A Platonic basis of Integers

TL;DR

This paper demonstrates that all integers can be expressed as a combination of four tetrahedral numbers and explores the modular periodicity of platonic numbers.

Contribution

It establishes a new representation theorem for integers using tetrahedral numbers and analyzes the modular properties of platonic numbers.

Findings

Every integer can be written as an integer combination of four tetrahedral numbers.

Computed the modular periodicity of platonic numbers.

Provided theoretical insights into the structure of platonic numbers.

Abstract

In this article, we prove that every integer can be written as an integer combination of exactly 4 tetrahedral numbers. Moreover, we compute the modular periodicity of platonic numbers.

Equations

$$t_n=\frac{n(n+1)(n+2)}{6}$$

$$o_n=\frac{n(2n^2+1)}{3}$$

$$c_n=n^3$$

$$i_n=\frac{n(5n^2-5n+2)}{2}$$

$$d_n=\frac{n(9n^2-9n+2)}{2}$$

$$\begin{array}[]{llllllllllllllll}t_{n}:&1,4,10,20,35,56,84,120,165,220,\dots\\ o_{n}:&1,6,19,44,85,146,231,344,489,670,\dots\\ c_{n}:&1,8,27,64,125,216,343,512,729,1000,\dots\\ i_{n}:&1,12,48,124,255,456,742,1128,1629,2260,\dots\\ d_{n}:&1,20,84,220,455,816,1330,2024,2925,4060,\dots\end{array}$$

$$\begin{array}[]{cccccccccccccccccccccc}t_{n}:&1&4&10&20&35&56&84&120&165&220&286&364&455&560&\dots\\ r_{n}:&1&0&0&0&1&0&0&0&1&0&0&0&1&0\dots\end{array}$$

$$y_n=4y_{n-1}-6y_{n-2}+4y_{n-3}-y_{n-4}.$$

$$\Delta y_n=y_{n+1}-y_n.$$

$${\Delta }^n{y}_n={\Delta }^{n-1}{y}_{n+1}-{\Delta }^{n-1}{y}_n.$$

$$\begin{array}[]{llllllllllllll}\begin{array}[]{cccccccccccccccccccccccc}t_{n}&\Delta t_{n}&\Delta^{2}t_{n}&\Delta^{3}t_{n}&\Delta^{4}t_{n}\\ \hline\cr 1&3&3&1&0\\ 4&6&4&1&0\\ 10&10&5&1&0\\ 20&15&6&1&0\\ 35&21&7&1&0\\ 56&28&8&1&0\\ 84&36&9&1&\vdots\\ 120&45&10&\vdots&\vdots\\ 165&\vdots&\vdots&\vdots&\vdots\\ \vdots&\vdots&\vdots&\vdots&\vdots\end{array}\par&&\begin{array}[]{cccccccccccccccccccccccc}o_{n}&\Delta o_{n}&\Delta^{2}o_{n}&\Delta^{3}o_{n}&\Delta^{4}o_{n}\\ \hline\cr 1&5&8&4&0\\ 6&13&12&4&0\\ 19&25&16&4&0\\ 44&41&20&4&0\\ 85&61&24&4&0\\ 146&85&28&4&0\\ 231&113&32&4&\vdots\\ 344&145&36&\vdots&\vdots\\ 489&181&\vdots&\vdots&\vdots\\ 670&\vdots&\vdots&\vdots&\vdots\\ \vdots&\vdots&\vdots&\vdots&\vdots\end{array}\par&&\begin{array}[]{cccccccccccccccccccccccc}c_{n}&\Delta c_{n}&\Delta^{2}c_{n}&\Delta^{3}c_{n}&\Delta^{4}c_{n}\\ \hline\cr 1&7&12&6&0\\ 8&19&18&6&0\\ 27&37&24&6&0\\ 64&61&30&6&0\\ 125&91&36&6&0\\ 216&127&42&6&0\\ 343&169&48&6&0\\ 512&217&52&6&\vdots\\ 729&271&58&\vdots&\vdots\\ 1000&\vdots&\vdots&\vdots&\vdots\\ \vdots&\vdots&\vdots&\vdots&\vdots\end{array}\\ \end{array}$$

$$\begin{array}[]{llllllllllllll}\begin{array}[]{cccccccccccccccccccccccc}i_{n}&\Delta i_{n}&\Delta^{2}i_{n}&\Delta^{3}i_{n}&\Delta^{4}i_{n}\\ \hline\cr 1&11&25&15&0\\ 12&36&40&15&0\\ 48&76&55&15&0\\ 124&131&70&15&0\\ 255&201&85&15&0\\ 456&286&100&15&0\\ 742&386&115&15&\vdots\\ 1128&501&130&\vdots&\vdots\\ 1629&631&\vdots&\vdots&\vdots\\ 2260&\vdots&\vdots&\vdots&\vdots\\ \vdots&\vdots&\vdots&\vdots&\vdots\end{array}\par&&\begin{array}[]{cccccccccccccccccccccccc}d_{n}&\Delta d_{n}&\Delta^{2}d_{n}&\Delta^{3}d_{n}&\Delta^{4}d_{n}\\ \hline\cr 1&19&45&27&0\\ 20&64&72&27&0\\ 84&136&99&27&0\\ 220&235&126&27&0\\ 455&361&153&27&0\\ 816&514&180&27&0\\ 1330&694&207&27&\vdots\\ 2024&901&234&\vdots&\vdots\\ 2925&1135&\vdots&\vdots&\vdots\\ 4060&\vdots&\vdots&\vdots&\vdots\\ \vdots&\vdots&\vdots&\vdots&\vdots\end{array}\end{array}$$

$$\Delta t_n=t_{n+1}-t_n$$

$$\Delta o_n=o_{n+1}-o_n$$

$$\Delta c_n=c_{n+1}-c_n$$

$$\Delta i_n=i_{n+1}-i_n$$

$$\Delta d_n=d_{n+1}-d_n$$

$$t_n-2t_{n+1}+t_{n+2}$$

$$o_n-2o_{n+1}+o_{n+2}$$

$$c_n-2c_{n+1}+c_{n+2}$$

$$i_n-2i_{n+1}+i_{n+2}$$

$$d_n-2d_{n+1}+d_{n+2}$$

$$t_{n+3}-3t_{n+2}+3t_{n+1}-t_n$$

$$o_{n+3}-3o_{n+2}+3o_{n+1}-o_n$$

$$c_{n+3}-3c_{n+2}+3c_{n+1}-c_n$$

$$i_{n+3}-3i_{n+2}+3i_{n+1}-i_n$$

$$d_{n+3}-3d_{n+2}+3d_{n+1}-d_n$$

$$t_{n+4}-4t_{n+3}+6t_{n+2}-4t_{n+1}+t_n$$

$$o_{n+4}-4o_{n+3}+6o_{n+2}-4o_{n+1}+o_n$$

$$c_{n+4}-4c_{n+3}+6c_{n+2}-4c_{n+1}+c_n$$

$$i_{n+4}-4i_{n+3}+6i_{n+2}-4i_{n+1}+i_n$$

$$d_{n+4}-4d_{n+3}+6d_{n+2}-4d_{n+1}+d_n$$

$$3t_n-8t_{n+1}+7t_{n+2}-2t_{n+3}=n.$$

$$2o_n-5o_{n+1}+4o_{n+2}-o_{n+3}=4n.$$

$$2c_n-5c_{n+1}+4c_{n+2}-c_{n+3}=6n,$$

$$5i_n-12i_{n+1}+9i_{n+2}-2i_{n+3}=45n.$$

$$5d_n-12d_{n+1}+9d_{n+2}-2d_{n+3}=54n.$$

$$\begin{array}[]{lllllll}86=85+1,&87=85+1+1,&88=85+1+1+1,&89=85+4,\\ 90=85+4+1,&91=85+4+1+1,&92=85+4+1+1+1,&93=85+8,\\ 94=85+8+1,&95=85+10,&96=85+10+1,&97=85+12,\\ 98=85+12+1,&99=85+12+1+1,&100=85+12+1+1+1,&101=85+12+4,\\ 102=85+12+4+1,&103=85+12+4+1+1,&104=85+10+8+1,&105=85+20,\\ 106=85+20+1,&107=85+20+1+1,&108=85+20+1+1+1,&109=85+20+4,\\ 110=85+20+4+1,&111=85+20+4+1+1,&112=85+27,&113=85+27+1,\\ 114=85+27+1+1,&115=85+27+1+1+1,&116=85+27+4,&117=85+27+4+1,\\ 118=85+27+4+1+1,&119=85+20+12+1+1\par\end{array}$$

$$p^{t}=\left\{\begin{array}[]{llllllllllll}6d&\mbox{if $d$ is divisible by $3$};\\ 2d&\mbox{otherwise.}\end{array}\right.$$

$$p^{t}=\left\{\begin{array}[]{llllllllllll}3d&\mbox{if $d$ is divisible by $3$};\\ d&\mbox{otherwise.}\end{array}\right.$$

Full text

1 Introduction

Figurate numbers related to the five platonic solids are called platonic numbers. In [5], Sir Frederick Pollock conjectured that every positive integer is the sum of at most five, seven, nine, thirteen, and twenty one tetrahedral numbers, octahedral numbers, cubes, icosahedral numbers, and dodecahedral numbers, respectively. See [1] for a detailed study of Platonic numbers. In this article, we discuss integer combinations of platonic numbers instead of sums. Thus we also include the operation of subtraction.

Let $t_{n},o_{n},c_{n},i_{n}$, and $d_{n}$ represent the tetrahedral numbers, octahedral numbers, cubes, icosahedral numbers, and the dodecahedral numbers, respectively. The following identities are discussed in [1] and [4]:

[TABLE]

We list a few numbers in the sequences:

[TABLE]

In Section 2, we prove that every integer can be written as an integer combination of exactly 4 tetrahedral numbers. Moreover, we prove that integers that are divisible by four, six, forty five, and fifty four, can be written as integer combinations of exactly four octahedral numbers, cubes, icosahedral numbers, and dodecahedral numbers, respectively. We also conjecture that every integer can be written as a sum of at most five different platonic numbers.

We say a sequence $s_{n}$ is periodic mod $d$ if there exists an integer $m$ such that, $s_{n+m}\equiv s_{n}$ mod $d$ for every $n$. The smallest such integer $m$ is called the period of $s_{n}$ mod $d$.

Example 1.1**.**

Let $r_{n}$ denote the residues of the the tetrahedral numbers $t_{n}$ modulo $2$:

[TABLE]

Observe that the reduced sequence $r_{n}$ repeats after every four terms. Consequently, $t_{n+4}\equiv t_{n}$ mod $2$. Hence $t_{n}$ has period $4$ mod $2$.

In Section 2, we show that the platonic numbers satisfy the following linear recurrence relation:

[TABLE]

It is well known that linear recurrence sequences are eventually periodic. See [3] and the references therein for a discussion of periodic sequences defined by linear recurrences. In Section 3, we derive the periods of the platonic numbers.

2 Forward Differences.

In this section, we use forward differences of tetrahedral numbers to prove that every integer can be written as an integer combination of exactly 4 tetrahedral numbers. Let $y_{n}$ denote a sequence of integers, then the first forward difference, denoted by $\Delta y_{n}$, is defined as

[TABLE]

$\Delta$ is called the forward difference operator. The differences of the first forward differences are called the second forward differences and are denoted by $\Delta^{2}y_{i}$. Continuing thus, the $n$-th forward difference is defined as

[TABLE]

We provide the forward differences tables of the five platonic numbers below.

[TABLE]

[TABLE]

We see the fourth forward differences of all the platonic numbers are zero. The following identities are derived from the first forward differences of the Platonic numbers.

[TABLE]

Since, $\Delta^{2}y_{n}=y_{n+2}-2y_{n+1}+y_{n}$, we get the following identities.

[TABLE]

Since $\Delta^{3}y_{n}=y_{n+3}-3y_{n+2}+3y_{n+1}-y_{n}$, we get the following identities.

[TABLE]

The fourth forward difference is given by $\Delta^{4}y_{n}=y_{n+4}-4y_{n+3}+6y_{n+2}-4y_{n+1}+y_{n}$. Consequently, we get the following identities.

[TABLE]

Consequently, it follows that all the platonic numbers satisfy the linear recurrence relation given by Equation 2.

From Equations 4 and 5, we get $\Delta^{2}t_{n}-2\Delta^{3}t_{n}=n$. Consequently,

[TABLE]

Equation 7 implies that every integer can be written as an integer combination of four tetrahedral numbers.

Again, from Equations 4 and 5, we get $\Delta^{2}o_{n}-\Delta^{3}o_{n}=4n$. Therefore,

[TABLE]

Therefore, we conclude that integers divisible by $4$ can be written as integer combinations of four octahedral numbers.

Similarly, since $\Delta^{2}c_{n}-\Delta^{3}c_{n}=6n$, we get

[TABLE]

which implies that the integers which are divisible by $6$ are integer combinations of four cubes.

From Equations 4 and 5, we get $3\Delta^{2}i_{n}-2\Delta^{3}i_{n}=45n$, therefore

[TABLE]

Hence integers that are divisible by $45$ can be written as integer combinations of four icosahedral numbers.

Finally, from Equations 4 and 5, we get $3\Delta^{2}d_{n}-2\Delta^{3}d_{n}=54n$, therefore

[TABLE]

Hence we conclude that integers that are divisible by $54$ are integer combinations of four dodecahedral numbers.

Note that since the second forward differences are linear in $n$, some combination of these differences might lead to a proof of the Pollock’s conjecture. We leave that to the reader to explore. In our study of the Pollock’s conjecture, we observed that we could write the numbers between two platonic numbers as sums of at most five Platonic numbers.

Example 2.1**.**

Integers between $o_{5}=85$ to $t_{8}=120$ written as sums of at most five different Platonic numbers.

[TABLE]

Thus, we add one more conjecture to the collection of the many beautiful unproved conjectures on platonic numbers.

Conjecture 2.1**.**

Every integer can be written as a sum of at most five different Platonic numbers.

3 Modular Periods of Platonic numbers.

In this section, we derive the modular periods of platonic numbers. We use the formulas (1) in Section 1 for the proof of the following proposition.

Proposition 3.1**.**

Let $d>1$ be an integer.

Consider the sequence of tetrahedral numbers $t_{n}$. Let $p^{t}$ denote the period of $t_{n}$ mod $d$.

1. Case 1:

$d$* is an even integer.*

[TABLE] 2. Case 2:

$d$* is an odd integer.*

[TABLE] 2. 2.

Let $p^{o}$ denote of the period of the sequence of octahedral numbers $o_{n}$ mod $d$.

[TABLE] 3. 3.

If $p^{c}$ denotes the period of the sequence of cubes mod $d$, then $p^{c}=d$. 4. 4.

Let $p^{i}$ denote the period of the sequence of icosahedral numbers mod $d$.

[TABLE] 5. 5.

Let $p^{d}$ denote the sequence of dodecahedral numbers, then

[TABLE]

Proof.

We derive the periods of the sequence of tetrahedral numbers.

1. Case 1:

$d$ is even.

Let $d$ be not divisible by $3$. Since $d$ is even, we write $d=2k$ for some $k$. For $n\geq 1$,

[TABLE]

Since $d$ is not divisible by $3$, and $t_{n+4k}-t_{n}$ is an integer, $3$ divides $(3n^{2}+12nk+6n+16k^{2}+12k+2)$. Hence $t_{n+4k}-t_{n}=2ks$ for some integer $s$. Consequently, $t_{n+2d}\equiv t_{n}$ mod $d$, which implies $p^{t}=2d$.

On the other hand, if $d$ is divisible by $3$, then $d=6k$ for some integer $k\geq 1$. For any $n\geq 1$,

[TABLE]

Hence, $t_{n+36k}\equiv t_{n}$ mod $6k$. Consequently, $p^{t}=6d$. 2. Case 2:

$d$ is an odd integer.

Let $d$ be not divisible by $3$. Since $d>1$ is odd, $d=2k+1$ for some integer $k\geq 1$. For $n\geq 1$,

[TABLE]

Now $d$ is not divisible by $2$ or $3$. Hence $2k+1$ is not divisible by $6$. Consequently, $6$ divides $(3n^{2}+6nk+9n+4k^{2}+10k+6)$ because $t_{n+(2k+1)}-t_{n}$ is an integer. Therefore $t_{n+2k+1}-t_{n}=(2k+1)s$ for some integer $s$. Thus, $p^{t}=d$.

Now consider the case when $d$ is divisible by $3$. Then $d=3(2k+1)$, for some $k\geq 0$. For $n\geq 1$,

[TABLE]

Since $t_{n+3d}-t_{n}$ is an integer, $2$ divides $(3n^{2}+54nk+33n+324k^{2}+378k+110)$. Therefore $t_{n+9(2k+1)}-t_{n}=3(2k+1)s$ for some integer $s$. Consequently, $p^{t}=3d$ in this case. 2. 2.

Next, we consider the sequence of octahedral numbers $o_{n}$.

Let $d$ be not divisible by $3$. For $n\geq 1$,

[TABLE]

Since $o_{n+d}-o_{d}$ is an integer, and $d$ is not divisible by $3$, we get $3$ divides $(6n^{2}+6nd+2d^{2}+1)$. Consequently, $o_{n+d}\equiv o_{d}$ mod $d$. Hence $p^{o}=d$ in this case.

On the other hand, if $d$ is divisible by $3$, then $d=3k$ for some $k\geq 1$. For $n\geq 1$,

[TABLE]

Consequently, $o_{n+3d}\equiv o_{d}$ mod $d$. Hence $p^{o}=3d$. 3. 3.

$p^{c}=d$ because for $n\geq 1$,

[TABLE] 4. 4.

We now compute $p^{i}$. Let $d$ be an odd integer. For $n\geq 1$,

[TABLE]

Since $d$ is odd, and $i_{n+d}-i_{n}$ is an integer, $2$ divides $(15n^{2}+15nd-10n+5d^{2}-5d+2)$. Consequently, $p^{i}=d$ in this case.

On the other hand, when $d$ is even, $d=2k$ for some $k\geq 1$. For $n\geq 1$,

[TABLE]

Thus $p^{i}=2d$. 5. 5.

Finally, we compute the periods of the sequence of dodecahedral numbers $d_{n}$. Let $d$ be an odd integer. For $n\geq 1$,

[TABLE]

Since $d$ is odd and $d_{n+d}-d_{n}$ is an integer, we get $2$ divides $(27n^{2}+27nd-18n+9d^{2}-9d+2)$. Consequently, we get $p^{d}=d$ in this case.

On the other hand, when $d$ is even, $d=2k$ for some $k\geq 1$. For $n\geq 1$,

[TABLE]

Thus, $p^{d}=2d$.

∎