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.

Parasitic Numbers

My inbox contained the following content from Puzzle A Day:

The number 105263157894736842 is fascinating. This is because it’s the smallest positive integer with a remarkable property that occurs when you move its final digit to the front: 210526315789473684. How is the new number related to the original?

It doesn't take long to see the new number is twice the original. I got Gemini to investigate further via this query:

The number 105263157894736842 is the smallest positive integer with a remarkable property that it doubles when you move its final digit to the front: 210526315789473684. This is quite a large number. Why are there no smaller numbers with this property? What is the next largest number? Is there any history behind the discovery of such numbers? There must be somebody who first discovered it.

Here was Gemini's response:

Numbers that multiply by an integer \(k\) when their trailing digit is shifted to the front are known in recreational number theory as \(k\)-parasitic numbers—and the smallest solutions for each multiplier are specifically called Dyson numbers. The 18-digit integer \(105{,}263{,}157{,}894{,}736{,}842\) is the smallest 2-parasitic number in base 10.

1. Why There Are No Smaller Solutions

To see why 18 digits is the absolute minimum, we can translate the digit-shifting operation into a modular congruence.

Suppose an \(m\)-digit number \(X\) ends with the units digit \(d\) (where \(1 \le d \le 9\)), and the remaining \(m-1\) leading digits form the number \(A\). We can express \(X\) as:\[X = 10A + d\]Moving the final digit \(d\) to the front of the \(m-1\) digits of \(A\) produces the shifted number:\[Y = d \cdot 10^{m-1} + A\]We require this shifted number to be exactly twice the original number (\(Y = 2X\)):\[ \begin{aligned} d \cdot 10^{m-1} + A &= 2(10A + d)  \\ \\ d \cdot 10^{m-1} + A &= 20A + 2d \\ \\ 19A &= d \, (10^{m-1} - 2) \end{aligned} \]Now solve for the original number \(X = 10A + d\) by multiplying both sides by \(19\):\[ \begin{aligned} 19X &= 19(10A + d) = 10(19A) + 19d \\ \\ 19X &= 10d \, (10^{m-1} - 2) + 19d = d \,(10^m - 20 + 19) \\ \\ X &= \frac{d \, (10^m - 1)}{19} \end{aligned} \]For \(X\) to be a positive integer, the numerator \(d \, (10^m - 1)\) must be divisible by \(19\):

• Since \(d\) is a single digit (\(1 \le d \le 9\)) and \(19\) is prime, \(\gcd(d, 19) = 1\).
• Therefore, \(19\) must divide \(10^m - 1\), which is equivalent to the congruence:\[10^m \equiv 1 \pmod{19}\]The smallest positive integer \(m\) that satisfies \(10^m \equiv 1 \pmod{19}\) is the multiplicative order of \(10\) modulo \(19\). By Fermat's Little Theorem, \(m\) must be a divisor of \(\phi(19) = 18\) (so \(m \in \{1, 2, 3, 6, 9, 18\}\)). Checking the smaller divisors modulo \(19\):

• \(10^1 \equiv 10 \pmod{19}\)
• \(10^2 \equiv 5 \pmod{19}\)
• \(10^3 \equiv 12 \pmod{19}\)
• \(10^6 \equiv (12)^2 \equiv 11 \pmod{19}\)
• \(10^9 \equiv 12 \cdot 11 \equiv 18 \equiv -1 \pmod{19}\)

Because \(10\) is a primitive root modulo \(19\) (and \(19\) is a full reptend prime in base 10), the decimal expansion of \(\frac{1}{19}\) has the maximum possible period length of \(19 - 1 = 18\). No power of \(10\) smaller than \(10^{18}\) leaves a remainder of \(1\) when divided by \(19\), making an 18-digit number the shortest possible length.

Furthermore, we cannot pick \(d = 1\). If \(d = 1\), then$$X = \frac{10^{18} - 1}{19} = 052{,}631{,}578{,}947{,}368{,}421$$which has a leading zero and is actually a 17-digit number whose shift fails unless padded with a leading zero. Thus, \(d = 2\) yields the true smallest positive solution.

2. What Are the Next Largest Numbers?

Every valid choice of units digit \(d \in \{2, 3, 4, 5, 6, 7, 8, 9\}\) in the formula:$$ X = \frac{d \,(10^{18} - 1)}{19}$$ generates an 18-digit solution. Because \(\frac{1}{19}\) is a cyclic number, all eight 18-digit solutions are simply integer multiples of \(\frac{10^{18}-1}{19}\) and are cyclic permutations of the exact same 18 digits:

Rank	Units Digit d	Fraction	2-Parasitic Number X	Doubled / Shifted Result 2X
1	\(d = 2\)	\(2/19\)	\(105{,}263{,}157{,}894{,}736{,}842\)	\(210{,}526{,}315{,}789{,}473{,}684\)
2	\(d = 3\)	\(3/19\)	\(157{,}894{,}736{,}842{,}105{,}263\)	\(315{,}789{,}473{,}684{,}210{,}526\)
3	\(d = 4\)	\(4/19\)	\(210{,}526{,}315{,}789{,}473{,}684\)	\(421{,}052{,}631{,}578{,}947{,}368\)
4	\(d = 5\)	\(5/19\)	\(263{,}157{,}894{,}736{,}842{,}105\)	\(526{,}315{,}789{,}473{,}684{,}210\)
5	\(d = 6\)	\(6/19\)	\(315{,}789{,}473{,}684{,}210{,}526\)	\(631{,}578{,}947{,}368{,}421{,}052\)
6	\(d = 7\)	\(7/19\)	\(368{,}421{,}052{,}631{,}578{,}947\)	\(736{,}842{,}105{,}263{,}157{,}894\)
7	\(d = 8\)	\(8/19\)	\(421{,}052{,}631{,}578{,}947{,}368\)	\(842{,}105{,}263{,}157{,}894{,}736\)
8	\(d = 9\)	\(9/19\)	\(473{,}684{,}210{,}526{,}315{,}789\)	\(947{,}368{,}421{,}052{,}631{,}578\)

• The immediate next largest number is the \(d = 3\) case: \(157{,}894{,}736{,}842{,}105{,}263\).
• Notice that the 3rd solution (\(d = 4\)) is \(210{,}526{,}315{,}789{,}473{,}684\)—which is the doubled result of the first solution!
• Beyond 18 digits: Because \(10^m \equiv 1 \pmod{19}\) holds whenever \(m\) is a multiple of \(18\) (\(m = 18, 36, 54, \dots\)), the next family of solutions occurs at 36 digits. You obtain them simply by concatenating any 18-digit solution with itself. The smallest 36-digit solution is:\[105263157894736842105263157894736842\]

3. History and Discovery

While the underlying cyclic period of \(\frac{1}{19}\) was known to 18th- and 19th-century number theorists studying repeating decimals and primitive roots (including Johann Heinrich Lambert in 1769, Carl Friedrich Gauss in Disquisitiones Arithmeticae in 1801, and William Shanks in the 1850s), the specific digit-shifting puzzle has a rich recreational pedigree:

1. Walter B. Gibson (1938) & Maurice Kraitchik (1942): The earliest documented versions of the digit-transposition puzzle in recreational literature trace to American magician and puzzle writer Walter B. Gibson in 1938, and Belgian mathematician Maurice Kraitchik, who analyzed cyclic digit-shifting multiplications in his 1942 classic Mathematical Recreations.
2. Freeman Dyson's Table Puzzle: In the mid-20th century, physicist Freeman Dyson famously posed the problem at a mathematical gathering: "Find the smallest positive integer that doubles when its last digit is moved to the front". Because Dyson could calculate the 18-digit number mentally in a few seconds—by starting with a trailing \(2\) and repeatedly doubling right-to-left with carries (\(2 \to 42 \to 842 \to 6842 \dots\) until reaching \(1\) with no carry)—the smallest \(n\)-parasitic numbers (OEIS sequence A092697) became known as Dyson numbers.
3. Leon Bernstein (1968): In his paper "Multiplicative twins and primitive roots" (Mathematische Zeitschrift, 1968), mathematician Leon Bernstein formalized the algebraic theory connecting these numbers directly to primitive roots and \(p\)-adic expansions.
4. Clifford A. Pickover (2000): The term "parasitic number" was coined and popularized by science and mathematics author Clifford Pickover in Chapter 28 of his 2000 book Wonders of Numbers.

Key takeaway: The size of the smallest \(k\)-parasitic number depends entirely on the multiplicative order of \(10\) modulo \((10k - 1)\). For \(k = 2\), \(10k - 1 = 19\) is a full reptend prime, forcing an 18-digit minimum. By contrast, for \(k = 4\), \(10k - 1 = 39 = 3 \times 13\), and since \(10^6 \equiv 1 \pmod{39}\), the smallest 4-parasitic number has just 6 digits: \(102{,}564 \times 4 = 410{,}256\).