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.

Another Palindromic Cyclops Number

A thousand and ten days ago, I made a post titled Palindromic Cyclops Numbers in which I focused on the number associated with my then diurnal age of 26062. Today I turned 27072 days old and I'm reminded of such numbers once again. They form OEIS  A138131:

A138131

Palindromic cyclops numbers.

The sequence follow a predictable pattern:

0, 101, 202, 303, 404, 505, 606, 707, 808, 909, 11011, 12021, 13031, 14041, 15051, 16061, 17071, 18081, 19091, 21012, 22022, 23032, 24042, 25052, 26062, 27072, 28082, 29092, 31013, 32023, 33033, 34043, 35053, 36063, 37073, 38083

Some of these numbers are prime, such as 101, but not all. 27072 is composite and in fact it has 42 divisors which qualifies it for membership of OEIS A175750$$27072 = 2^6 \times 3^2 \times 47 \rightarrow \text{ 42 divisors}$$

A175750

Numbers with 42 divisors.

The initial members of the sequence are as follows:

2880, 4032, 4800, 6336, 7488, 9408, 9792, 10944, 11200, 13248, 14580, 15552, 15680, 16704, 17600, 17856, 20412, 20800, 21312, 23232, 23328, 23616, 24768, 27072, 27200, 30400, 30528, 32076, 32448, 33984, 34496, 35136, 36450, 36800, 37908, 38592, 38720, 40768

I've marked in red the numbers in this sequence that are palindromic. As can be seen, 27072 is the only palindromic cyclops number that has 42 divisors. If we extend the range up to ten million, there are only these very few palindromic numbers with 42 divisors: 2308032, 4099904, 6714176 and 8820288.

Another of 27072's claim to fame is that it is 100-gonal number and thus a member of OEIS A261276. Such numbers are generated from the formula$$ \begin{align} \text{number } &= \frac{(s-2) \times n \times(n-1)}{2}+ n \\ &= \frac{98 \times n \times(n-1)}{2}+ n \text{   since } s=100 \end{align} $$The initial members of the sequence are (permalink):

0, 1, 100, 297, 592, 985, 1476, 2065, 2752, 3537, 4420, 5401, 6480, 7657, 8932, 10305, 11776, 13345, 15012, 16777, 18640, 20601, 22660, 24817, 27072, 29425, 31876, 34425, 37072, 39817, 42660, 45601, 48640, 51777, 55012, 58345, 61776, 65305, 68932, 72657, 76480

The number 27072 is also "bipronic" which is a term I've not encountered before, although I'm familiar with the term "pronic". The former term is an extension of the later so that bipronic numbers are of the form:$$ x \times (x+1) \times y \times (y+1) \\ \text{ where }x \text{ and } y \text{ are distinct integers}$$In the case of 27072 we have$$27072 = 3 \times 4 \times 47 \times 48$$These types of bipronic numbers form OEIS A053990 and if we relax the condition that \(x\) and \(y\) need to be distinct, then we have OEIS A072389. Combining the bipronics from OEIS A072389 with the palindromes, we get OEIS A346919:

A346919

Numbers that are both palindromes (A002113) and terms of A072389.

The initial members of this sequence are:

0, 4, 252, 2112, 2772, 6336, 21012, 27072, 42924, 48384, 48984, 63036, 252252, 297792, 407704, 2327232, 2572752, 2747472, 2774772, 2958592, 4457544, 4811184, 6378736, 6396936, 25777752, 27633672, 29344392, 63099036, 63399336, 404080404, 409757904, 441525144

The number 27072 also arises when we consider the sum of the divisors of the number of partitions of \(n\). These sums form OEIS A139041:

A139041

Sum of divisors of the number of partitions of \(n\).

The initial members of this sequence are:

1, 3, 4, 6, 8, 12, 24, 36, 72, 96, 120, 96, 102, 240, 372, 384, 480, 576, 1026, 960, 2340, 2016, 1512, 3224, 3240, 6720, 6336, 6588, 6048, 13104, 11232, 12768, 17784, 22176, 22344, 17978, 27072, 35112, 69696, 87552, 74496, 87048, 104544, 97216, 137088, 214896

In the case of 27072, we have \(n=37\) with the the number of partitions of 37 being equal to 21637 and with the divisors [1, 7, 11, 77, 281, 1967, 3091, 21637] adding to the number. There are of course many other properties associated with this number but I'll end off there.