Truncated Pyramid

I was surprised to discover that the number associated with my diurnal age today (27930) has a connection to the volume of a truncated square pyramids. Firstly, let's recall the formula for the volume of such figure:$$V=\frac{1}{3} (a^2+ab+b^2) \, h$$where \(a\) is the side length of the square base, \(b\) is the side length of the top square and \(h\) is the height of the truncated pyramid. The formula is easily confirmed by integration from first principles without relying on the formula for the volume of a pyramid. 

Now 27930 is a member of OEIS A027444: \( \text{a}(n) = n^3 + n^2 + n\) with the following interesting comment attached to the entry:

For \(n>1\), a(\(n\)) is the volume of a truncated square pyramid with height \(n\) and base  lengths \(n+2\) and \(n-1\). See Figure 1.

Figure 1

Now if we substitute \(a=n+2\), \(b=n-1\) and \(n=h\) into our earlier formula we find the volume \(V\) of the truncated pyramid becomes:$$V=n^3+n^2+n$$When \(n=30\), the volume is 27930 cubic units. Here are the volumes for various values of \(n>1\):

  n    n-1   n+2   V 

  2    1     4     14

  3    2     5     39

  4    3     6     84

  5    4     7     155

  6    5     8     258

  7    6     9     399

  8    7     10    584

  9    8     11    819

  10   9     12    1110

  11   10    13    1463

  12   11    14    1884

  13   12    15    2379

  14   13    16    2954

  15   14    17    3615

  16   15    18    4368

  17   16    19    5219

  18   17    20    6174

  19   18    21    7239

  20   19    22    8420

  21   20    23    9723

  22   21    24    11154

  23   22    25    12719

  24   23    26    14424

  25   24    27    16275

  26   25    28    18278

  27   26    29    20439

  28   27    30    22764

  29   28    31    25259

  30   29    32    27930

  31   30    33    30783

  32   31    34    33824

  33   32    35    37059

Seven Eleven Rules

I was struggling to find something that caught my fancy regarding the number associated with my diurnal age today: 27929. I thought I'd look at its reverse, 92972, and compare their factorisations. The results were:$$ \begin{align} 27929 =  11 \times 2539 \\ 92972 ==2^2 \times 11 \times 2113$$Clearly, the number and its reverse share a common prime factor of 11. I then realised that 27929 has a digit sum of 29 and the two digits, when added together, give 11. So I then decided to look for numbers with the following properties:

• number is divisible by 11
• its reverse is also divisible by 11
• its sum of digits gives a number whose digits sum to 11

It turns out that there are only 30 numbers that satisfy these criteria in the range up to 40000. They are:

20999, 21989, 22979, 23969, 24959, 25949, 26939, 27929, 28919, 29909, 30899, 30998, 31889, 31988, 32879, 32978, 33869, 33968, 34859, 34958, 35849, 35948, 36839, 36938, 37829, 37928, 38819, 38918, 39809, 39908

The details are (permalink):

  number   factors              reverse   factors              digit sum   sum

  20999    11 * 23 * 83         99902     2 * 11 * 19 * 239    29          11
  21989    11 * 1999            98912     2^5 * 11 * 281       29          11
  22979    11 * 2089            97922     2 * 11 * 4451        29          11
  23969    11 * 2179            96932     2^2 * 11 * 2203      29          11
  24959    11 * 2269            95942     2 * 7^2 * 11 * 89    29          11
  25949    7 * 11 * 337         94952     2^3 * 11 * 13 * 83   29          11
  26939    11 * 31 * 79         93962     2 * 11 * 4271        29          11
  27929    11 * 2539            92972     2^2 * 11 * 2113      29          11
  28919    11^2 * 239           91982     2 * 11 * 37 * 113    29          11
  29909    11 * 2719            90992     2^4 * 11^2 * 47      29          11
  30899    11 * 53^2            99803     11 * 43 * 211        29          11
  30998    2 * 11 * 1409        89903     11^2 * 743           29          11
  31889    11 * 13 * 223        98813     11 * 13 * 691        29          11
  31988    2^2 * 11 * 727       88913     11 * 59 * 137        29          11
  32879    7^2 * 11 * 61        97823     11 * 8893            29          11
  32978    2 * 11 * 1499        87923     11 * 7993            29          11
  33869    11 * 3079            96833     11 * 8803            29          11
  33968    2^4 * 11 * 193       86933     7 * 11 * 1129        29          11
  34859    11 * 3169            95843     11 * 8713            29          11
  34958    2 * 7 * 11 * 227     85943     11 * 13 * 601        29          11
  35849    11 * 3259            94853     11 * 8623            29          11
  35948    2^2 * 11 * 19 * 43   84953     11 * 7723            29          11
  36839    11 * 17 * 197        93863     7 * 11 * 23 * 53     29          11
  36938    2 * 11 * 23 * 73     83963     11 * 17 * 449        29          11
  37829    11 * 19 * 181        92873     11 * 8443            29          11
  37928    2^3 * 11 * 431       82973     11 * 19 * 397        29          11
  38819    11 * 3529            91883     11 * 8353            29          11
  38918    2 * 11 * 29 * 61     81983     11 * 29 * 257        29          11
  39809    7 * 11^2 * 47        90893     11 * 8263            29          11
  39908    2^2 * 11 * 907       80993     11 * 37 * 199        29          11

The algorithm can be modified to search for prime numbers other than 11. For example, there are 80 numbers in the range up to 40000 that satisfy these criteria:

• number is divisible by 7
• its reverse is also divisible by 7
• its sum of digits gives a number whose digits sum to 7

These numbers are (permalink):

259, 952, 1078, 1708, 2527, 2779, 3346, 3598, 4165, 5614, 5866, 6433, 6685, 7252, 8071, 8701, 8953, 9079, 9709, 9772, 10087, 10717, 10969, 11536, 11788, 12103, 12355, 13174, 13804, 14623, 14875, 15442, 15694, 17017, 17269, 17962, 18088, 18718, 19537, 19789, 20545, 20797, 21364, 22183, 22813, 23884, 24451, 25207, 25459, 26026, 26278, 26908, 26971, 27097, 27727, 27979, 28546, 28798, 29113, 29365, 30121, 30373, 31129, 31192, 31822, 32641, 32893, 33649, 34216, 34468, 35035, 35287, 35917, 36736, 36988, 37303, 37555, 38122, 38374, 39823

The details are (permalink):

  number   factors             reverse   factors                digit sum   sum

  259      7 * 37              952       2^3 * 7 * 17           16          7
  952      2^3 * 7 * 17        259       7 * 37                 16          7
  1078     2 * 7^2 * 11        8701      7 * 11 * 113           16          7
  1708     2^2 * 7 * 61        8071      7 * 1153               16          7
  2527     7 * 19^2            7252      2^2 * 7^2 * 37         16          7
  2779     7 * 397             9772      2^2 * 7 * 349          25          7
  3346     2 * 7 * 239         6433      7 * 919                16          7
  3598     2 * 7 * 257         8953      7 * 1279               25          7
  4165     5 * 7^2 * 17        5614      2 * 7 * 401            16          7
  5614     2 * 7 * 401         4165      5 * 7^2 * 17           16          7
  5866     2 * 7 * 419         6685      5 * 7 * 191            25          7
  6433     7 * 919             3346      2 * 7 * 239            16          7
  6685     5 * 7 * 191         5866      2 * 7 * 419            25          7
  7252     2^2 * 7^2 * 37      2527      7 * 19^2               16          7
  8071     7 * 1153            1708      2^2 * 7 * 61           16          7
  8701     7 * 11 * 113        1078      2 * 7^2 * 11           16          7
  8953     7 * 1279            3598      2 * 7 * 257            25          7
  9079     7 * 1297            9709      7 * 19 * 73            25          7
  9709     7 * 19 * 73         9079      7 * 1297               25          7
  9772     2^2 * 7 * 349       2779      7 * 397                25          7
  10087    7 * 11 * 131        78001     7 * 11 * 1013          16          7
  10717    7 * 1531            71701     7 * 10243              16          7
  10969    7 * 1567            96901     7 * 109 * 127          25          7
  11536    2^4 * 7 * 103       63511     7 * 43 * 211           16          7
  11788    2^2 * 7 * 421       88711     7 * 19 * 23 * 29       25          7
  12103    7^2 * 13 * 19       30121     7 * 13 * 331           7           7
  12355    5 * 7 * 353         55321     7^2 * 1129             16          7
  13174    2 * 7 * 941         47131     7 * 6733               16          7
  13804    2^2 * 7 * 17 * 29   40831     7 * 19 * 307           16          7
  14623    7 * 2089            32641     7 * 4663               16          7
  14875    5^3 * 7 * 17        57841     7 * 8263               25          7
  15442    2 * 7 * 1103        24451     7^2 * 499              16          7
  15694    2 * 7 * 19 * 59     49651     7 * 41 * 173           25          7
  17017    7 * 11 * 13 * 17    71071     7 * 11 * 13 * 71       16          7
  17269    7 * 2467            96271     7 * 17 * 809           25          7
  17962    2 * 7 * 1283        26971     7 * 3853               25          7
  18088    2^3 * 7 * 17 * 19   88081     7 * 12583              25          7
  18718    2 * 7^2 * 191       81781     7^2 * 1669             25          7
  19537    7 * 2791            73591     7 * 10513              25          7
  19789    7 * 11 * 257        98791     7 * 11 * 1283          34          7
  20545    5 * 7 * 587         54502     2 * 7 * 17 * 229       16          7
  20797    7 * 2971            79702     2 * 7 * 5693           25          7
  21364    2^2 * 7^2 * 109     46312     2^3 * 7 * 827          16          7
  22183    7 * 3169            38122     2 * 7^2 * 389          16          7
  22813    7 * 3259            31822     2 * 7 * 2273           16          7
  23884    2^2 * 7 * 853       48832     2^6 * 7 * 109          25          7
  24451    7^2 * 499           15442     2 * 7 * 1103           16          7
  25207    7 * 13 * 277        70252     2^2 * 7 * 13 * 193     16          7
  25459    7 * 3637            95452     2^2 * 7^2 * 487        25          7
  26026    2 * 7 * 11 * 13^2   62062     2 * 7 * 11 * 13 * 31   16          7
  26278    2 * 7 * 1877        87262     2 * 7 * 23 * 271       25          7
  26908    2^2 * 7 * 31^2      80962     2 * 7 * 5783           25          7
  26971    7 * 3853            17962     2 * 7 * 1283           25          7
  27097    7^3 * 79            79072     2^5 * 7 * 353          25          7
  27727    7 * 17 * 233        72772     2^2 * 7 * 23 * 113     25          7
  27979    7^2 * 571           97972     2^2 * 7 * 3499         34          7
  28546    2 * 7 * 2039        64582     2 * 7^2 * 659          25          7
  28798    2 * 7 * 11^2 * 17   89782     2 * 7 * 11^2 * 53      34          7
  29113    7 * 4159            31192     2^3 * 7 * 557          16          7
  29365    5 * 7 * 839         56392     2^3 * 7 * 19 * 53      25          7
  30121    7 * 13 * 331        12103     7^2 * 13 * 19          7           7
  30373    7 * 4339            37303     7 * 73^2               16          7
  31129    7 * 4447            92113     7 * 13159              16          7
  31192    2^3 * 7 * 557       29113     7 * 4159               16          7
  31822    2 * 7 * 2273        22813     7 * 3259               16          7
  32641    7 * 4663            14623     7 * 2089               16          7
  32893    7 * 37 * 127        39823     7 * 5689               25          7
  33649    7 * 11 * 19 * 23    94633     7 * 11 * 1229          25          7
  34216    2^3 * 7 * 13 * 47   61243     7 * 13 * 673           16          7
  34468    2^2 * 7 * 1231      86443     7 * 53 * 233           25          7
  35035    5 * 7^2 * 11 * 13   53053     7 * 11 * 13 * 53       16          7
  35287    7 * 71^2            78253     7^2 * 1597             25          7
  35917    7^2 * 733           71953     7 * 19 * 541           25          7
  36736    2^7 * 7 * 41        63763     7 * 9109               25          7
  36988    2^2 * 7 * 1321      88963     7 * 71 * 179           34          7
  37303    7 * 73^2            30373     7 * 4339               16          7
  37555    5 * 7 * 29 * 37     55573     7 * 17 * 467           25          7
  38122    2 * 7^2 * 389       22183     7 * 3169               16          7
  38374    2 * 7 * 2741        47383     7^2 * 967              25          7
  39823    7 * 5689            32893     7 * 37 * 127           25          7

Some Interesting Properties of 39

My daughter-in-law turned 39 yesterday and so I was prompted to investigate some of its mathematical properties. One of its properties is its membership in OEIS A055233:

A055233: composite numbers equal to the sum of the primes from their smallest prime factor to their largest prime factor.

The only members of this sequence in the range up to 40000 are 10, 39, 155 and 371. All are semiprimes and factorise as follows:

• \(10 = 2 \times 5 \text{ with } 2 + 3 + 5 = 10 \)
• \(39 = 3 \times 13 \text{ with } 3 + 5 + 7 + 11+13 = 39\)
• \(155 = 5 \times 31 \text{ with } 5 + 7 + \ldots + 29 + 31=155\)
• \(371 = 7 \times 53 \text{ with } 7 + 11 + \ldots + 47 + 53=371\)

Because they are semiprimes they are thus equal to the product of their smallest and largest prime factors. However, this is not the case for the next member of the sequence: 2935561623745. The reason is that it is not a semiprime.

• \(2935561623745= 5 \times 19 \times 53 \times 61 \times 9557887\)

The next member of the sequence 454539357304421 is a semiprime and thus follows the pattern of the first four members of the sequence:

• \(454539357304421 = 3536123 \times 128541727\)

So we see that 39 by virtue of its membership in OEIS A055233 is rather special. Of course, it has some other interesting qualities. For example, it can be constructed from the first three powers of 3:$$39=3+3^2+3^3$$Gemini also mentions the following number properties:

Beyond these patterns, 39 is also classified as a \( \textbf{Perrin number}\) and a \( \textbf{Størmer number}\), placing it within specialized mathematical sequences that are far from intuitive. 

The number also has an \( \textbf{aliquot sum}\) of 17, which is a prime number, a unique characteristic that links it to a specific aliquot sequence. 

In the realm of number partitions, 39 is notable as the smallest natural number to have three distinct partitions into three parts that all yield the same product, 1200. These partitions are:

• {25, 8, 6} 

• {24, 10, 5} 

• {20, 15, 4}. 

Lastly, in analytic number theory, the \( \textbf{Mertens function}\) returns a value of 0 when given 39, a property that suggests a form of numerical equilibrium or stability, a concept that finds intriguing parallels in other domains. See blog post Zeroes of the Mertens Function.

39 is also what's termed a \( \textbf{perfect totient number} \) because the sum of its iterated totients equals the number itself. Let's confirm this:$$ \begin{align} \phi(39) &=24 \\ \phi(24) &=8 \\ \phi(8) &=4 \\ \phi(4) &=2 \\ \phi(2) &=1 \end{align} $$The sum of these iterated totients equals 39:$$24 + 8 + 4 + 2 + 1 =39$$The perfect totient numbers are listed in OEIS A082897 (permalink):

3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, 5571, 6561, 8751, 15723, 19683, 36759, 46791, 59049, 65535, 140103, 177147, 208191, 441027, 531441, 1594323, 4190263, 4782969, 9056583, 14348907, 43046721

Tricky Entrance Exam Questions

I came across this problem on a YouTube channel. The problem was purported to be a Harvard University entrance exam question.$$ \text{Simplify } \sqrt{\sqrt{121}-\sqrt{120}}$$Once you see the method, it's easy enough so let's start to simplify:$$

\begin{align}

\sqrt{\sqrt{121}-\sqrt{120}} &= \sqrt{11 - 2 \cdot \sqrt{30}} \\

&= \sqrt{11 - 2 \cdot \sqrt{6} \cdot \sqrt{5}} \\

&= \sqrt{6 - 2 \cdot \sqrt{6} \cdot \sqrt{5} + 5} \\

&= \sqrt{(\sqrt{6})^2 - 2 \cdot \sqrt{6} \cdot \sqrt{5} + (\sqrt{5})^2} \\

&= \sqrt{(\sqrt{6} - \sqrt{5})^2} \\

&= \sqrt{6} - \sqrt{5}

\end{align}

$$Here's another one:$$ \text{Simplify } \sqrt{\sqrt{36}-\sqrt{20}}$$The approach is exactly the same:

$$
\begin{align}

\sqrt{\sqrt{36}-\sqrt{20}} &= \sqrt{6 - 2 \cdot \sqrt{5}} \\

&= \sqrt{6- 2 \cdot \sqrt{5} \cdot \sqrt{1}} \\

&= \sqrt{5 - 2 \cdot \sqrt{5} \cdot \sqrt{1} + 1} \\

&= \sqrt{(\sqrt{5})^2 - 2 \cdot \sqrt{5} \cdot \sqrt{1} + (\sqrt{1})^2} \\

&= \sqrt{(\sqrt{5} - \sqrt{1})^2} \\

&= \sqrt{5} - \sqrt{1}\\
&=\sqrt{5}-1

\end{align}

$$

A Special Date

 I came across this article today that discusses today's date: the 16th of September 2025:

Once a century, a very special day comes along. That day is today — 9/16/25.

Pi Day (3/14) often comes with sweet treats; Square Root Day (4/4/16 or 5/5/25, for example) has a certain numerical rhyme. But the particular string of numbers in today's date may be especially delightful to the brains of mathematicians and the casual nerds among us.

First, "all three of the entries in that date are perfect squares — and what I mean by that is \(9\) is equal to \(3^2\), \(16\) is equal to \(4^2\), and \(25\) is equal to \(5^2,\)" says Colin Adams, a mathematician at Williams College who was first tipped off about today's special qualities during a meeting with his former student, Jake Malarkey.

Next, those perfect squares come from consecutive numbers — three, four, and five.

But perhaps most special of all is that three, four, and five are an example of what's called a Pythagorean triple.

"And what that means," explains Adams, "is that if I take the sum of the squares of the first two numbers, \(3^2 + 4^2\), which is \(9 + 16\) is equal to \(25\), which is \(5^2\), so \(3^2 + 4^2 = 5^2\)."

This is the Pythagorean Theorem: \(a^2 + b^2 = c^2\). "And that in fact is the most famous theorem in all of mathematics," says Adams.

It's a theorem that means something geometrically, too. Any Pythagorean triple — including 3, 4, and 5 — also gives the lengths of the three sides of a right triangle. That is, the squares of the two shorter lengths add up to the square of the final, longer side (the hypotenuse).

There are no other dates this century that meet all these conditions, so most of us will experience it just once in our lifetime.

(Fun bonus: It turns out the full year, \(2025\), is also a perfect square: \(45 \times 45\).)

In any case, Adams says that if it were up to him, he'd call the day Pythagorean Triple Square Day. And he plans on celebrating with a rectangular cake cut along the diagonal to yield two right triangles.

"If I have any luck at all, if I can find a cake with the right dimensions, it'll look like a 3, 4, 5 cake, namely edge length 3, edge length 4, and edge length 5," he says. In the middle, he intends to have the date inscribed in icing.

"This date is hiding one of the most beautiful coincidences we will ever encounter," says Terrence Blackman, chair of the mathematics department at Medgar Evers College in the City University of New York. "Those numbers, they tell a story that goes back to ancient Greece."

Blackman says the Pythagorean Theorem is used frequently by carpenters and architects. But for him, as a mathematician, today's date captures a special elegance.

"It reveals some kind of hidden mathematical poetry that is sitting there — just like walking and coming upon a beautiful flower," he says.

In a world that can feel chaotic, Blackman feels that a day like today shows that math can provide a source of comfort.

"It reminds us that beauty and meaning can be found anywhere and everywhere," he says. "We just have to continue to look for it."

Sequences Involving SOD and POD

The number associated with my diurnal age today, 27923, has the interesting property that its sum of digits (23) is equal to the last two digits of the number. This number is part of a sequence of consecutive numbers that all share this same property. The numbers are 27920 up to 27929. In the range of numbers up to 40000, there are 440 numbers with this property. They are (permalink):

SOD = Concatenation of Last Two Digits

910, 911, 912, 913, 914, 915, 916, 917, 918, 919, 1810, 1811, 1812, 1813, 1814, 1815, 1816, 1817, 1818, 1819, 2710, 2711, 2712, 2713, 2714, 2715, 2716, 2717, 2718, 2719, 3610, 3611, 3612, 3613, 3614, 3615, 3616, 3617, 3618, 3619, 4510, 4511, 4512, 4513, 4514, 4515, 4516, 4517, 4518, 4519, 5410, 5411, 5412, 5413, 5414, 5415, 5416, 5417, 5418, 5419, 6310, 6311, 6312, 6313, 6314, 6315, 6316, 6317, 6318, 6319, 7210, 7211, 7212, 7213, 7214, 7215, 7216, 7217, 7218, 7219, 8110, 8111, 8112, 8113, 8114, 8115, 8116, 8117, 8118, 8119, 9010, 9011, 9012, 9013, 9014, 9015, 9016, 9017, 9018, 9019, 9920, 9921, 9922, 9923, 9924, 9925, 9926, 9927, 9928, 9929, 10810, 10811, 10812, 10813, 10814, 10815, 10816, 10817, 10818, 10819, 11710, 11711, 11712, 11713, 11714, 11715, 11716, 11717, 11718, 11719, 12610, 12611, 12612, 12613, 12614, 12615, 12616, 12617, 12618, 12619, 13510, 13511, 13512, 13513, 13514, 13515, 13516, 13517, 13518, 13519, 14410, 14411, 14412, 14413, 14414, 14415, 14416, 14417, 14418, 14419, 15310, 15311, 15312, 15313, 15314, 15315, 15316, 15317, 15318, 15319, 16210, 16211, 16212, 16213, 16214, 16215, 16216, 16217, 16218, 16219, 17110, 17111, 17112, 17113, 17114, 17115, 17116, 17117, 17118, 17119, 18010, 18011, 18012, 18013, 18014, 18015, 18016, 18017, 18018, 18019, 18920, 18921, 18922, 18923, 18924, 18925, 18926, 18927, 18928, 18929, 19820, 19821, 19822, 19823, 19824, 19825, 19826, 19827, 19828, 19829, 20710, 20711, 20712, 20713, 20714, 20715, 20716, 20717, 20718, 20719, 21610, 21611, 21612, 21613, 21614, 21615, 21616, 21617, 21618, 21619, 22510, 22511, 22512, 22513, 22514, 22515, 22516, 22517, 22518, 22519, 23410, 23411, 23412, 23413, 23414, 23415, 23416, 23417, 23418, 23419, 24310, 24311, 24312, 24313, 24314, 24315, 24316, 24317, 24318, 24319, 25210, 25211, 25212, 25213, 25214, 25215, 25216, 25217, 25218, 25219, 26110, 26111, 26112, 26113, 26114, 26115, 26116, 26117, 26118, 26119, 27010, 27011, 27012, 27013, 27014, 27015, 27016, 27017, 27018, 27019, 27920, 27921, 27922, 27923, 27924, 27925, 27926, 27927, 27928, 27929, 28820, 28821, 28822, 28823, 28824, 28825, 28826, 28827, 28828, 28829, 29720, 29721, 29722, 29723, 29724, 29725, 29726, 29727, 29728, 29729, 30610, 30611, 30612, 30613, 30614, 30615, 30616, 30617, 30618, 30619, 31510, 31511, 31512, 31513, 31514, 31515, 31516, 31517, 31518, 31519, 32410, 32411, 32412, 32413, 32414, 32415, 32416, 32417, 32418, 32419, 33310, 33311, 33312, 33313, 33314, 33315, 33316, 33317, 33318, 33319, 34210, 34211, 34212, 34213, 34214, 34215, 34216, 34217, 34218, 34219, 35110, 35111, 35112, 35113, 35114, 35115, 35116, 35117, 35118, 35119, 36010, 36011, 36012, 36013, 36014, 36015, 36016, 36017, 36018, 36019, 36920, 36921, 36922, 36923, 36924, 36925, 36926, 36927, 36928, 36929, 37820, 37821, 37822, 37823, 37824, 37825, 37826, 37827, 37828, 37829, 38720, 38721, 38722, 38723, 38724, 38725, 38726, 38727, 38728, 38729, 39620, 39621, 39622, 39623, 39624, 39625, 39626, 39627, 39628, 39629

A plot of these numbers shows a quite regular pattern. See Figure 1.

Figure 1: permalink

Addendum: I noticed that OEIS A052018: numbers \(k\) with the property that the sum of the digits of \(k\) is a substring of \(k\) is a more general version of what I've just described.

If we consider the product of digits instead and exclude numbers with the digit 0, then only 31 numbers satisfy in the range up to 40000. These are (permalink):

POD = Concatenation of Last Two Digits

236, 315, 324, 612, 1236, 1315, 1324, 1612, 2136, 2312, 3115, 3124, 3212, 6112, 11236, 11315, 11324, 11612, 12136, 12312, 13115, 13124, 13212, 16112, 21136, 21312, 23112, 31115, 31124, 31212, 32112

Figure 2 shows a graph of these numbers.

Figure 2: permalink

Even Numbers as Sums of Three Pronic Numbers

I saw this problem mentioned in a post on math.stackexchange.com:

See if you can prove that every even number is the sum of three pronic numbers (if needed, we say 0 is also pronic) 

I put the problem to Gemini and got the response that "it's impossible to prove that every even number is the sum of three pronic numbers". I wasn't convinced so I consulted Grok. Here was its response.

Every even non-negative integer can indeed be expressed as the sum of three pronic numbers, where pronic numbers are of the form \(n(n+1)\) for non-negative integers \(n \geq 0\) (thus including 0 as the pronic number for \(n=0\)).

To prove this, first recall that a pronic number \(p_n = n(n+1) = 2 \cdot T_n\), where \(T_n = \frac{n(n+1)}{2}\) is the \(n\)th triangular number (with \(T_0 = 0\)). Let \(E\) be an arbitrary even non-negative integer, so \(E = 2m\) for some non-negative integer \(m\). The claim is equivalent to showing that every non-negative integer \(m\) can be written as the sum of three triangular numbers: \(m = T_a + T_b + T_c\) for some non-negative integers \(a, b, c\). This is a known theorem (attributed to Gauss): every non-negative integer is the sum of at most three triangular numbers. The proof relies on a connection to the three squares theorem and modular arithmetic, as follows. Step 1: Relate triangular numbers to squares.

Observe that \(8T_n + 1 = (2n + 1)^2\). Thus, every number of the form \(8T_n + 1\) is an odd perfect square, and conversely, every odd perfect square is of this form for some \(n\). Step 2: Reformulate the problem.

Suppose \(m = T_a + T_b + T_c\). Multiplying by 8 and adding 3 gives:$$ \begin{align} 8m + 3 &= (8T_a + 1) + (8T_b + 1) + (8T_c + 1) \\ &= (2a + 1)^2 + (2b + 1)^2 + (2c + 1)^2 \end{align} $$so \(8m + 3\) is the sum of three odd squares. The converse also holds: if \(8m + 3\) is the sum of three odd squares, then \(m\) is the sum of three triangular numbers.

Thus, it suffices to show that every number of the form \(8m + 3\) (for \(m \geq 0\)) is the sum of three odd squares. Step 3: Use the three squares theorem

Legendre's three squares theorem states that a positive integer can be expressed as the sum of three squares if and only if it is not of the form \(4^k(8l + 7)\) for non-negative integers \(k, l\). Numbers of the form \(8m + 3\) are congruent to 3 modulo 8. Check if such numbers can be of the forbidden form \(4^k(8l + 7)\):

• For \(k = 0\): \(8l + 7 \equiv 7 \pmod{8}\), not 3.
• For \(k = 1\): \(4(8l + 7) = 32l + 28 \equiv 4 \pmod{8}\), not 3.
• For \(k \geq 2\): \(4^k(8l + 7) \equiv 0 \pmod{8}\), not 3.

Since \(8m + 3\) is never of the forbidden form, every such number is the sum of three squares.

Step 4: Ensure the squares are odd

It remains to show these three squares must be odd. Consider sums of three squares modulo 8. Perfect squares modulo 8 are 0, 1, or 4:

• Odd squares: \(\equiv 1 \pmod{8} \)
• Even squares: \( \equiv 0 or 4 \pmod{8} \)

The possible sums modulo 8 are 0, 1, 2, 3, 4, 5, or 6 (never 7). Specifically for \(\equiv 3 \pmod{8} \)

• Three odd squares: \(1 + 1 + 1 = 3\)
• Two odd, one even: \(1 + 1 + 0 = 2\) or \(1 + 1 + 4 = 6\).
• One odd, two even: \(1 + 0 + 0 = 1, 1 + 0 + 4 = 5\), or \(1 + 4 + 4 = 9 \equiv 1\).
• Three even: \(0 + 0 + 0 = 0, 0 + 0 + 4 = 4\), \( 0 + 4 + 4 = 8 \equiv 0\), or \(4 + 4 + 4 = 12 \equiv 4\).

The only combination yielding \(3 \pmod{8} \) is three odd squares. Therefore, since every \(8m + 3\) is the sum of three squares and \(\equiv 3 \pmod{8}\), it must be the sum of three odd squares. This completes the proof. For any even \(E = 2m\), express \(m = T_a + T_b + T_c\), then$$ \begin{align} E &= 2T_a + 2T_b + 2T_c \\ &= p_a + p_b + p_c \end{align} $$where each \(p_i \) is pronic.

This is a nice straightforward proof. I've written about pronic numbers before in posts titled:

• Even Numbers as Sums and Differences of Two Pronic Numbers 
• Revisiting Sums and Differences of Two Pronic Numbers
• My Yearly Pronic Number
• Pronic Pandigital Numbers and Beyond
• Midway Between Cubics and Pronics
• Some Special Pronic Numbers
• Infinite Sums of Reciprocals of Pronic Numbers

A Range Limit of 40000: Why?

In many of my posts when I'm considering sequences, I only look at sequence members whose values do not exceed 40000. Why? The answer to this question relates to the reason that I set this blog up in the first place. It was designed primarily to post about interesting sequences associated with the numbers marking my diurnal age.

If we divide 40000 by 365.2425 (the average number of days in a year) we get slightly more than 109.5 years and there are not many people who live to that ripe old age. Here are milestones, one might say, on the road to oblivion.

Quite a few people won't see 30000 days let alone 40000 but many will and hence the realistic upper limit to the numbers that I normally investigate. The focus of my blog posts is generally the number associated with my diurnal age and the sequences that it can be connected to. For example, today I am \( \textbf{27920} \) days old.

This number found its way into a sequence that I created that involves gapful numbers with the property that not only does the number formed by the concatenation of the first and last digit divide the number but this concatenated number is also the sum of the number's digits. Thus we have:$$ \begin{align} \frac{27920}{20} &=1396 \\ \\ 2 + 7 + 9 + 2 + 0 &= 20 \end{align}$$I described this sequence in a post titled Gapgul Numbers in December of 2024. Interestingly, 27920 also has the property that it has 20 divisors. Only 19 numbers satisfy this additional criterion in the range up to 40000. The conditions to be met are:

• the number is gapful meaning the number formed by concantenating the first and last digits divides the number without remainder
• the sum of the number's digits equals the number formed by the concatenated first and last digits
• the number of divisors of the number equals its sum of digits (and the concatenated number)

Here are the numbers:

1548, 1812, 1908, 10188, 10548, 11268, 12252, 12612, 12708, 13428, 14052, 14412, 15138, 18108, 21984, 26480, 27920, 29360, 39996

Here are the details:

  number        factor          concat   dividend   S0D   divisors

  1548     2^2 * 3^2 * 43         18       86         18    18
  1812     2^2 * 3 * 151          12       151        12    12
  1908     2^2 * 3^2 * 53         18       106        18    18
  10188    2^2 * 3^2 * 283        18       566        18    18
  10548    2^2 * 3^2 * 293        18       586        18    18
  11268    2^2 * 3^2 * 313        18       626        18    18
  12252    2^2 * 3 * 1021         12       1021       12    12
  12612    2^2 * 3 * 1051         12       1051       12    12
  12708    2^2 * 3^2 * 353        18       706        18    18
  13428    2^2 * 3^2 * 373        18       746        18    18
  14052    2^2 * 3 * 1171         12       1171       12    12
  14412    2^2 * 3 * 1201         12       1201       12    12
  15138    2 * 3^2 * 29^2         18       841        18    18
  18108    2^2 * 3^2 * 503        18       1006       18    18
  21984    2^5 * 3 * 229          24       916        24    24
  26480    2^4 * 5 * 331          20       1324       20    20
  27920    2^4 * 5 * 349          20       1396       20    20
  29360    2^4 * 5 * 367          20       1468       20    20
  39996    2^2 * 3^2 * 11 * 101   36       1111       36    36

6-P-6 Primes And Beyond

What I mean by a 6-P-6 prime is a prime, greater than 2,  whose two adjacent composite numbers contain exactly six prime factors with multiplicity. There are only 30 such primes in the range up to 40000 and they are (permalink):

1889, 3079, 4591, 5023, 7649, 12689, 13751, 18089, 19249, 19889, 22193, 22639, 23057, 23311, 23561, 26839, 27919, 28027, 28751, 30449, 30941, 31121, 32993, 33641, 33967, 36251, 38177, 38431, 39799, 39929

Here are the details:

  previous                prime   next

  2^5 * 59                1889    2 * 3^3 * 5 * 7
  2 * 3^4 * 19            3079    2^3 * 5 * 7 * 11
  2 * 3^3 * 5 * 17        4591    2^4 * 7 * 41
  2 * 3^4 * 31            5023    2^5 * 157
  2^5 * 239               7649    2 * 3^2 * 5^2 * 17
  2^4 * 13 * 61           12689   2 * 3^3 * 5 * 47
  2 * 5^4 * 11            13751   2^3 * 3^2 * 191
  2^3 * 7 * 17 * 19       18089   2 * 3^3 * 5 * 67
  2^4 * 3 * 401           19249   2 * 5^3 * 7 * 11
  2^4 * 11 * 113          19889   2 * 3^2 * 5 * 13 * 17
  2^4 * 19 * 73           22193   2 * 3^4 * 137
  2 * 3 * 7^3 * 11        22639   2^4 * 5 * 283
  2^4 * 11 * 131          23057   2 * 3^3 * 7 * 61
  2 * 3^2 * 5 * 7 * 37    23311   2^4 * 31 * 47
  2^3 * 5 * 19 * 31       23561   2 * 3^2 * 7 * 11 * 17
  2 * 3^3 * 7 * 71        26839   2^3 * 5 * 11 * 61
  2 * 3^3 * 11 * 47       27919   2^4 * 5 * 349
  2 * 3^4 * 173           28027   2^2 * 7^2 * 11 * 13
  2 * 5^4 * 23            28751   2^4 * 3 * 599
  2^4 * 11 * 173          30449   2 * 3 * 5^2 * 7 * 29
  2^2 * 5 * 7 * 13 * 17   30941   2 * 3^4 * 191
  2^4 * 5 * 389           31121   2 * 3^2 * 7 * 13 * 19
  2^5 * 1031              32993   2 * 3^3 * 13 * 47
  2^3 * 5 * 29^2          33641   2 * 3^3 * 7 * 89
  2 * 3^3 * 17 * 37       33967   2^4 * 11 * 193
  2 * 5^4 * 29            36251   2^2 * 3^2 * 19 * 53
  2^5 * 1193              38177   2 * 3^3 * 7 * 101
  2 * 3^2 * 5 * 7 * 61    38431   2^5 * 1201
  2 * 3^3 * 11 * 67       39799   2^3 * 5^2 * 199
  2^3 * 7 * 23 * 31       39929   2 * 3 * 5 * 11^3

The algorithm linked to above is easily modified to find 7-P-7 primes that are surrounded by two composite numbers with exactly seven prime factors with multiplicity. There are only four in the range up to 40000 are these are 10529, 15391, 32561 and 35153. The details are (permalink):

  previous            prime   next

  2^5 * 7 * 47        10529   2 * 3^4 * 5 * 13
  2 * 3^4 * 5 * 19    15391   2^5 * 13 * 37
  2^4 * 5 * 11 * 37   32561   2 * 3^5 * 67
  2^4 * 13^3          35153   2 * 3^4 * 7 * 31

There are no 8-P-8 primes in the range up to 40000 but if we extend the range to 100000 we find one (permalink):

  previous         prime   next

  2^6 * 11 * 107   75329   2 * 3^5 * 5 * 31

These number properties of certain primes are not base-dependent. Obviously as the numbers get bigger there will be instances of 9-P-9 primes and beyond.

The Catenary

How do we find the equation for the shape of a hanging rope? Such a shape is called a catenary and this YouTube video does a good job of explaining it. There are notes attached to this video and they can be read by following this link. I also asked Gemini the following question:

Can you create a detailed report on the history and properties of the catenary. You should mention the persons involved in the investigation of its properties and who finally formulated its equation. Any special properties that this curve has and any real world applications should also be discussed.

Here is the report: 

An Expert Report on the History, Properties, and Applications of the Catenary

1. Introduction: The Catenary as a Principle of Form

The catenary, a term derived from the Latin catenaria meaning "chain," is the elegant and profound curve assumed by a perfectly flexible, uniform, and inextensible chain or cable suspended from two points and hanging under its own weight. This seemingly simple form, ubiquitous in the natural and engineered world, is far more than a mere arc; it is a physical manifestation of a fundamental principle: the minimization of gravitational potential energy. A hanging chain, left to its own devices, will naturally settle into the shape that represents the lowest possible energy state, a concept that places the catenary at the heart of the calculus of variations.

This report will serve as a definitive guide to the catenary curve, tracing its journey from a historical problem that perplexed some of the greatest minds of the scientific revolution to its modern-day role as a cornerstone of engineering and architecture. The analysis will progress from the historical narrative of its discovery to a deep exploration of its unique mathematical and physical properties, culminating in a survey of its diverse and often surprising real-world applications.

2. The Historical Pursuit of the "Chain-Curve"

2.1 Galileo's Approximation and Initial Confusion

The quest to understand the "chain-curve" began in earnest in the 17th century, though its properties had been intuitively recognized for millennia. In his 1638 book Two New Sciences, Galileo Galilei was one of the first to formally address the curve's nature. He correctly deduced that the shape of a hanging chain was distinct from a parabola. However, he concluded that it was an approximation of a parabola, observing that the accuracy of this approximation improved as the curve's sag diminished. This initial observation, though not entirely correct, laid the groundwork for a more rigorous investigation by later mathematicians. The fact that the curve was not a parabola was definitively proven posthumously in 1669 by Joachim Jungius.

2.2 The Anagram of Robert Hooke: An Architectural Precedent

While the mathematical community grappled with the problem, the English polymath Robert Hooke arrived at a solution from an architectural and mechanical perspective. In the 1670s, during the rebuilding of St. Paul's Cathedral, Hooke intuitively grasped a fundamental truth about arches.⁷ He announced to the Royal Society in 1671 that he had solved the problem of the optimal arch shape and, in 1675, published an encrypted solution in the form of a Latin anagram: ut pendet continuum flexile, sic stabit contiguum rigidum inversum. The solution, which was revealed posthumously in 1705, translates to "As hangs a flexible cable so, inverted, stand the touching pieces of an arch". This profound statement demonstrated Hooke's understanding that an inverted hanging chain provides the "true mathematical and mechanical form" for a stable arch, a form that directs all forces along the curve itself, eliminating bending moments. This crucial insight predated the derivation of the curve’s explicit mathematical equation, highlighting an elegant synergy between empirical observation and structural theory.

2.3 The Triumvirate of Solutions (1691)

The challenge of defining the catenary with a precise equation became a major point of interest with the advent of the new infinitesimal calculus. The pivotal moment arrived in 1691 when Jakob Bernoulli, a prominent Swiss mathematician, publicly posed the problem of finding the equation for the "chain-curve". This challenge was not merely a physical curiosity; it was a foundational test designed to demonstrate the power of calculus, a new and revolutionary mathematical language. The problem served as a proving ground, inviting the world's leading minds to showcase the utility and elegance of the infinitesimal methods championed by Gottfried Leibniz and Isaac Newton. The successful resolution of the problem was a powerful validation of calculus as the definitive language for describing complex physical phenomena.

In a remarkable display of intellectual prowess, three of the era's greatest mathematicians—Gottfried Leibniz, Christiaan Huygens, and Jakob's brother, Johann Bernoulli—derived the equation independently and nearly simultaneously. Their solutions were published in the prestigious journal

Acta Eruditorum in June 1691. Huygens, in a letter to Leibniz in 1690, was the first to formally use the term "catenary," which has been the standard nomenclature ever since.

Leibniz's approach to the problem was particularly revealing of the intellectual tensions of the time. While he used calculus to arrive at his solution, his published work presented the final answer not as a modern analytical formula but as a classical Euclidean construction. He chose to conceal the derivation itself, revealing it only in a private letter. This paradoxical presentation reflects the shift from the traditional geometric canon to the more abstract and powerful methods of calculus, a new paradigm that was not yet universally accepted as the sole basis for rigorous proof. The fact that the problem, once deemed unsolvable by traditional means, was conquered by three separate mathematicians using the same new tool cemented calculus's place in mathematical history and established its authority as the primary language for describing the mechanics of the physical world.

3. The Mathematical and Physical Foundations of the Catenary

3.1 Derivation from First Principles

The elegant simplicity of the catenary's form belies the sophisticated calculus required for its derivation. The equation is a direct consequence of balancing the forces acting on a segment of a hanging chain that is in static equilibrium. Consider a chain segment starting from the lowest point (the apex) of the curve, where the tension is purely horizontal ( \( T_0 \)​ ), and extending to an arbitrary point \( P \). The forces acting on this segment are the horizontal tension at the apex, the tension at point \( P \) (which is tangent to the curve), and the downward force of gravity.

By splitting the forces into horizontal and vertical components, it can be shown that the horizontal tension component remains constant throughout the chain, equaling the tension at the apex ( \(T \cos \theta=T_0 \) ​). The vertical component of tension ( \( T\sin \theta \) ) must balance the weight of the chain segment, which is proportional to its arc length, \( s \). This physical balance of forces leads to a first-order differential equation that relates the slope of the curve to its arc length. The solution of this differential equation, which requires a substitution involving hyperbolic functions, yields the final equation of the catenary.

3.2 The Hyperbolic Cosine and its Properties

The standard Cartesian equation for the catenary, when centered on the y-axis, is given by \( y=a \cosh(x/a) \). This equation is defined in terms of the hyperbolic cosine function, cosh, which is itself a simple combination of exponential functions: \( \cosh(x)=(e^x+e^{−x})/2 \).

The appearance of this transcendental function in the solution to a simple physical problem is not a coincidence; it is a profound demonstration of the deep connections between pure mathematics and the physical world. The curve of a freely hanging chain is a physical manifestation of this abstract function, providing a powerful example of how mathematical forms govern the fundamental behavior of nature. This link provides a tangible bridge between the abstract realm of mathematics and the observable phenomena of the natural world.

3.3 Special Properties of the Catenary

Beyond its defining equation, the catenary possesses a number of unique and fascinating geometric properties:

• Arc Length and Slope: The slope of the catenary at any point is directly proportional to its arc length measured from the lowest point.

• Evolute and the Tractrix: The catenary is the evolute of a related curve known as the tractrix. This means that the catenary is the locus of the centers of curvature of the tractrix curve.

• The Catenoid: A Minimal Surface: One of the most remarkable properties of the catenary is its relationship to the catenoid. A catenoid is a three-dimensional surface of revolution formed by rotating a catenary curve around its directrix, which is its lowest, horizontal axis. The catenoid holds the distinction of being the first non-trivial minimal surface discovered after the plane, a fact proved by Leonhard Euler in 1744.

A minimal surface is a shape that occupies the least possible area for a given boundary. A classic physical demonstration of this principle is a soap film stretched between two coaxial circular rings. This soap film naturally forms the shape of a catenoid because it minimizes its surface energy, which is directly proportional to its surface area. This physical phenomenon offers a tangible parallel to the catenary itself, which minimizes its gravitational potential energy. The fact that both the catenary and the catenoid are solutions to a minimization problem—gravitational potential energy for the curve and surface area for the surface—reveals a powerful, unifying principle. It signifies that the catenary is not an isolated mathematical curiosity but a representative of a broader class of forms that embody optimal, stable states in nature, governed by the principles of the calculus of variations.

4. The Catenary and the Parabola: A Critical Distinction

4.1 The Visual Illusion and Physical Disparity

A common point of confusion is the visual similarity between the catenary and the parabola. This visual illusion, especially for curves with a small sag-to-span ratio, contributed to Galileo's initial misidentification and continues to be a source of error. However, the two curves are fundamentally different, both in their governing physics and their mathematical form.The distinction lies in the distribution of the load or weight. A catenary is a curve of uniform weight per unit of its arc length. This is the physical model for a simple, free-hanging chain whose weight is evenly distributed along its physical form. A parabola, on the other hand, is the curve of uniform weight per unit of horizontal distance. This physical model applies to the main cables of a suspension bridge, which bear the weight of a flat roadway deck that is distributed evenly along the horizontal axis.

4.2 The Mathematical and Practical Differences

The physical disparity is reflected in the curves' defining equations. The parabola is described by a simple quadratic equation, typically \(y=kx^2 \), while the catenary is governed by the hyperbolic cosine function, \(y=a\cosh(x/a) \). This difference is most apparent over long spans, where the catenary’s exponential growth causes its “arms” to rise far more steeply than the parabola’s quadratic growth.

The frequent confusion between these two curves underscores a critical principle in applied science: the mathematical model must accurately reflect the underlying physical assumptions. A seemingly minor difference in how weight is distributed (per unit length vs. per unit horizontal distance) leads to two fundamentally different mathematical solutions. An engineer who misapplies the simpler parabolic formula to a system governed by catenary principles (or vice-versa) risks structural failure. The distinction is not an academic nicety but a matter of structural integrity and safety.

The table below provides a clear, side-by-side comparison of the two curves.

Characteristic	Catenary	Parabola
Defining Equation	\(y=a\cosh(x/a) \)	\(y=kx^2\)
Governing Physical Principle	Minimum Potential Energy	Static Equilibrium (Uniform Horizontal Load)
Load Distribution	Uniform weight per unit of arc length	Uniform weight per unit of horizontal distance
Visual Distinction (at long spans)	Exponential growth (steeper ends)	Quadratic growth (less steep ends)
Typical Applications	Free-hanging chains, power lines, simple suspension bridges	Projectile trajectories, suspension bridge cables with flat decks

5. Real-World Applications Across Disciplines

5.1 Structural Architecture

The catenary principle is widely applied in architecture and engineering to create structures that are both aesthetically pleasing and exceptionally strong. The most famous application is the inverted catenary arch. Inverting the shape of a hanging chain transfers its uniform tensile forces into uniform compressive forces, making it the ideal form for a compression-only structure.³ This shape naturally directs all forces along the curve, eliminating bending moments and making the structure inherently stable and durable.

• The St. Louis Gateway Arch: The iconic Gateway Arch in St. Louis, Missouri, designed by architect Eero Saarinen, is a prominent example of this principle. While often referred to as a simple catenary, it is technically a "weighted catenary". This sophisticated variation was chosen to account for the arch's varying thickness and load distribution, resulting in a slightly flatter curve that is both structurally efficient and visually graceful.

• The Genius of Antoni Gaudí: The Spanish Catalan architect Antoni Gaudí was a master of the catenary principle. He pioneered the "funicular method" in his designs, such as the Church of Colònia Güell and the Sagrada Família. He built elaborate, upside-down scale models using weighted strings and chains to physically determine the optimal, compression-only forms for his complex, organic structures. By changing the length, weight, or anchor points of the chains, he could instantly "recompute" a new, structurally sound geometry. This process, governed by the immutable laws of physics, was an early form of what is now known as parametric design, allowing him to create his fluid forms while ensuring their inherent stability. This ingenuity showcases a continuous lineage of human design that links physical principles to modern computational paradigms.

5.2 Civil and Marine Engineering

The catenary is a critical component in various civil and marine engineering applications. While modern suspension bridge cables are typically parabolic due to the flat roadway deck, the catenary remains the true form for simple, non-loaded suspension bridges where the roadway follows the cable. This also holds true for the sag of overhead power and telephone lines, where the cables hang under their own weight.

In the marine and offshore industries, the catenary curve is essential for the design of steel catenary risers, which are pipelines that connect oil platforms to the seabed. The curve’s shape provides stability and flexibility in the face of currents and platform movement. Similarly, the slack of a marine mooring line forms a catenary, which enhances anchor holding power by lowering the angle of pull on the anchor.

The table below summarizes some of the key applications of the catenary principle.

Structure/Application	Location	Designer	Catenary Principle Applied
St. Louis Gateway Arch	St. Louis, Missouri	Eero Saarinen	Weighted catenary arch to account for variable thickness
Casa Milà	Barcelona, Spain	Antoni Gaudí	Catenary arches for roof supports in the attic
Church of Colònia Güell	Santa Coloma de Cervelló, Spain	Antoni Gaudí	Used funicular models of hanging chains to determine structural forms
Kiln Arches	General Application	Various	Inverted catenary arch for structural efficiency and stability
Steel Catenary Risers	Offshore Oil Platforms	Various	Catenary shape for pipeline stability and flexibility

6. Conclusion: Synthesis and Future Directions

The journey of the catenary, from a source of confusion for Galileo to a foundational principle of modern science, is a perfect microcosm of how a simple physical observation can catalyze profound human innovation. The problem of the "chain-curve" served as a crucible for the new calculus, providing a public forum for Leibniz, Huygens, and the Bernoullis to demonstrate its power. The elegant appearance of the hyperbolic cosine in the solution reveals a deep and direct correspondence between abstract mathematical functions and the physical reality of our world.

Furthermore, the catenary’s ability to solve a minimization problem—whether for gravitational potential energy in a curve or surface area in a catenoid—establishes it as a representative of a broader class of forms that embody optimal, stable states in nature. The principle of the inverted catenary arch, intuitively understood by Robert Hooke and masterfully applied by Antoni Gaudí, continues to inform structural design today, a testament to its timeless efficiency. The catenary represents a bridge between theory and practice, a timeless example of how elegant physical principles can inform and shape our world.