Power Towers

The number associated with my diurnal age today, \( \textbf{28144}\), has the property that its power tower of digits is equal to its product of digits. That is to say:

2 ^ 8 ^ 1 ^ 4 ^ 4 = 2 x 8 x 1 x 4 x 4 = 36

Such numbers are relatively rare. In the range beyond 28144 up to 40000, there are only the following numbers that satisfy: 28182, 31111, 33113 and 33131.

Meanwhile, I thought it was time to clarify the reasons behind the right associativity of power towers and so I consulated Gemini. Here was the response:

Let's break down the mechanics of power towers, tetration, and the crucial role of associativity in how we evaluate these operations.

Associativity: The Order of Operations

In mathematics, associativity dictates how we group numbers when an expression contains multiple instances of the same operator.

• Left-Associative: Operations are evaluated from left to right. Subtraction and division are left-associative by convention.
  For example, 10 − 5 − 2 is evaluated as (10 − 5) − 2 = 3.
• Right-Associative: Operations are evaluated from right to left.

Exponentiation is mathematically defined as right-associative. When you see a sequence of exponents, you evaluate them from the top down.

Here is why this convention exists: if exponentiation were left-associative, it would be mathematically redundant. Let's look at 2³².

• If it were left-associative: (2³)² = 8² = 64.
  By the rules of exponents, (a^b)^c = a^{b × c}. Therefore, a left-associative power tower is just a disguised way of multiplying exponents together (2^{3 × 2} = 2⁶ = 64). We already have multiplication, so we don't need a new convention for this.
• Because it is right-associative: 2⁽³²⁾ = 2⁹ = 512.
  Evaluating from the top-down creates a fundamentally new, uniquely powerful operation that cannot be simplified by basic multiplication.

Power Towers

A power tower is simply a visual stack of exponents, written as a^bcd.

Because of the right-associative rule, you always start at the highest point of the tower and work your way down to the base.

For example, to evaluate 2³²²:

1. Start at the top: 2² = 4
2. Move down one level: 3⁴ = 81
3. Calculate the final base: 2⁸¹

The result is 2,417,851,639,229,258,349,412,352.

Tetration: Iterated Exponentiation

Tetration is a specific type of power tower where the base number is identical throughout the entire stack.

Just as multiplication is repeated addition, and exponentiation is repeated multiplication, tetration is repeated exponentiation. It is the fourth fundamental hyperoperation in mathematics (following addition, multiplication, and exponentiation).

Because standard exponentiation notation becomes visually cumbersome when stacked high, mathematicians use specific notations for tetration:

• Knuth's Up-Arrow Notation: a ↑↑ n
• Rudy Rucker's Superscript Prefix: ⁿa

In both notations, a is the base and n is the height of the tower.

Examples of Tetration:

• ²3 (or 3 ↑↑ 2) = 3³ = 27
• ³3 (or 3 ↑↑ 3) = 3³³ = 3²⁷ = 7,625,597,484,987

The Computational Horizon

Tetration generates exponential growth at a scale that rapidly breaks the limits of standard computation.

If we take just one more step up the tower to ⁴3 (or 3 ↑↑ 4), the expression becomes 3³³³, which evaluates to 3^{7,625,597,484,987}.

This number has over 3.6 trillion decimal digits. While theoretical mathematics can easily define and manipulate the expression ⁴3, calculating the exact integer quickly leaves the realm of standard datatypes. Numbers at this scale become highly unrenderable in their full decimal form; storing just the digits of ⁴3 as a standard text file would require terabytes of disk space. When computational number theory investigates numbers at or beyond this threshold, it relies on custom algorithms, modulo arithmetic, or symbolic representation frameworks to analyze the properties of the digits without attempting to hold the entire integer in memory.

Digit Equations Continued

My post of March 2024 titled Forming Equations from the Digits of a Number expanded an idea that I'd broached in a far earlier post in August of 2013. I'm relating in this current post on my interaction with Gemini in helping me to determine all the numbers between 1 and 40000 that can't be rendered as digit equations. Firstly let's recap what the rules for rendering are:

• only digits can be manipulated not combinations of digits, that is no concatenations
• the order of the digits cannot be changed
• only the operations of addition, subtraction, multiplication, division and exponentiation are allowed
• division can be divided into e.g. 2 | 8 or divided by e.g. 8 / 2
• an unlimited number of brackets can be used
• unary operations are allowed meaning any digit can be changed into its negative.

What I got Gemini to do was to create a list of all the numbers from 1 to 40000 that CANNOT be rendered as digit equations. There are 4839 numbers that qualify in that range and of course the early numbers predominate. Figure 1 shows a graph of the distribution.

Figure 1

What is striking about the graph is that between 10979 and 20294, there are only two numbers that CANNOT be rendered as digit equations. These numbers are 15795 and 15975, each a permutation of the other's digits. This means that, out of the 9313 numbers from 10980 to 20293 inclusive, there are only two that CANNOT be rendered as digit equations. The other 9311 can be. There is another smaller gap between 21027 and 22525 (exclusive) in which there are only three numbers (21037, 21049 and 21059) that cannot be rendered.

Here is a link to the Gemini chat that I had which was long and involved: 

https://gemini.google.com/share/d704238c21fb.

Here are the numbers that CANNOT be rendered as digit equations from 28000 to 40000:

28027, 28049, 28108, 28120, 28210, 28255, 28270, 28290, 28308, 28383, 28395, 28429, 28474, 28494, 28558, 28585, 28672, 28708, 28759, 28849, 28908, 28959, 29049, 29059, 29109, 29120, 29130, 29169, 29210, 29212, 29229, 29230, 29239, 29240, 29260, 29269, 29280, 29284, 29292, 29293, 29296, 29309, 29379, 29392, 29397, 29409, 29410, 29420, 29432, 29433, 29460, 29467, 29479, 29480, 29490, 29494, 29509, 29510, 29514, 29530, 29537, 29559, 29569, 29572, 29573, 29577, 29587, 29589, 29590, 29595, 29596, 29598, 29599, 29609, 29659, 29673, 29679, 29692, 29697, 29739, 29749, 29769, 29779, 29793, 29794, 29796, 29797, 29809, 29859, 29937, 29959, 30292, 30295, 30424, 30464, 30497, 30592, 30637, 30727, 30738, 30757, 30794, 30797, 30828, 30837, 30848, 30857, 30868, 30938, 30949, 30959, 30968, 31027, 31607, 31667, 31677, 31707, 31708, 31717, 31767, 31778, 31787, 31788, 31807, 31808, 31818, 31877, 31878, 31887, 31898, 31908, 31977, 31987, 31988, 31998, 32535, 32597, 32737, 32957, 32979, 33585, 33597, 33727, 33737, 33747, 33828, 33858, 34647, 34737, 34746, 34747, 34757, 34758, 34847, 34858, 34949, 34959, 35105, 35106, 35235, 35253, 35325, 35352, 35358, 35385, 35397, 35405, 35445, 35450, 35470, 35477, 35499, 35527, 35605, 35606, 35650, 35656, 35665, 35670, 35747, 35775, 35835, 35838, 35853, 35868, 35874, 35885, 35886, 35905, 35927, 35949, 35959, 35995, 36105, 36106, 36107, 36474, 36505, 36506, 36556, 36560, 36565, 36566, 36590, 36706, 36707, 36717, 36760, 36766, 36767, 36776, 36780, 36807, 36868, 36885, 36886, 37027, 37106, 37107, 37108, 37117, 37167, 37176, 37177, 37178, 37187, 37188, 37198, 37207, 37210, 37237, 37240, 37255, 37270, 37273, 37295, 37299, 37306, 37308, 37327, 37337, 37347, 37372, 37373, 37374, 37437, 37447, 37457, 37464, 37473, 37474, 37475, 37484, 37507, 37508, 37547, 37574, 37592, 37606, 37607, 37608, 37617, 37618, 37650, 37666, 37667, 37670, 37671, 37676, 37680, 37690, 37698, 37699, 37806, 37807, 37808, 37817, 37818, 37855, 37860, 37870, 37871, 37877, 37878, 37887, 37888, 37890, 37891, 37907, 37908, 37929, 37952, 37953, 37979, 37997, 38107, 38108, 38109, 38118, 38120, 38167, 38168, 38177, 38178, 38187, 38188, 38189, 38197, 38198, 38208, 38283, 38307, 38309, 38310, 38328, 38340, 38355, 38356, 38358, 38360, 38365, 38370, 38382, 38383, 38385, 38388, 38390, 38408, 38458, 38535, 38538, 38548, 38558, 38562, 38568, 38574, 38583, 38584, 38585, 38586, 38607, 38608, 38652, 38658, 38668, 38685, 38686, 38707, 38708, 38709, 38717, 38718, 38745, 38755, 38760, 38777, 38778, 38780, 38781, 38787, 38788, 38790, 38907, 38908, 38909, 38917, 38918, 38950, 38960, 38961, 38965, 38966, 38967, 38970, 38980, 38981, 38988, 38989, 38998, 38999, 39027, 39108, 39109, 39178, 39188, 39198, 39199, 39279, 39292, 39297, 39409, 39449, 39459, 39494, 39495, 39509, 39528, 39537, 39549, 39558, 39559, 39594, 39595, 39708, 39729, 39779, 39792, 39797, 39808, 39809, 39817, 39818, 39855, 39860, 39867, 39870, 39871, 39888, 39889, 39890, 39891, 39898, 39899, 39927

In the SageMath program on my Jupyter notebook, I've added some additional code, courtesy of Gemini, that will render the number associated with my diurnal age as a digit equation or announce failure if a rendering is not possible. Of course, I'll try to create the equation myself before looking at the program's output. Today I'm 28138 days old:

Forming Digit Equations: A Game

Source

It occurred to me how forming digit equations from five digits might make for an interesting game. After all, Wordle consists of guessing the five letters that make up a hidden word. In the game I'm conceiving of, a five digit number between 10000 and 99999 would be generated. Let's say the number generated is 25529. The challenge is to use only the following operators to form a digit equation without altering the order of the digits:

+    addition

-    subtraction

x    multiplication

/    divide by

|    divide into

^    exponentiation

(     left bracket

)    right bracket

Figure 1 shows one possible representation:

Figure 1

The same digit equation would look mathematically as follows:$$ \Big ( 2 + \frac{5}{5} \Big )^2=9$$This is somewhat on the difficult side. Obviously an effective game nowadays involves a catchy user interface so coloured balls that pop up to start the game would be effective. The operators shown above would need to be able to be dragged and dropped between the digits in order to generate the equation. The successful creation of an equation would need to be displayed in mathematically readable way such as was done with the example of 25529.

The game could have different levels starting at the fairly elementary level. For example, 10348 could be rendered as:$$1+0+3+4=8$$This equation involves only the a single operation, addition, and the correct placement of the equal sign. By contrast, 25529 is considerably more challenging and involves the use of brackets, addition, division and exponentiation. Of course, for some numbers it is impossible to form an equation within the imposed constraints. An example is 27497 which is unsolvable as far as I can see. This is a long term project but something that I will keep thinking about or perhaps I'll discover that someone has already created such a game. Who knows?

For earlier posts on this theme see my posts titled The Number Plate Game from the 29th June 2024 and Forming Equations from the Digits of a Number on 14th March 2024. To generate some random five digits numbers follow this permalink.

Forming Equations from the Digits of a Number

There was a post that I made to my Pedagogical Posturing blog in August of 2013 before I created this mathematical blog in the second half of 2015. The title was "Forming Equations from Integer Sequences" and that title was perhaps a little misleading. At the time, I wasn't aware of the OEIS or Online Encyclopedia of Integer Sequences. What I meant was the sequence of digits that define a number. For example, today my diurnal age is 27374 and so the sequence of digits is 2, 7, 3, 7, 4. Concatenation of the digits is not allowed. Thus we cannot have 27, 3, 7, 4 for example. 

In that long ago post, I wrote:

Recently I've been using Twitter to create a daily tweet that records my "day count" (number of days I've been alive) plus its factors (if not prime) and some interesting facts about the number itself or one of its factors. Sometimes there's little to say about the number and in such cases I've found that I can usually form an equation by inserting mathematical operators between one or more of the digits. 

For example, yesterday the count was 23518 and 23 - 5 = 18. Today the count is 23519 and 2 + 3 + 5 - 1 = 9. I was wondering if it's always possible to create an equation from five digits using the standard mathematical operators (addition, subtraction, multiplication, division and exponentiation in combination with brackets). Obviously with just two digits, it's only possible when the digits are repeated e.g. 99 becomes 9=9. With three digits, it's sometimes possible e.g. 819 becomes 8 + 1 = 9 but generally it isn't e.g. 219. With four digits, it's more likely e.g. 2119 becomes -2 + 11 = 9 but I'm doubtful whether this is always so. There must come a point however, where the number of digits is sufficient to ensure that it's always so. Maybe five digits is that point.

From now on, I'll try each day to form an equation to test out this theory. For example, tomorrow the count is 23520 which becomes 2 + 3 - 5 = 2 x 0 and it works for tomorrow but beyond that let's see.

Needless to say I didn't "try each day to form an equation to test out this theory" but it might be time to give it a go. The 27374 of my diurnal age today is an easy one:$$2 \times 7 -3=7+4$$However, yesterday's number, 27373, doesn't prove so easy. It seems that having two 3's and two 7's in the number makes things difficult. The conditions that I imposed in the original blog post were the use of only the standard mathematical operators of addition, subtraction, multiplication, division and exponentiation in combination with brackets. These operators needed to be applied to the digits in the order in which they occurred.

One modification that I will make here is to allow \(x \, | \,y\) meaning \(x\) is divided into \(y\) as opposed to \(x/y\) meaning \(x\) is divided by \(y\). This seems quite reasonable as it still only involves the operation of division but allows more flexibility. Its application doesn't seem to help in the case of 27373. If we allow the operator \(x\) // \(y\) meaning return the whole number part of the dividend, then an equation is possible:$$2|(7-3)=7//3$$If we allow // then we could allow \(x\) % \( y\) meaning return the remainder as a whole number when \(x\) is divided by \(y\). Thus 7 % 3 = 1.  This might be termed modulo division.

I can't see any way to create an equation from 27373 without extending the original conditions. Even concatenation of the digits doesn't seem to help. A common symbol for concatenation is || and thus 2 || 7 = 27. However, I've stipulated that the digits are to be treated as separate so I'll adhere to that condition. The best approach is to stick to the original conditions and if a solution is not possible, then // and % can be resorted to.

In the case of 27374, there is more than one way to create an equation. Here is another way:$$ 2 \times 7 -(3+7)=4 $$So what I'll try to do is to reassert my original goal of trying each day to form an equation and see what patterns emerge.

This activity of forming a "digit equation" is not all that different from one of Quanta's mathematical games called "Hyperjumps". See Figure 1.

Figure 1

Representing Numbers With Digits

My previous post focused on the number 1089 in which I made reference to a newly discovered blogging site at https://math1089.in/ and in particular to a post about the number 1089. In that post it was noted that:$$ \begin{align} 1089 &= 12 \times 3^4 + 5 \times 6 + 78 + 9\\1089 &= 987 + 65 + 4 + 32 + 1 \end{align}$$In another post about the number 108, it was noted that:$$ \begin{align} 108 &= 1 + 2 + 3 + 4 + 5 + 6 + 78 + 9\\108 &= 9 + 8 \times 7 + 6 \times 5 + 4 \times 3 + 2 ‒ 1 \end{align}$$This got me thinking about whether it was possible to represent every number at least once in terms of consecutive single or concatenated digits separated by the basic operations of arithmetic combined with exponentiation and brackets.

For example, my diurnal age today is 27339. Is such a feat possible for this number? The determination is not easy because there are just  so many possible ways to combine the digits from 1 to 9. I spent quite some time playing around with the possibilities and I did come close but not close enough. The exercise is oddly addictive. There's no serious mathematics involved in the exploration. It's more in the nature of a puzzle, like Sudoku. 

There are a variety of strategies, one of which is to establish base points with as few digits as possible. An example of this is:$$35016=1+2+3+4+5+6^7/8+9$$Here the digits 6, 7 and 8 combine to form 34992 and the remaining digits are free to be manipulated. It's true that we can use 34567 to get to a similar number but here five digits are tied up and there's little that can be done with the remaining digits (1, 2 and 8, 9) since they are separated. Here is an example of how we can get close to 27339 using the earlier mentioned base point:$$28084=-(12^3 \times 4+5)+6^7/8+9$$Another approach is to use factors. We know that 27339 = 3 x 13 x 701 and it's easy enough to create the first two factors using 1 + 2 = 3 and 3 x 4 - 5 + 6 = 13. However, we are then left with 789:$$ \begin{align} 27339 &=3 \times 13 \times 701\\30771 &= (1+2) \times (3 \times 4 - 5 +6) \times 789\\ &=3 \times 13 \times 789 \end{align}$$While this works fine for 30771, it's not of much use for 27339. In general, this approach is quite restrictive and the more promising approaches will involve additions and subtractions along with exponentiation, multiplication and division.

At the moment I don't have a solution to the specific problem of representing 27339 in terms of sequential digits and I certainly don't have an answer to the general problem of whether such a representation is always possible or only sometimes possible. Certainly there's an upper limit on the number size and this is imposed by the nine digit restriction but such a limit is huge and my investigation is focusing on numbers in the region of 30000. My suspicion is that it's not always possible within the restrictions imposed. If we relax the requirement that the digits need to be in sequential order or we allows square roots, factorials etc. then maybe it's possible but for the moment I'll keep within the earlier rules and keep revisiting the problem from time to time.

What's Special About 516493?

5 1 6 4 9 3

I got to thinking about primes that remain prime under the operation of prime + digit sum. The acronym "sod" standing for sum of digits is commonly used in this context. There are many primes that satisfy. For example, 11 has a digit sum of 2 and 11 + 2 = 13 which is prime. If we repeat the operation we get 13 + 4 = 17 which is still prime. However, repeating the operation again yields 17 + 8 = 25 which is composite.

The primes that remain prime for only one application of the prime + sod operation constitute OEIS A048523 (permalink):

A048523

Primes for which only one iteration of 'Prime plus its digit sum equals a prime' is possible.

The initial members are:

13, 19, 37, 53, 71, 73, 97, 103, 127, 163, 181, 233, 271, 307, 383, 389, 431, 433, 499, 509, 563, 587, 631, 701, 743, 787, 811, 857, 859, 947, 1009, 1049, 1061, 1087, 1153, 1171, 1223, 1283, 1423, 1483, 1489, 1553, 1597, 1601, 1607, 1733, 1801, 1861, 1867

There are 7939 such primes in the range up to one million. Notice that 11 is not included here because it produces a prime again. The primes, like 11, that remain prime for only two applications of the prime + sod operation constitute OEIS A048524 (permalink):

A048524

Primes for which only two iterations of 'Prime plus its digit sum equals a prime' are possible.

The initial members of the sequence are:

11, 59, 101, 149, 167, 257, 293, 367, 419, 479, 547, 617, 727, 839, 1409, 1579, 1847, 2039, 2129, 2617, 2657, 2837, 3449, 3517, 3539, 3607, 3719, 4217, 4637, 4877, 5689, 5807, 5861, 6037, 6257, 6761, 7027, 7517, 8039, 8741, 8969, 9371, 9377, 10667

There are 1111 such primes in the range up to one million. The primes that remain prime for only three applications of the prime + sod operation constitute OEIS A048525 (permalink):

A048525

Primes for which only three iterations of 'Prime plus its digit sum equals a prime' are possible.

The initial members of the sequence are:

277, 1559, 5779, 7489, 11279, 15091, 22093, 37811, 43579, 46279, 48541, 49957, 53479, 54751, 60589, 68473, 72883, 74821, 83621, 85621, 90793, 91921, 93901, 97501, 107981, 110899, 111799, 120193, 153379, 157739, 170299, 180731, 184441

There are 136 such primes in the range up to one million. The primes that remain prime for only four applications of the prime + sod operation constitute OEIS A048526 (permalink):

A048526

Primes for which only four iterations of 'Prime plus its digit sum equals a prime' are possible.

The initial members of the sequence are:

37783, 85601, 259631, 268721, 350941, 371939, 378901, 516521, 665111, 733331, 883331, 967781, 1047929, 1056521, 1081721, 1258811, 1427411, 1480573, 1515929, 1584901, 1614929, 1842131, 1875311, 1885981, 2027801, 2044873, 2450531

There are only 12 such primes in the range up to one million. The primes that remain prime for only five applications of the prime + sod operation constitute OEIS A048527 (permalink):

A048527

Primes for which only five iterations of 'Prime plus its digit sum equals a prime' are possible.

The initial members of the sequence are:

516493, 1056493, 1427383, 1885943, 3166183, 3805183, 4241593, 6621283, 7646953, 12912283, 17987839, 32106493, 107152093, 120224773, 131144473, 133210873, 139388891, 142782877, 150326173, 155382923, 177865819, 184081943, 227795839, 242376877, 264174877

There is only one such prime in the range up to one million and that is 516493 and that is why it is deserving of this special post. The progression is shown below where True = Prime and False = Composite:

516493 True

516521 True

516541 True

516563 True

516589 True

516623 True

516646 False

516493 is the only prime in the range up to one million that generates five successive primes under the prime + sum of digits operation

There are primes that go further than five iterations. For example, 286330897 survives seven iterations (link). This is the progression:

286330897 True

286330943 True

286330981 True

286331021 True

286331047 True

286331081 True

286331113 True

286331141 True

286331170 False

The number 56676324799 survives eight iterations (link). This is the progression:

56676324799 True

56676324863 True

56676324919 True

56676324977 True

56676325039 True

56676325091 True

56676325141 True

56676325187 True

56676325243 True

56676325292 False

Presumably there is no limit to the number of iterations.

Forming Equations from Integers

Read on!

Once upon a time, before I started this exclusively mathematical blog in 2015, I used to post occasional mathematical content to my Pedagogical Posturing blog at https://voodoo-guru.blogspot.com. I was looking back at some of these posts and noticed this one, from Saturday, 24 August 2013, titled Forming Equations from Integer Sequences. It's only short so I'll quote it in full:

Recently I've been using Twitter to create a daily tweet that records my "day count" (number of days I've been alive) plus its factors (if not prime) and some interesting facts about the number itself or one of its factors. Sometimes there's little to say about the number and in such cases I've found that I can usually form an equation by inserting mathematical operators between one or more of the digits. 

For example, yesterday the count was \(23518\) and \(23 - 5 = 18.\) Today the count is \(23519\) and \(2 + 3 + 5 - 1 = 9\). I was wondering if it's always possible to create an equation from five digits using the standard mathematical operators (addition, subtraction, multiplication, division and exponentiation in combination with brackets). Obviously with just two digits, it's only possible when the digits are repeated e.g. \(99\) becomes \(9 = 9\). With three digits, it's sometimes possible e.g. \(819\) becomes \(8 + 1 = 9\) but generally it isn't e.g. \(219\). With four digits, it's more possible e.g. \(2119\) becomes \(-2 + 11 = 9\) but I'm doubtful whether this is always so. There must come a point however, where the number of digits is sufficient to ensure that it's always so. Maybe five digits is that point.

From now on, I'll try each day to form an equation to test out this theory. For example, tomorrow the count is \(23520\) which becomes \(2+3-5=2 \times 0 \) and it works for tomorrow but beyond that let's see.

Well, I didn't keep my promise of trying to form an equation each day from the digits making up my diurnal age. I have however, written about selfie numbers in a post of Friday, 27th March 2020 to my Mathematical Meanderings blog site. These are somewhat similar in spirit. In that post, I also mention Friedman numbers that can be described as follows:

Consider \(28547 =(8+5)^4−(7 \times 2)\) expressed in base 10, both sides use the same digits. An integer is a Friedman number if it can be put into an equation such that both sides use the same digits but the right hand side has one or more basic arithmetic operators (addition, subtraction, multiplication, division, exponentiation) interspersed. Brackets, as usual, are essential to clarify the order of operations. These numbers are named after Erich Friedman, Assoc. Professor of Mathematics at Stetson University. With the help of his students he has researched Friedman numbers in bases 2 through 10 and even with Roman numerals. When both sides use the digits in the same order, the number is called a ”nice” or ”strong” Friedman number. For example, \( 3125=(3+[1\times2])^5.\)

My approach is similar to this except I'm trying to create an equation from the digits (using them in the same order as they appear in the number). As another example, today I'm \(26121\) days old and this is an easy one because \( (2 \times 6)/12=1 \) if we allow concatenation of digits. If only individual digits are allowed, then \(-(2-6)=1+2+1 \) satisfies. It's probably better to use the individual digits as is done with the Friedman numbers. I'll try to include this as part of my daily number analysis. 

Going back a few days, we have:

• \(26120 \text{ --> } 2 = \frac{6}{1 + 2 +0} \)
• \(26119 \text{ --> } 2 + 6 \times 1 = -1 + 9 \)
• \(26118 \text{ --> } 2 + 6 + 1 = 1 + 8 \)
• \(26117 \text{ --> } 2 + 6 = 1 \times 1 + 7 \)
• \(26116 \text{ --> } 2 + 6 = 1 + 1 + 6 \)
• \(26115 \text{ --> } -2 + 6 + 1 \times 1 = 5 \)
• \(26114 \text{ --> } -2 + 6 = 1 - 1 + 4 \)
• \(26113 \text{ --> } -2 + 6 = 1 \times 1 + 3 \)
• \(26112 \text{ --> } -2 + 6 = 1 + 1 + 2 \)
• \(26111 \text{ --> } 2 = \frac{6}{1+1+1}\)
• \(26110 \text{ --> } 2 \times 6 \times (1-1) = 0 \)

It seems that it's always possible to form an equation using only brackets and the basic arithmetic operators of addition, subtraction, multiplication, division and exponentiation. Perhaps even the exponentiation is not needed, as the examples above show. Let's see if this assumption holds true for future numbers. This is hardly high level mathematics but it's a simple yet oddly satisfying activity.