223, 33333, 33363, 33373, 33388, 33533, 33656, 33732, 33823, 33882, 33908, 33944, 34083, 34323, 34344, 34384, 34437, 34453, 34553, 34737, 34774, 34834, 35443, 35506, 35543, 35837, 36316, 36333, 36494, 36616, 36667, 36677, 36776, 36777, 37333, 37457, 37522, 37577, 37667, 37673, 37744, 37757, 37767, 37783, 37877, 38043, 38434, 38773, 38884, 39334, 39964

Let's take the last number listed above, 39964, as an example:$$39964 \times 46993 = 1878028252$$Of these 160 numbers, 13 are prime. Specifically:

20123, 21323, 22573, 24223, 30133, 32233, 32833, 33223, 33533, 35543, 35837, 36677, 37783

Remarkably, the left and right hand side of the equation have no digits in common. We can reverse this and look for numbers in the same range that multiply with their reversals to generated numbers with identical digits but not necessarily with the same frequency. There are 33 numbers in the range between 10000 and 40000 that satisfy this criterion. They are:

10000, 10003, 10026, 10030, 10062, 10204, 10279, 10300, 10309, 10749, 10804, 12086, 13042, 15228, 15600, 15860, 18730, 21068, 21168, 24031, 24062, 26001, 26042, 28134, 28579, 30001, 30010, 30100, 31024, 34217, 34880, 37810, 39025

Take the last number in the list above, 39025, as an example:$$39025 \times 52093 = 2032929325$$Of the 33 numbers, only two are prime viz. 28579 and 34217.

Visual Display of Numbers: Part 2

Following on from my previous post, Visual Display of Numbers: Part 1, this post will deal with another two ways to visually represent numbers:

• Hasse Diagram
• Logarithmic Divisor-Pair Chord Diagram

Hasse Diagram

A Hasse divisor diagram transforms a number’s factors into a geometric shape. Instead of listing divisors in a flat row, it stacks them in vertical layers from bottom to top, showing how prime numbers multiply together to build the target number.

1. The Four Visual Rules of a Hasse Diagram

Every Hasse divisor lattice follows four rules:

1. Start at the Bottom, End at the Top: The number $1$ always sits at the bottom (Level 0), and the number $n$ sits at the very top.

2. One Step Up = Multiply by One Prime: Every upward line connects a divisor to a larger divisor obtained by multiplying by a single prime factor.

3. Parallel Lines = Same Prime Factor: All lines moving in the same direction (and color) represent multiplying by the exact same prime number.

4. Each New Prime Adds a New Dimension: When a number is made of distinct primes (no repeated primes like $2^2$ ), each new prime factor doubles the number of divisors by copying the previous shape and shifting it along a new direction:

Number Type	Prime Recipe	Example	Total Divisors	Geometric Shape
Prime	$p$	$28349$	$2^1 = 2$	1D Line Segment (see below)
Biprime	$p \times q$	$28339 = 17 \times 1667$	$2^2 = 4$	2D Square - Diamond (see below)
Triprime	$p \times q \times r$	$28338 = 2 \times 3 \times 4723$	$2^3 = 8$	3D Cube (see below)
Tetraprime	$p \times q \times r \times s$	$28329 = 3 \times 7 \times 19\times 71$	$2^4 = 16$	4D Hypercube (Tesseract) (see below)

When there are repeated factors such as with \(28341=3^2 \times 47 \times 67\), the Hasse diagram is as shown below:

Here is a permalink: just enter the number whose visual representation you want to look at.

2. How to Read Calculations Off the Shape

Once the shape is drawn, you can find the Greatest Common Divisor (GCD) and Least Common Multiple (LCM) of any two integers in the diagram just by following the lines:

• To find the GCD of two numbers: Trace downward along the lines from both integers. The highest point where the two downward paths meet is their GCD. 

• To find the LCM of two numbers: Trace upward along the lines from both integers. The lowest point where the two upward paths meet is their LCM. 

• Factor Pairs (Opposite Corners): Any two integers sitting in exact opposite positions across the center of the shape multiply together to equal to the number.

Logarithmic Divisor-Pair Chord Diagram

A Logarithmic Divisor-Pair Chord Diagram arranges all the divisors of a number $n$ around a circle like a clock face and connects each factor pair ($d$ and $\frac{n}{d}$) with a straight line (a chord). Because the circle uses a logarithmic scale rather than a regular counting scale, every factor pair sits at the exact same vertical height on opposite sides of the circle—meaning every connecting chord is perfectly horizontal and parallel.

Why Use a Logarithmic Scale Around the Circle

If you placed the divisors of $30$ ($\{1, 2, 3, 5, 6, 10, 15, 30\}$) on a standard ruler from $1$ to $30$, the first five divisors ($1, 2, 3, 5, 6$) would be crammed into the first $20\%$ of the line, while the last $80\%$ would be mostly empty space.

A logarithmic scale spaces numbers by multiplication instead of addition. In high school math, the logarithm rule for multiplication says:

$$\log(a \times b) = \log(a) + \log(b)$$

To place each divisor $d$ around a circle from $0\%$ to $100\%$ of the way around, we compute its fractional position $t$:$$t(d) = \frac{\log(d)}{\log(n)}$$

This simple formula creates three visual properties on the circle:

1. Top of the Circle ($0\%$ and $100\%$): Because $\log(1) = 0$, the number $1$ always starts at the top-right ($t = 0$), and the number $n$ always ends at the top-left ($t = 1$), since: $$\dfrac{\log(n)}{\log(n)} = 1$$

2. Bottom of the Circle ($50\%$ — The Square Root): Halfway around the circle ($t = 0.5$) sits the exact square root $\sqrt{n}$, because $\log(\sqrt{n}) = \log(n^{0.5}) = 0.5 \log(n)$.

   • Every divisor smaller than $\sqrt{n}$ sits on the Right Arc ($0 \le t < 0.5$).

   • Every divisor larger than $\sqrt{n}$ sits on the Left Arc ($0.5 < t \le 1$).

3. Why Every Factor-Pair Chord Is Horizontal: If two divisors multiply to give $n$ ($d \times \frac{n}{d} = n$), their fractional positions around the circle always add up to $1$:

   $$t(d) + t\left(\frac{n}{d}\right) = \frac{\log(d) + \log(n/d)}{\log(n)} = \frac{\log(n)}{\log(n)} = 1$$

   Because a point at distance $t$ clockwise down the right side of the circle and a point at distance $1 - t$ clockwise around to the left side are mirror images across the vertical centerline, connecting them draws a strictly horizontal chord.

2. Progression from Prime to Tetraprime

Each time you multiply a number by a new distinct prime factor, the number of divisors doubles, and the number of horizontal chords across the circle doubles as well (in the table below D stands for the number of Divisors and C stands for the number of chords):

Number Category	Example n	D	C	What the Chord Diagram Looks Like
1. Prime $p$	$28349$	$2$	$1$	Single Top Bar: Only one chord at the very top ($1 \times 28349 = 28349$). The rest of the circle is completely empty, showing $28349$ cannot be broken down further. See below.
2. Biprime $p \times q$	$28339 = 17 \times 1667$	$4$	$2$	Two-Rung Ladder: A top chord ($1 \times 28339 = 28339$) and a lower chord ($17 \times 1667 = 28339$) hovering above the bottom pole ($\sqrt{28339} \approx 168.35$). See below.
3. Triprime $p \times q \times r$	$28338 = 2 \times 3 \times 4723$	$8$	$4$	Four-Rung Ladder: Four horizontal chords ($1 \times 28338$, $2 \times 14169$, $3 \times 9446$, and $4763 \times 6$) descending smoothly toward $\sqrt{28338} \approx 168.3$. See below.
4. Tetraprime $p \times q \times r \times s$	$28329 = 3 \times 7 \times 19\times 71$	$16$	$8$	Eight-Rung Ladder: Eight parallel horizontal chords filling the circle from $1 \times 28329$ at the top down to $133 \times 213 = 28329$ right at the bottom pole ($\sqrt{28329} \approx 168.3$). See below.

Here is a permalink: just enter the number whose visual representation you want to look at.

Visual Display of Numbers: Part 1

In my previous post titled Binary Display, I borrowed an idea from Numbers Aplenty about the visual display of a positive integer expressed in terms of its binary digits. This gave me the idea for a display of the number in terms of its decimal digits. I got Gemini to develop an algorithm to achieve this and Figure 1 shows the result for 28338, my diurnal age today (permalink):

Figure 1

There is a small text-based display:

  | 2   8   3   3   8  

--+--------------------

9 |                    

8 |     ██          ██ 

7 |                    

6 |                    

5 |                    

4 |                    

3 |         ██  ██     

2 | ██                 

1 |                    

0 |                    

The larger diagram below this in the output can then be screen-shot and annotated as shown in Figure 2.

Figure 2

On this basis, the number 28338 might be associated with Twin Peaks (with an intervening valley). This sort of depiction of the digits of a number can be used to visually depict so called \( \textbf{mountain numbers}\) that belong to OEIS A134941. These numbers are defined as:

The first digit is 1. The last digit is 1. The first digits are in increasing order. The last digits are in decreasing order. The numbers only have one largest digit. This sequence is finite. The last term is 12345678987654321. The total number of terms is 21846.

19871 is an example of such a number:

  | 1   9   8   7   1  

--+--------------------

9 |     ██             

8 |         ██         

7 |             ██     

6 |                    

5 |                    

4 |                    

3 |                    

2 |                    

1 | ██              ██ 

0 |                    

There are also generalized mountain numbers described in OEIS A134853 as:

There is only one largest digit which represents the top of the mountain. This sequence is finite. The last member is 123456789876543210. The sequence is a supersequence of A134941, because the restriction that both feet of the mountain are at "sea level" (first and last digit equal 1) is dropped here. There are 173247 terms in this sequence.

28320 is an example of such a number:

  | 2   8   3   2   0  
--+--------------------
9 |                    
8 |     ██             
7 |                    
6 |                    
5 |                    
4 |                    
3 |         ██         
2 | ██          ██     
1 |                    
0 |                 ██  

This is just a rough start for a general theme of how to represent a decimal number. I've addressed this issue before in a post titled What Does A Number Look Like from July of 2018. Using the site Numbermatics, the representation of the number 28338 is shown in Figure 3:

Figure 3

The commentary on the display is as follows:

In this visualization, the outer ring segments show the relative proportions of its three prime factors. Inside, the large and small discs symbolize the relationship between these factors (center) and eight divisors.

This representation is more informative and very different to the display shown in Figure 1. I've gotten Gemini to investigate visual representations more using this prompt:

I'm interested in exploring different ways to represent a positive integer such as 28338. Numbermatics (https://numbermatics.com/n/28338) shows one way and the display that I've just gotten Gemini to create shows another. What are some other visual ways to represent numbers that ideally show information not just about the digits but also its factors, divisors etc. Maybe you suggest modifications or improvements to existing methods. See what you can come up with. Links to other sites like Numbermatics can naturally be included in your report.

Using Gemini's Argon set to high level thinking, the response that I got was very comprehensive and I won't attempt to summarise it in this post and I'm still digesting what it came up with. I'll reserve further comment to my next blog post. 

Binary Display

I noticed that Numbers Aplenty had the display shown in Figure 1 as part of its output for the number associated with my diurnal age today (28337) with the comment that "when arranged in a rectangular grid, its binary digits exhibit some kind of symmetry". The binary digits are 110111010110001 and there are 15 of them so they can be arranged in a 5 x 3 grid.

Figure 1

I quite liked this output and decided to incorporate it into the program for my daily number analysis. The output is as shown in Figure 2.

Figure 2

Every day, from this day forwards, there will be a similar display and I can decide whether there is anything of interest regarding the display, symmetrical or otherwise. Figure 3 shows the results for 27334 where there is also lateral symmetry.

Figure 3

The range of numbers for which there are 15 binary digits is 16,384 to 32,767 corresponding to \(2^{14}\) and \(2^{15}-1\). Prior to 16384, there is a range of numbers with 14 binary digits that can also be arranged in a rectangular grid. This range is from 8192 to 16383 corresponding to \(2^{13}\) and \(2^{14}-1\). Figure 4 shows the display for 16382:

Figure 4

Above 32767, we enter the realm of 16 binary digits and no grid will be displayed. The output for 32768 would be:

Integer : 32768
Binary  : 1000000000000000
Length  : 16 binary digits

Result  : Bit length 16 (factorization: 2^4) is NOT a biprime. 

No grid generated.

Of course, a 4 x 4 or 8 x 2 grid could be generated with a tweak of the algorithm but I'll consider that as I approach 32768 by which point I'll be 89 years old.