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