Fermat Polygonal Number Theorem

In a recent post, I discussed centered tetrahedral numbers and so a recent Numberphile video naturally attracted my attention with the title 343867 and Tetrahedral Numbers. It turns out that there is a conjecture that every number can be written as the sum of at most five tetrahedral numbers and 343867 is the first number to require all five tetrahedral numbers. However, it can be represented by five tetrahedral numbers in 322 different ways. See Figure 1 where three of these ways are shown.

Figure 1: link

It should be borne in mind that the tetrahedral numbers are given by the formula:$$T_n=\frac{n \cdot (n+1) \cdot (n+2)}{6}$$The initial members are:

1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540, 1771, 2024, 2300, 2600, 2925, 3276, 3654, 4060, 4495, 4960, 5456, 5984, 6545, 7140, 7770, 8436, 9139, 9880, 10660, 11480, 12341, 13244, 14190, 15180, 16215, 17296, 18424, 19600, 20825, 22100, 23426, 24804, 26235, 27720, 29260, 30856, 32509, 34220, 35990, 37820, 39711, 41664, 43680, 45760, 47905, 50116, 52394, 54740, 57155, 59640, 62196, 64824, 67525, 70300, 73150, 76076, 79079, 82160, 85320, 88560, 91881, 95284, 98770, 102340, 105995, 109736, 113564, 117480, 121485, 125580, 129766, 134044, 138415, 142880, 147440, 152096, 156849, 161700, 166650, 171700

Another interesting fact was mentioned in the video, namely that any number \(n\) can be represented by at most \(n\)-gonal numbers where \(n \gt= 3\). This means that any number can be represented by at most three 3-gonal (triangular) numbers, any number can be represented by at most four 4-gonal (square) numbers, any number can be represented by at most five 5-gonal (pentagonal) numbers and so forth.

This theorem is named the Fermat polygonal number theorem with the following details from Wikipedia:

The theorem is named after Pierre de Fermat, who stated it, in 1638, without proof, promising to write it in a separate work that never appeared. Joseph Louis Lagrange proved the square case in 1770, which states that every positive number can be represented as a sum of four squares, for example, 7 = 4 + 1 + 1 + 1. Gauss proved the triangular case in 1796, commemorating the occasion by writing in his diary the line "ΕΥΡΗΚΑ! num = Δ + Δ + Δ", and published a proof in his book Disquisitiones Arithmeticae. For this reason, Gauss's result is sometimes known as the Eureka theorem. The full polygonal number theorem was not resolved until it was finally proven by Cauchy in 1813.

Palindromic Day 27272

It's a palindromic day again, which happens every one hundred days during my current millennium (27000 to 27999). The number associated with my diurnal age today (27272) has a connection to triangular numbers, a topic that I wrote about in two recent posts titled Happy Triangular Numbers on the 22nd November 2023 and Four Fun Facts About Triangular Numbers on the 25th November 2023.

The connection of 27272 to triangular numbers arises via OEIS A340953:

A340953

Number of ways to write \(n\) as an ordered sum of eight nonzero triangular numbers.

So when \(n=55\) it turns out that it can be written as an ordered sum of eight nonzero triangular numbers in 27272 different ways. The triangular numbers less or equal to 55 are as follows:$$1, 3, 6, 10, 15, 21, 28, 36, 45, 55$$Using these numbers, and only these number, one possible sum would be:$$3 + 3 + 6 + 6 + 6 + 6 +10 + 15 = 55$$What makes the number of possibilities so high is that the order of the terms in the sum is being taken into account.

The initial members of the sequence are (permalink):

1, 0, 8, 0, 28, 8, 56, 56, 70, 176, 84, 336, 196, 448, 492, 504, 953, 616, 1456, 960, 1814, 1792, 1904, 3032, 2100, 4144, 3052, 4768, 4670, 5264, 6720, 5936, 8876, 7112, 10620, 9648, 11718, 12720, 13216, 15960, 15261, 19608, 17164, 23296, 21226, 25424, 26796, 27272, 32844, 30480, 38640, 34160, 43512

Take the case of \(n=10\). There are eight possible ordered sums and they are:$$ \begin{align} 1+1+1+1+1+1+1+3 =55\\1+1+1+1+1+1+3+1 =55\\1+1+1+1+1+3+1+1 =55\\1+1+1+1+3+1+1+1 =55\\1+1+1+3+1+1+1+1 =55\\1+1+3+1+1+1+1+1 =55\\1+3+1+1+1+1+1+1 =55\\3+1+1+1+1+1+1+3 =55 \end{align}$$Another property of 27272 that is not immediately obvious is that it stands in the middle of a run of so-called iban numbers. These are numbers that do not contain the letter "i" when written using the letters of the alphabet. The run is:$$27270, 27271, 27272, 27273, 27274$$I've written about these in my post titled Iban Numbers on July 31st 2023. In the case of 27272, it is written as:

Two Thousand Two Hundred Seventy Two

What other interesting properties does 27272 have? Well, for one, it belongs to a sequence of composite numbers whose arithmetic derivatives have no digits in common with the originating number. $$ \begin{align} 27272 &= 2 \times 2 \times 2 \times 7 \times 487 \\ 27272' &= 44860 \end{align}$$Here is a permalink to the calculation. 27272 is also what is called a nude number because it divisible by every one of its digits. However, if we look more closely, we see that the number is also divisible by every one of the digits in its prime factors (2, 4, 7 and 8).$$27272=2^3 * 7 * 487$$There are only 337 such numbers in the range up to 40,000 (none seem to end in 3, 7 or 9 except for the initial single digit 9) and they are:

1, 4, 6, 8, 9, 12, 15, 22, 24, 33, 36, 44, 48, 55, 66, 77, 88, 99, 112, 126, 128, 132, 135, 144, 162, 168, 175, 216, 224, 264, 288, 312, 315, 324, 336, 366, 384, 396, 432, 448, 624, 648, 672, 735, 777, 784, 864, 936, 1116, 1155, 1176, 1197, 1248, 1266, 1296, 1344, 1368, 1395, 1448, 1464, 1488, 1575, 1715, 1764, 1848, 1944, 2112, 2184, 2196, 2232, 2248, 2688, 2744, 2772, 2916, 3132, 3144, 3168, 3276, 3312, 3432, 3444, 3612, 3864, 3888, 4116, 4128, 4212, 4224, 4344, 4368, 4392, 4416, 4464, 4644, 4968, 5115, 5355, 5775, 6132, 6144, 6192, 6216, 6264, 6288, 6312, 6336, 6624, 6696, 6762, 6864, 6888, 6912, 7112, 7119, 7224, 7266, 7371, 7644, 7728, 8112, 8136, 8184, 8232, 8424, 8448, 8688, 8736, 8832, 8928, 9126, 9216, 9288, 9324, 9396, 9432, 9936, 11112, 11115, 11184, 11196, 11232, 11316, 11424, 11616, 11664, 11848, 11916, 12144, 12168, 12222, 12264, 12288, 12312, 12366, 12384, 12432, 12624, 12636, 12666, 12712, 12816, 12996, 13122, 13248, 13326, 13377, 13392, 13416, 13488, 13662, 13755, 13776, 13797, 13824, 13896, 13932, 13995, 14112, 14224, 14328, 14364, 14448, 14488, 14616, 14784, 16128, 16164, 16224, 16236, 16332, 16368, 16416, 16464, 16488, 16632, 16848, 17136, 17199, 17248, 17262, 17472, 17724, 17955, 18144, 18216, 18248, 18384, 18424, 18432, 18648, 18816, 18864, 18936, 19224, 19368, 19719, 19971, 21144, 21168, 21222, 21264, 21336, 21384, 21492, 21648, 21672, 21888, 21924, 22122, 22128, 22176, 22224, 22326, 22368, 22392, 22464, 22632, 22764, 22848, 22932, 22968, 23136, 23166, 23184, 23232, 23322, 23328, 23424, 23436, 23616, 23688, 23832, 24192, 24276, 24288, 24336, 24444, 24624, 24696, 24864, 26124, 26136, 26244, 26364, 26496, 26832, 27216, 27272, 27384, 27636, 27762, 27888, 27972, 28224, 28296, 28344, 28448, 28728, 29232, 29412, 31122, 31248, 31266, 31311, 31332, 31416, 31464, 31488, 31644, 31896, 32112, 32184, 32292, 32328, 32364, 32448, 32664, 32832, 33144, 33192, 33222, 33264, 33336, 33444, 33696, 33726, 33768, 34224, 34272, 34416, 34776, 34848, 34992, 35595, 36126, 36288, 36372, 36432, 36612, 36666, 36792, 36864, 36936, 37128, 37212, 37296, 37317, 37464, 37632, 38136, 38448, 38688, 39312, 39366, 39375, 39816

Four Fun Facts About Triangular Numbers

In my previous post (Happy Triangular Numbers) I made reference to a website Fascinating Triangular Numbers and this post I'd like to mention just four more of the "fun facts" mentioned there.

FUN FACT 1: 

The sum of two consecutive triangular numbers is a square number. This is easily proven as follow:$$ \begin{align} T_n+T_{n+1} &= \frac{n(n+1)}{2}+ \frac{(n+1)(n+2)}{2}\\ &= \frac{n+1}{2} \cdot (2n+2)\\ &= (n+1)^2 \end{align} $$FUN FACT 2:

The sum of the squares of two consecutive triangular numbers is also a triangular number. Again this is easily proven as follows:$$ \begin{align} \big (T_n \big )^2+ \big (T_{n+1} \big )^2&= \big (\frac{n(n+1)}{2} \big )^2+ \big ( \frac{(n+1)(n+2)}{2} \big )^2\\ &= \Big (\frac{n+1}{2} \Big )^2 \cdot \Big ( n^2+(n+2)^2 \Big )\\ &= \Big (\frac{n+1}{2} \Big )^2 \cdot \Big ( 2n^2+4n+4 \Big ) \\ &= \frac {(n^2 +2n+1) \cdot (n^2+2n+2)}{2} \\ &=T_{(n+1)^2} \end{align} $$FUN FACT 3:

There are infinitely many triangular numbers, which are also squares as given by the series 1, 36, 1225, 41616, 1413721, 48024900, 1631432881, 55420693056 etc. These can be termed as square triangular numbers. The \(n\)th Square Triangular number \(K_n\) can easily be obtained from the recursive formula: $$K_n = 34 \times K_{n-1} - K_{n-2} + 2$$So knowing the first two square triangular numbers i.e. \(K_1 = 1\) and \(K_2 = 36\) , all other successive Square Triangular numbers can be obtained. For example:$$ \begin{align} K_3 &= 34 \times K_2 - K_1 + 2 \\ &= 34 \times 36 -1 + 2 \\ &= 1225 \\ &= 35^2 \\ K_4 &= 34 \times K_3 - K_2 + 2 \\ &= 34 \times 1225 - 36 + 2 \\ &= 41616 \\ &=204^2 \end{align}$$FUN FACT 4:

There exist infinite triangular numbers that are simultaneously the sum, the difference and the product of two other triangular numbers. Here is a list of the initial such numbers:$$ \begin {align} 990 &= 1035 - 45 = 780 + 210 = 66 \times 15\\

1540 &= 1711 - 171 = 1485 + 55 = 55 \times 28\\

2850 &= 3003 - 153 = 2415 + 435 = 190 \times 15\\

4851 &= 5151 - 300 = 3081 + 1770 = 231 \times 21\\

8778 &= 10731 - 1953 = 7875 + 903 = 2926 \times 3\\

11781 &= 12246 - 465 = 11628 +153 = 561 \times 21\\

15400 &= 18721 - 3321 = 14365 + 1035 =1540 \times 10\\

26796 &= 27261 - 465 = 26565 + 231 = 406 \times 66\\

43956 &= 44551 - 595 = 41328 + 2628 = 666 \times 66 \end{align} $$Here are the initial triangular numbers along with their associated indices (permalink):

[(1, 1), (3, 2), (6, 3), (10, 4), (15, 5), (21, 6), (28, 7), (36, 8), (45, 9), (55, 10), (66, 11), (78, 12), (91, 13), (105, 14), (120, 15), (136, 16), (153, 17), (171, 18), (190, 19), (210, 20), (231, 21), (253, 22), (276, 23), (300, 24), (325, 25), (351, 26), (378, 27), (406, 28), (435, 29), (465, 30), (496, 31), (528, 32), (561, 33), (595, 34), (630, 35), (666, 36), (703, 37), (741, 38), (780, 39), (820, 40), (861, 41), (903, 42), (946, 43), (990, 44), (1035, 45), (1081, 46), (1128, 47), (1176, 48), (1225, 49), (1275, 50), (1326, 51), (1378, 52), (1431, 53), (1485, 54), (1540, 55), (1596, 56), (1653, 57), (1711, 58), (1770, 59), (1830, 60), (1891, 61), (1953, 62), (2016, 63), (2080, 64), (2145, 65), (2211, 66), (2278, 67), (2346, 68), (2415, 69), (2485, 70), (2556, 71), (2628, 72), (2701, 73), (2775, 74), (2850, 75), (2926, 76), (3003, 77), (3081, 78), (3160, 79), (3240, 80), (3321, 81), (3403, 82), (3486, 83), (3570, 84), (3655, 85), (3741, 86), (3828, 87), (3916, 88), (4005, 89), (4095, 90), (4186, 91), (4278, 92), (4371, 93), (4465, 94), (4560, 95), (4656, 96), (4753, 97), (4851, 98), (4950, 99), (5050, 100), (5151, 101), (5253, 102), (5356, 103), (5460, 104), (5565, 105), (5671, 106), (5778, 107), (5886, 108), (5995, 109), (6105, 110), (6216, 111), (6328, 112), (6441, 113), (6555, 114), (6670, 115), (6786, 116), (6903, 117), (7021, 118), (7140, 119), (7260, 120), (7381, 121), (7503, 122), (7626, 123), (7750, 124), (7875, 125), (8001, 126), (8128, 127), (8256, 128), (8385, 129), (8515, 130), (8646, 131), (8778, 132), (8911, 133), (9045, 134), (9180, 135), (9316, 136), (9453, 137), (9591, 138), (9730, 139), (9870, 140), (10011, 141), (10153, 142), (10296, 143), (10440, 144), (10585, 145), (10731, 146), (10878, 147), (11026, 148), (11175, 149), (11325, 150), (11476, 151), (11628, 152), (11781, 153), (11935, 154), (12090, 155), (12246, 156), (12403, 157), (12561, 158), (12720, 159), (12880, 160), (13041, 161), (13203, 162), (13366, 163), (13530, 164), (13695, 165), (13861, 166), (14028, 167), (14196, 168), (14365, 169), (14535, 170), (14706, 171), (14878, 172), (15051, 173), (15225, 174), (15400, 175), (15576, 176), (15753, 177), (15931, 178), (16110, 179), (16290, 180), (16471, 181), (16653, 182), (16836, 183), (17020, 184), (17205, 185), (17391, 186), (17578, 187), (17766, 188), (17955, 189), (18145, 190), (18336, 191), (18528, 192), (18721, 193), (18915, 194), (19110, 195), (19306, 196), (19503, 197), (19701, 198), (19900, 199), (20100, 200), (20301, 201), (20503, 202), (20706, 203), (20910, 204), (21115, 205), (21321, 206), (21528, 207), (21736, 208), (21945, 209), (22155, 210), (22366, 211), (22578, 212), (22791, 213), (23005, 214), (23220, 215), (23436, 216), (23653, 217), (23871, 218), (24090, 219), (24310, 220), (24531, 221), (24753, 222), (24976, 223), (25200, 224), (25425, 225), (25651, 226), (25878, 227), (26106, 228), (26335, 229), (26565, 230), (26796, 231), (27028, 232), (27261, 233), (27495, 234), (27730, 235), (27966, 236), (28203, 237), (28441, 238), (28680, 239), (28920, 240), (29161, 241), (29403, 242), (29646, 243), (29890, 244), (30135, 245), (30381, 246), (30628, 247), (30876, 248), (31125, 249), (31375, 250), (31626, 251), (31878, 252), (32131, 253), (32385, 254), (32640, 255), (32896, 256), (33153, 257), (33411, 258), (33670, 259), (33930, 260), (34191, 261), (34453, 262), (34716, 263), (34980, 264), (35245, 265), (35511, 266), (35778, 267), (36046, 268), (36315, 269), (36585, 270), (36856, 271), (37128, 272), (37401, 273), (37675, 274), (37950, 275), (38226, 276), (38503, 277), (38781, 278), (39060, 279), (39340, 280), (39621, 281), (39903, 282), (40186, 283), (40470, 284), (40755, 285), (41041, 286), (41328, 287), (41616, 288), (41905, 289), (42195, 290), (42486, 291), (42778, 292), (43071, 293), (43365, 294), (43660, 295), (43956, 296), (44253, 297), (44551, 298), (44850, 299), (45150, 300)]

The results for FUN FACT 4 could be written in terms of these indices. For example:$$ \begin {align} 990 &= 1035 - 45 = 780 + 210 = 66 \times 15\\ \text{T}_{44} &= \text{T}_{45} -\text{T}_{9} = \text{T}_{39} + \text{T}_{20} = \text{T}_{11} \times \text{T}_{5} \end{align}$$

Happy Triangular Numbers

I have to confess to treating triangular numbers with some complacency over the years. Let's recall that triangular numbers are of the form:$$ \frac{n \, (n+1)}{2} \text{ with } n \geq 1$$The first triangular numbers are:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431

Figure 1: Triangular numbers appear in Pascal's Triangle.
In fact 3rd diagonal of Pascal's Triangle, gives all triangular numbers as shown

Similarly I've grown rather complacent about happy numbers that have the property that repeated iterations of the sum of digits squared lead to 1. For example:$$ 27261 \rightarrow 94, 97, 130, 10, 1$$The reason that I chose 27261 is that it's the number associated with my diurnal age today.

It turns out however, that numbers that are both triangular and happy are rather rare. These types of numbers comprise OEIS A076712 and the initial members of the sequence are:

1, 10, 28, 91, 190, 496, 820, 946, 1128, 1275, 2080, 2211, 2485, 3321, 4278, 8128, 8256, 8778, 9591, 9730, 11476, 12090, 12880, 13203, 13366, 13530, 15753, 16471, 17205, 17578, 20910, 21115, 21321, 22791, 24753, 25651, 27261, 29890, 30135, 31626, 33670, 35245

As can be seen, it will be quite some time before I meet the next such number: 29890. The main reason for creating this post was to draw attention to an interesting website titled Fascinating Triangular Numbers run by Shyam Sunder Gupta. It was begun on October 26th 2002 but remains active. There is a plethora of information on this site about various properties of triangular numbers so it's a wonderful resource.

What follows are just two examples from the site:

Example 1: The only known example of a Pythagorean triangle (\(a, b, c\)) where \(a, b, c\) are triangular numbers is (8778, 10296, 13530): $$ \begin{align} 8778^2 + 10296^2 &= 13530^2 \\ (\text{T}_{132})^2 + (\text{T}_{143})^2 &= (\text{T}_{164})^2 \end{align} $$Example 2: The only known examples of a Pythagorean triangle such that both Perimeter as well as Area are triangular numbers are: $$ \begin{align} &3312, 14091, 14475) \\ \text{with Perimeter } &= 31878 = T_{252} \\ \text{and Area } &= 23334696 = T_{6831}\\  \\&3405996, 8013265, 8707079 \\ \text{with Perimeter } &= 20126340 = T_{6344} \\ \text{and Area } &= 13646574268470 = T_{5224284} \end{align} $$

Natural Numbers associated with the Eisenstein Integers

 My number associated with my diurnal age today, 26908, is a member of OEIS A118886:

A118886

Numbers expressible as \(x^2 + x \cdot y + y^2, 0 \leq x \leq y \), in two or more ways.

In the case of 26908, there are three ways viz. 

• \(x=4 \) and \(y=162\) so that \(4^2+4 \cdot 162+162^2=26908\)
• \(x=62\) and \(y=124\) so that \(62^2+62 \cdot 124+124^2=26908\)
• \(x=74\) and \(y=114\) so that \(74^2+74 \cdot 114+114^2=26908\)

As stated in the OEIS comments, the numbers in this sequence represent squares of distances between two points in the triangular lattice in two or more non-trivially different ways. The triangular lattice being referred here is that formed by the so-called Eisenstein integers. Such integers are complex numbers of the form:$$z=a+ b \omega\\ \text{ where } a \text{ and } b \text{ are integers and }\\ \omega=\frac{-1+i \sqrt{3}}{2}=e^{2\pi i/3}$$Thus we have:$$a^2+ab+b^2=(a-\omega)(b-\omega^2) \text{ because }1+\omega+\omega^2=0$$Numbers of the form \(a^2-ab+b^2 \) represent the squared modulus of an Eisenstein integer:

\(|a+b\omega|^2=(a-\frac{1}{2}b)^2+\frac{3}{4}b^2=a^2-ab+b^2\)

Figure 1: from Wikipedia

Thus both numbers of the form \(a^2+ab+b^2 \) and \(a^2-ab+b^2 \) are associated with Eisenstein integers. Looking at the latter type of number we can write:$$ \begin{align} a^2+ab+b^2 &= a^2-2ab+b^2+3ab\\&=(a-b)^2+(\sqrt{3ab})^2\\&=(a-b-i\sqrt{3ab})(a-b+\sqrt{3ab}) \\&=(a-b-i\sqrt{3} \sqrt{ab})(a-b+i\sqrt{3} \sqrt{ab}) \\ &=(a-b-(2\omega+1)\sqrt{ab})(a-b+(2\omega+1)\sqrt{ab}) \\&=(a-b-\sqrt{ab}-2\sqrt{ab} \cdot \omega)(a-b+\sqrt{ab}+2\sqrt{ab} \cdot \omega) \\&=(c_1-d\omega)(c_2+d\omega)\end{align}$$which represents the product of two Eisenstein integers which could be evaluated for 26908 since we know three different sets of values for \(a\) and \(b\). For example, for the case of \(a=4 \) and \(b=162\), we have \( \sqrt{ab}=18\sqrt{2}\). There's a lot more that could be said here of course but the important takeaway is the association with the Eisenstein integers.

Getting back to the original OEIS A118886, we find that there are numbers that are of the form \(x^2+xy+y^2\) in eight or more ways. Such numbers include 53599, 63973, 74347, 84721 and 105469. Let's take the last number as an example. The \(x,y\) values associated with this number are (15, 317), (33, 307), (53, 295), (100, 263), (108, 257), (145, 227), (153, 220) and (187, 188).

Using a Jupyter notebook, I was able to find these numbers, in the range up to a little beyond one million, that can be represented in nine or more ways:

157339, 229957, 256711, 306397, 356083, 375193, 427063, 447811, 472017, 505141, 520429, 548359, 554827, 593047, 604513, 612157, 629356, 654199, 654493, 689871, 696787, 730639, 738283, 760627, 770133, 803257, 810901, 831649, 849121, 852943, 872053, 883519, 894691, 902629, 919191, 919828, 956137, 966511, 967603, 1013467, 1018381, 1026844, 1067857, 1101373, 1125579, 1173991, 1204567

Here is a permalink to the code being used but it will time out if run in SageMathCell. Further investigation yielded the numbers shown below that can be represented in ten or more ways. However, now that the list was considerably shorter, it was possible to test each of these numbers to see how many representations are possible and these numbers are shown in brackets after each number. As can be seen, the maximum is twelve in the range up to a little over one million.

375193 (12), 447811 (12), 520429 (12), 593047 (10), 696787 (12), 730639 (12), 738283 (12), 810901 (12), 831649 (12), 849121 (12), 883519 (12), 956137 (12), 966511 (12), 967603 (10), 1013467 (12), 1018381 (12), 1067857 (12), 1101373 (12), 1125579 (12)

SageMathCell makes it easy to determine the number of possible pairs for each number. It's just necessary to tset up the equation. Let's use 1125579 as an example:$$x^2 + x \cdot y + y^2 =1125579$$Here is a permalink to the code that solves this Diophantine equation and the output is:

Figure 2 shows the graph of this elliptic function and there are exactly twenty four points on this graph with coordinates that are integers. Why 24 when we have only 12 above? Well, a restriction had been applied such that \(x \leq y\) but no such restriction need apply for this ellipse. The case of A = (422, 785) and B = (785, 422) is shown below.

Figure 2: using GeoGebra

My Yearly Pronic Number

Pronic numbers are numbers of the form \(n \times (n+1) \) where \(n\) is an integer \( \geq 1\). Thus the first such number is 2. Here are the pronic numbers up to 40,000:

2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462, 506, 552, 600, 650, 702, 756, 812, 870, 930, 992, 1056, 1122, 1190, 1260, 1332, 1406, 1482, 1560, 1640, 1722, 1806, 1892, 1980, 2070, 2162, 2256, 2352, 2450, 2550, 2652, 2756, 2862, 2970, 3080, 3192, 3306, 3422, 3540, 3660, 3782, 3906, 4032, 4160, 4290, 4422, 4556, 4692, 4830, 4970, 5112, 5256, 5402, 5550, 5700, 5852, 6006, 6162, 6320, 6480, 6642, 6806, 6972, 7140, 7310, 7482, 7656, 7832, 8010, 8190, 8372, 8556, 8742, 8930, 9120, 9312, 9506, 9702, 9900, 10100, 10302, 10506, 10712, 10920, 11130, 11342, 11556, 11772, 11990, 12210, 12432, 12656, 12882, 13110, 13340, 13572, 13806, 14042, 14280, 14520, 14762, 15006, 15252, 15500, 15750, 16002, 16256, 16512, 16770, 17030, 17292, 17556, 17822, 18090, 18360, 18632, 18906, 19182, 19460, 19740, 20022, 20306, 20592, 20880, 21170, 21462, 21756, 22052, 22350, 22650, 22952, 23256, 23562, 23870, 24180, 24492, 24806, 25122, 25440, 25760, 26082, 26406, 26732, 27060, 27390, 27722, 28056, 28392, 28730, 29070, 29412, 29756, 30102, 30450, 30800, 31152, 31506, 31862, 32220, 32580, 32942, 33306, 33672, 34040, 34410, 34782, 35156, 35532, 35910, 36290, 36672, 37056, 37442, 37830, 38220, 38612, 39006, 39402, 39800

I've marked the pronic number 26732 = 163 x 164 in bold because that is my diurnal age today (June 11th 2022) and this fact is what prompted me to make this post. The previous such number (26406 = 162 x 163) occurred on Tuesday, July 20th 2021 and the next (27060 = 164 x 165) will occur on Friday, May 5th 2023. So at the moment, a pronic number appearing as my diurnal age is pretty much a yearly thing and as such should be celebrated.

Pronic numbers are also called oblong numbers, rectangular numbers or heteromecic numbers. Interestingly, the sum of the reciprocals of the pronic numbers is 1. Thus:$$\sum_{n=1}^{\infty} \frac{1}{n(n+1)}=1$$I've written about numbers of this sort before in a post titled Pronic Pandigital Numbers and Beyond on July 23rd 2021. Over 80% of pronic numbers are abundant but 26732 is deficient. In fact, of the 199 numbers in the list above, only 35 are deficient. These are:

2, 110, 182, 506, 1406, 1892, 2162, 2756, 3422, 3782, 4556, 5402, 6806, 7310, 8930, 9506, 11342, 11990, 14042, 14762, 17030, 17822, 18632, 20306, 21170, 22052, 22952, 24806, 26732, 27722, 29756, 31862, 32942, 36290, 37442

This sequence of numbers forms part of OEIS A077804:

A077804

Deficient oblong numbers.

The generating function for the pronic numbers is:$$\frac{2x}{(1-x)^3}=2x+6x^2+12x^3+20x^4+ \dots$$Pronic numbers are also figurate numbers of the form:$$P_n=2T_n=n(n+1)$$where \(T_n\) is the \(n^{th}\) triangular number. A very few pronic numbers are palindromic. The first few are listed below:

2, 6, 272, 6006, 289982, 2629262, 6039306, 27999972, 28233282, 2704884072, 20278187202, 20591819502, 2592587852952, 2936231326392, 21809166190812, 27237788773272, 229145919541922, 233552101255332, 250087292780052, 2243922442293422, 2570769009670752, 20333113431133302, 27785925652958772

These numbers form OEIS A028337:

A028337

Palindromes of the form n(n+1).

Loeschian Numbers

Loeschian numbers are numbers of the form \(x^2+xy+y^2\) where \(x\) and \(y\) are integers. I came across these today when looking for interesting properties associated with my diurnal age of 26575 days. The Wikipedia article states that:

They are a set of whole numbers, including zero, and having prime factorisation in which all primes congruent to 2 mod 3 have even powers (there is no restriction of primes congruent to 0 or 1 mod 3).

Now \(26575 =5^2 \times 1063\) and we see that \(5 \equiv 2 \hspace{-4pt} \mod{3}\) is raised to an even power while \(1063 \equiv 1 \hspace{-4pt} \mod{3}\). These numbers are relatively frequent. For example of the first 1000 integers, 277 (or 27.7%) of them are Loeschian. Some have only one representation such as \(26575 = 15^2+15 \times 155+155^2\) while others have more than one. For example:$$ \begin{align} 637 &=4^2+4 \times 23+ 23^2\\ &=7^2+7 \times 21+21^2 \\ &=12^2+12 \times 17+ 17^2 \\931&=1^2+1 \times 30+ 30^2 \\&=14^2+14 \times 21+21^2\\ &=9^2+ 9 \times 25+25^2 \end{align}$$It was in this context that the famous taxi cab number 1729 popped up again. I've written about this number before in a post titled The Original Taxi Cab Number in a New Light on December 21st 2019. In that post, I listed several of its properties but not the fact that it is a member of OEIS A198775:

A198775

Numbers having exactly four representations by the quadratic form \(x^2+xy+y^2\) with \(0 \leq x \leq y\).

We find that the first member of this sequence is 1729:

1729, 2821, 3367, 3913, 4123, 4459, 4921, 5187, 5551, 5719, 6097, 6517, 6643, 6916, 7189, 7657, 8029, 8113, 8463, 8827, 8911, 9139, 9331, 9373, 9709, 9919, 10101, 10507, 10621, 10633, 11137, 11284, 11557, 11739, 12369, 12649, 12691, 12901, 13237, 13377, ...

It has the following representations:$$ \begin{align} 1729 &= 23^2 +23 \times 25+25^2  \\&=3^2+3 \times 40+40^2  \\ &=15^2+15 \times 32+32^2 \\  &=8^2+ 8 \times 37+37^2 \end{align}$$

The Loeschian numbers are named after August Lösch whose Wikipedia entry remarks:

Overall, Lösch made a plenitude of significant findings in the world of economics, but his main contributions were to regional economics, specifically, pioneering the location theory, spatial equilibrium analysis and hierarchical spatial systems displaying a hexagonal pattern.

Figure 1 shows the triangular or, when combined into groups of six, the hexagonal lattice formed by the Eisenstein integers which Lösch must have used in his economic analysis. 

Figure 1

It turns out the Loeschian numbers are the norms of the Eisenstein integers. In mathematics, Eisenstein integers (named after Gotthold Eisenstein), occasionally also known as Eulerian integers (after Leonhard Euler), are complex numbers of the form:$$ \begin{align} z &= x + y\omega \text{ where }x \text{ and } y \text{ are integers }\\ \text{ and where }\omega &= \frac{-1 + i \hspace{2pt} \sqrt{3}}{2} = e^{i\frac{2\pi}{^3}} \end{align}$$ The 2-norm of an Eisenstein integer is just its squared modulus, and is given by:$$ \begin{align} \left|x + y\;\!\omega\right|^2 \,&= \, (x - \tfrac{1}{2} y)^2 + \tfrac{3}{4} y^2 \, \\ &= \, x^2 - xy + y^2 \end{align} $$It can be seen that we have a \(-xy\) instead of a \(+xy\) term but then again \(x\) and \(y\) are no longer restricted to being positive. For example, if we allow \(x\) and \(y\) to be negative as well as positive, then 26575 can be written as:$$ \begin{align} 26575 &=15^2+15 \times 155+155^2 \\ &=155^2-155 \times 170+170^2 \end{align}$$So that will do it for now but there is clearly much more that could be said about Loeschian numbers. More at a later date.