THE CHONG-3 SQUARES OF SQUARES
—from intriguing opacity to radical clarity

SUMMARY

Dissatisfied with a sequence of arithmetic operations which started with addition, continued with multiplication and stopped at exponentiation, theorists have come up with proposals to universalize these operations. Several of these proposals now belong to the theory of hyperoperations; my own proposal for a universal hierarchy of operations was introduced in The Chong Operators — a universalization in arithmetic. In this hierarchy, addition (without iteration) is found on the zero level, multiplication on the first, exponentiation on the second and iterated exponentiation on the third level of iteration. It is on this third level where the first nontrivial sequence generated wholly and solely by the chong-3 operator is a sequence which, after two initial numbers, contains only squares of squares (while having no parallel or counterpart among hyperoperations). I will provide the definition and calculation formulas of this unique sequence and discuss the squares which are, or may be, the most interesting from different perspectives. While all of this concerns pure number theory, the same sequence of squares of squares will be playing a central role in a universal integrated linguistic-mathematical numeral supersystem as well.

INTRODUCTION

In geometry squares and cubes are very popular objects; and so are questions such as How many 1-by-1 squares do the first five perfect squares count? and How many small cubes does it take to create one perfect large cube with a width, length and height of three cubes? This geometrical interest is immediately related to the attention paid to second powers or 'squares' and third powers or 'cubes' in algebra. But why should we pay special attention to the one collection of squares, or cubes, for that matter, and not to the other? So long as we confine ourselves to a traditional algebra which treats exponentiation as an operation at the highest and last iterative level there is, indeed, no reason to think of such triples of 'perfect' squares like {4,16,256}, {9,81,6561} and {25,625,390625} as belonging to fundamentally different mathematical categories. Can the introduction of higher levels of iteration affect this view; and, if so, how?

In order to be able to answer this question, we should have a closer look at the arithmetic operations which come after exponentiation. The arithmetic operators involved should all be part of one system of universalization, which, as far as binary operators are concerned, starts with addition. However, there is no such 'one system'! There is today's established sequence of hyperoperators and there is the sequence of chong operators which were introduced in my paper The Chong Operators — a universalization in arithmetic thirteen years ago. These two universalizations differ in several respects. I will confine myself here to the most important difference, a difference which may remind one of the distinction between countries, with legal right-side driving and those with legal left-side driving exclusively. The idea behind this is that in motorized traffic you cannot leave it to the individual which side of the road to choose (or to stay in the middle). In a mathematical universalization of operations you are forced to do something similar: when repeating a process on the level of iterated exponentiation (and higher) that iteration is either left- or right-associative. For example, without any further arrangement or agreement 3^3^3 means 27^3 = 19,683, or it means 3^27 = 7,625,597,484,987. In other words, 3^3^3=(3^3)^3, that is, left-associative, or 3^3^3=3^(3^3), that is, right-associative. Now, in traffic we must choose between the one or the other. In mathematics, however, there is only a convention of right-associativity in the absence of brackets or 'parentheses'. However, to treat this convention as the one and only possibility is a narrow-minded position to take: the least one should do is to look at the advantages and disadvantages of the one approach, and the advantages and disadvantages of the other. In mathematics there is not even a need to prescribe the right and proscribe the left, or vice versa, so long as the two are not mixed up.

A thorough (unpublished) study of the consequences of choosing for left- versus right-associativity led me to favor the chong operators even more over the hyperoperators; and, incidentally, to the subject of this article. Among other things, the chong operators provide us with the answer to the question why a quadruple such as {2,4,16,256}, and the entire sequence of which it is the beginning, has a characteristic distinguishing it fundamentally from quadruples such as {3,9,81,6561} and {5,25,625,390625}, and the entire sequences of which the latter are the beginnings.

THE CHONG OPERATORS

In The Chong Operators i^* have shown 'what a universal system of arithmetic operations based on their level of iteration can or will look like', as summarized in the first paragraph. (The word iteration refers to systematic repetition. The prefix re- in reiteration and its cognates ought to be reserved for iterations on the level of exponentiation or higher where there is really a higher-level iteration of a lower-level one.) My own universalization in arithmetic concerns binary operators only and is characterized by:

• a distinction between positive, zero and negative levels, whereby there is, strictly speaking, no iteration on the zero level, while the operator on a positive level is the inverse of the one on the corresponding negative level, and vice versa;
• a 'plus-zero' level for addition and a 'minus-zero' level for subtraction;
• a left-associative approach from iterated exponentiation on, where the difference between left- and right-associativity matters
• a break with the terminological past of mathematics by deriving a new morpheme and symbol from a natural language on the basis of its inherent suitability, regardless of its connection with Greco-Roman antiquity.

The inclusion of both positive and negative levels into a universal hierarchy of iteration is of tremendous importance, as it stretches the hierarchy from the infinitesimally small to the infinitely large. It will help to recover those suffering from an integer bias against what they call "fractions"; or worse, from the BIC, the Big Integer Craze. Nonetheless, for the purpose of the present article a discussion of the chong-plus operators suffices. Should the chong operators be incorporated into one universal linguistic-mathematical system for both operators and numbers, then the Chinese chong morpheme and 重 character will, like all Greek and Latin morphemes and characters, have to make way for the morphemes and symbols of a synthetic language for these numbers and operators in which only substantive syntactic, (inter)morphemic, (inter)phonemic and (inter)orthographical criterions count.

In the box below you will find a recapitulation of the definitions and main equations for the chong-plus operators.

The Basic Formulas of the Chong-plus Operators

THE CHONG-PLUS-3 TABLE

The table itself on the menu

There is no shortage of tables in arithmetic, but a great many so-called 'tables' are as disappointingly unalike real tables as plain chairs. When a dictionary defines table as tabular arrangement of data and tabular as of, relating to or arranged in a table this unenlightening circularity reveals at least that we are talking about a data table; and when it defines chart as form designed to record or provide information quickly and simply: table, graph, diagram it shows that it does not make a sharp distinction between a chart and a table, which is not very helpful either. Some call an addition table a "chart" too, which is odd, since addition, and also multiplication tables contain, for the time being, no changing information whatsoever, the sort of data you would expect in a chart. Another oddity is that after the addition and multiplication tables it is not an 'exponentiation table' which follows, but a power or 'exponent' table or chart, or some graphic display which is neither a table nor a chart. While all these tables and nontabular displays involve only one operation, an exponential table involves two operations: multiplication and exponentiation, because it represents an exponential function, that is, a funcion of the form y=a×b^x, in which a is the initial value and b the constant ratio.

What is a chong table?

Clearly, it does not seem a luxury to define explicitly what a chong table is supposed to be:

• it is a data table in the sense of a systematic two-dimensional arrangement in parallel columns and rows in which the left-most column (the y-axis, as it were) lists the (axial) values of one variable one by one and the top-row (the x-axis) the (axial) values of a different variable one by one;
• the (nonaxial) values in the cells where another column than the first one and another row than the top one cross each other express a relation, if any, between the two variables;
• in a chong-l table the variable on the far left is the a in a重^lb or chong(a,l,b), the variable at the top is the b in that expression.

From this general definition it should be clear what a chong-plus-2 or 'chong-2' table is supposed to be. It is the table which expresses the integer values of a重²b or chong(a,2,b) with a≥1 and b≥1. In other words, an exponentiation table pure and simple.

How to calculate the values in a chong-3 table

Just as the chong-2 operator systematically repeats, that is, iterates the chong-1 operation (multiplication) a certain number of times, so the chong-3 operator iterates the chong-2 operation (exponentiation) a certain number of times. (Keep in mind tho^** that repeating something b times for b=1 is not repeating anything and leaves a unchanged.) So, in the chong-2 table the value in cell (a,b), for a=2 and b=2, is chong(2,2,2)=2×2=4, the outcome of 2^2; in the chong-3 table the value in the same cell will be chong(2,3,2)=2^2=4 too. In the chong-2 table the value in cell (2,3) is chong(2,2,3)=(2×2)×2=8, the outcome of 2^3; in the chong-3 table the value will be chong(2,3,3)=(2^2)^2=16. In cell (2,4) chong(2,2,4)=((2×2)×2)×2=16 and chong(2,3,4)=((2^2)^2)^2=256. In cell (2,5) chong(2,2,5)=(((2×2)×2)×2)×2=32 and chong(2,3,5)=(((2^2)^2)^2)^2=65,536. These are all cells in the row where a=2. Two more examples should illustrate what happens in the cells (a,b), for a=3 and b=2, and for b=3. In cell (3,2) chong(3,2,2)=3×3=9, the outcome of 3^2, and chong(3,3,2)=3^3=27. In cell (3,3) chong(3,2,3)=(3×3)×3=27, the outcome of 3^3, and chong(3,3,3)=(3^3)^3=19,683. It may be impossible in practice, but in this way all nonaxial values in the chong-3 table can, in principle, be calculated, from a=1 to infinity and from b=1 to infinity.

First the squares of squares, then the cubes of cubes

What is being generated in the chong-2 table for a=2 is all values of chong(2,2,b), values which together constitute nothing else than a geometric sequence with 2 as its first term and 2 as its factor or 'common ratio'. For a=3, chong(3,2,b) generates all terms of a geometric sequence with 3 as its first term and 3 as its factor. Now, what does the chong-3 operator generate? It will not be a geometric sequence, let alone a so-called 'arithmetic' one. On the basis of its definition it will be a chong-2 sequence, but one of which the calculated values form a chong-3-generated sequence. (See my article A Substantive Terminology for Sequences and Cycles with a discussion of the difference between a definition formula and a calculation formula of a sequence.) The very first nontrivial sequence generated in the chong-3 table is a sequence of squares, apart from 2, the first term. However, it is not an ordinary set of squares such as the totally meaningless set starting with the five so-called 'perfect' squares 1, 4, 9, 16 and 25! (This talk of 'perfection' is in the eye of the beholder, someone with an adult integer bias.) The very first sequence generated in the chong-3 table is, apart from its first two terms (2 and 4) a sequence of squares of squares, and it is the sole such chong-generated sequence of squares. It is not followed by a sequence of squares, even squares of squares, such as {3,9,81,6561,43046721,...}; it is followed by a sequence of cubes, the sequence {3,27,19683,7625597484987,...}. The top part of the chong-3 table should be enough to attest to this.

DEFINITION AND CALCULATION FORMULAS

Defining versus calculating

Like any sequence, a chong-generated sequence has a definition formula and a calculation formula, as there is an essential difference between defining a sequence of numbers and calculating its terms or 'members'. While defining a sequence of numbers, size or other properties of the individual terms do not play a role. It is the relation between the terms which count. (See the paragraph on definition and calculation in my article A Substantive Terminology for Sequences and Cycles.)

The definition formula

The definition formula tells you how a new term x_n+1 follows after and from (usually) one or more previous terms. Its general form is x_n+1= f_d(x_n, x_n-1, x_n-2, ... x₁, x₀) in which d stands for definition. The definition formula may be considered an 'indirect access' formula, since it does not give the value of a term in a direct way, but only via the previous term or one or more other terms preceding it. A chong-generated sequence on the lth level of iteration, that is, a chong-l-generated sequence has a definition formula of which the operator is of level l-1. In the first chong-3 sequence (after the trivial 1重³b one) the definition formula is x_n+1= x_n重²2, with n≥0 and x₀=2. Note that the value 2 occurs in the chong-3 table as the value for a=2 and b=1. This initial value is the source of the sequence generated and cannot be left out of the formula, even if we are interested in squares, and especially squares of squares, only. Note also that the index s which will be used for the squares of squares, starting with 0 for the number 4 and 1 for the number 16, is not the same as the index n in the definition formula, starting with 0 for the number 2, 1 for the number 4 and 2 for the number 16! Since the definition formula uses merely a chong-2 operator, which is a pre-universalization operator, we can express the definition formula simply as x_n+1=x_n².

The calculation formula

The general form of a calculation formula is x_n= f_c(n) in which c stands for calculation. This may be considered a 'direct access' formula as it enables one to calculate the value of a term without making use of the value of any other term in the sequence, with the exception of the initial value. The other values may be and remain completely unknown. To find the calculation formula itself, however, we must first mount the index ladder until we have a clear enough view of what happens in the process. It starts with x₀=2=2重³1. For n=1, x₁=4=2重³2. For n=2, x₂=16=2重³3. For n=3, x₃=256=2重³4. For n=4, x₄=65536=2重³5. At this stage we should have sufficient information to conclude that the calculation formula is x_n=2重³(n+1). Because the chong-3 operator is a universal one the question arises of how to use this operator in traditional calculations, if possible at all. One method is to forget about those traditional calculations and use or write a computer program with a left-associative recursive function which can do the job for any level of iteration. I have done this myself in the preparation for the chong tables, of which the chong-3 table above is only a lower- or intermediate-level one. Another method, however, is to reduce the expression with the chong-3 operator to an expression in which only chong-2 and/or lower-level operators occur. Let's pursue that path here. We know that the chong-3 level is the level of reiterative, that is, systematically repeated, exponentiation, and therefore one of the (eight) exponent rules will probably be very helpful: the Power-raised-to-power Rule or 'the Power-of-power Law'. According to this rule (a^m)ⁿ= a^m×n, (a^m×n)^p= a^(m×n)×p, (a^m×n×p)^q= a^(m×n×p)×q, and so on. The a is the same as in the chong-3 formula: a=2. All we have to do now is to keep track of the number of as. The calculation formula tells us that x_n=2重³(n+1). We need an index and substitute a for 2: a重³(n+1)= ((((a₁^a₂) ^a₃) ... ) ^a_n) ^a_n+1= a₁^ (a₂×a₃× ... ×a_n×a_n+1)= a^(a^n)= 2^(2^n). This is precisely the kind of formula one would expect, a formula with exponentiation within the exponent. Keep in mind tho that n is the index of the whole sequence originally generated, from 2 on.

The only chong-generated sequence of squares of squares

We have discovered that while there may be an infinite number of sequences of squares, even of squares of squares, there is only one 'chong-generated sequence', that is, a sequence generated by a single chong operator, which contains the and all consecutive squares of squares. The complete sequence generated starts with 2, 4 and 16, but 2 itself is not a square of integers, to which we have confined ourselves here. The term 4 is a square of integers but not a square of a square, like 16, the first term that is. Hence, if we want to justify our speaking of 'the sequence of squares of squares', 16 should be the first member of that sequence, the one with index 1, while 4 may be considered a member with index 0. The head-16 power-2 'sequence' is a subsequence of the head-2 power- or 'chong-2 functor'-2 sequence —on the definitory chong-2 level power is the functor— with its own index for which i will use the letter symbol s. Hence, s=x-1, and to obtain specifically the value of a square of squares the formula x_s=2^(2^(s+1)) must be used. For example, the value of 'square 4', the fourth square (of squares), is 2^(2^(4+1))= 2^(2^5)= 2^32= 4,294,967,296.

Two types of chong-generated subsequence

One should take care not to confuse the head-16 power-2 subsequence of squares of squares with the head-16 power-16 sequence which is a standard chong-3-generated sequence like the head-2 power-2 sequence. The former is generated by 2重³b, whereas the latter is generated by 16重³b! The head-16 power-16 sequence starts with the value 16 (the initial number) and continues with 16^16, (16^16)^16, ((16^16)^16)^16, and so on, ad infinitum. It can be proved that the calculation formula for this sequence is 16重³n= 2^(2^(2+4×(n-1)), which means that 16重³n= 2^(2^(4n-2)). When we compare this with the 2重³b-generated sequence, the difference of 2 in the second exponent accounts for the fact that 16 occurs at b=3 in the head-2 power-2 sequence and at b=1 in the head-16 power-16 sequence. Compared with the calculation formula 2^(2^n) the index is also multiplied by 4, which proves that the head-16 power-16 sequence is also a subsequence of the head-2 power-2 sequence (just like the head-4 power-4 sequence with the calculation formula 4重³n=2^(2^(2n-1))). Nonetheless, the head-16 chong-generated sequence (and the head-4 chong-generated sequence) is a very different type of subsequence: it skips three of the elements of the head-2 chong-generated sequence at evey step (but without changing the order of the elements). The head-16 subsequence of the squares of squares, however, skips only the first two elements of the head-2 chong-generated sequence and then includes all its elements until infinity consecutively. Such a subsequence is called a "substring". The head-16 substring is almost the same as the complete sequence itself; no more than a slight adjustment of it at the beginning.

IN THE WAY OF THE RUT

Anything special about the squares of squares?

The fact that the 4-16-256 sequence is single-chong-level-generated may be a unique feature of that sequence among sequences of squares, it does not mean, as we have seen, that there is not an infinite number of other single-chong-level generated sequences, even an infinite number of other chong-3-generated ones, such as the 27-19,683-7,625,597,484,987 sequence of cubes of cubes. Is there, perhaps, something that makes the sequence of squares of squares even stand out among (almost) all chong-generated sequences as well? To tackle this issue we should analyze a part of the squares of squares individually, because a JavaScript program which takes us from 4 to a number with the approximate value of 1.3e154 (or 1.3E+154) is not likely to be the final means to a definite answer, if at all possible.

Approximations are not exact numbers

Let's first take the way of the rut. In a literal sense this 'rut' is the deep narrow track worn by the many wheels which have followed the same course thru a field for many months or years, perhaps, since time immemorial. In a figurative sense applied to arithmetic it is the impress left on the human mind by the use of a particular age-old, time-honored numeral system in calculations and the languages of speech communities. The rut which we will follow here is that of the established radix or 'base' 10 numeral system, presently the one most frequently employed in human intercourse. Even with a simple JavaScript program it should be no big deal to get the notation of the fifth square of squares in denary figures: it is 18,446,744,073,709,551,616. This is the exact number satisfying the present standard notation of numbers: the real fifth square does not even deviate from it by 1 or 0.0000000000000000054%, if only because its last digit must be 6! From the sixth square, or so, on exact numbers may not be given anymore, but a so-called 'standard (index) form' or 'scientific form/notation' is used — a misnomer, because the only standard/scientific form is the exact number to the last digit. Of course, towards the infinitely large and the infinitesimally small the standard notation of numbers is just becoming too long to write it out in full. So, what we get instead of the really standard and scientific form (the exact number) is a (scientific) approximation in the shape of 3.402823669209384634633746e+38 meaning 3.402823669209384634633746×(10^(+38)). (The small e may also be a capital E, and the + sign may be left out where no confusion is possible.) In tables i may abbreviate this to 3.4e38, provided that this much shorter approximation (which is something like 10^23 times less accurate) does not eradicate the difference with any other number in the same table.

When even the exponents become approximations

For the time being you may be lucky that you get to the ninth square, for that is about where a JavaScript program may confront you with 'Infinity'. When you continue with a scientific calculator 'Infinity' turns out to be approximately of the size 1.797693134862315907729305e+308 or, less accurately, 1.8e308. If you are lucky again your calculator or general-purpose computer may spew out the approximation 1.415461031044954789001553e+9864, say 1.4e9864, for the fourteenth square, just before the fifteenth where it cannot handle it anymore. Now it is not 'Infinity' but 'Overflow' which stops you. However, even without the shelter of a well-funded exact sciences organization with a supercomputer at its disposal, you need not despair (yet). There are websites on the internet which will let you go on to at least the 21st square of squares as far as exact numbers are concerned, and to at least the 1101st square of squares as far as approximations are concerned. But in the latter case even the powers of 10 in the approximations, which were always exact numbers in the beginning, have turned into approximations themselves.

Beyond one googol, but not yet one googolplex

The table below lists 101 squares of squares and gives the exact numbers of the first five (squares 1 to 5) and of the twenty-first (square 21, in the Appendix). Of squares 6 to 21, square 65, square 81 and square 101 the approximations are given. Of all other squares neither an exact number nor an approximation is known (by me), and they have not been calculated (yet) because, supposedly, they will only show more of the same of a size between a smaller and a larger square of which the exact number or approximation is already given. Those interested in, familiar with, or obsessed by googols and googolplexes ought to understand that radix-10 numbers or values like those cannot be found under the squares of squares, but they may be pleased to know that the number googol (1.0e100 and actually 1e100) is located somewhere between square 7, of which the approximation is 1.2e77, and square 8, of which the approximation is 1.3e154. None of the squares in this table has a value anywhere near the value of googolplex: the googolplex value is exactly 10^(10^100), whereas the value of square 101 is approximately 10^(10^30.2), as i will show later. Why, then, a table of 101 squares: why not many less, or many more? This choice is, indeed, nothing to do with the sizes of the squares. It is everything to do with the final-digit cycles in their radix-10 notations. More about that after the presentation of this table.

FINAL DIGIT CYCLES

Getting the last ten digits of squares 1 to 7

The above table with its one hundred and one squares of squares shows the last one, two, three, four and five digits of each square, regardless of its complete radix-10 notation being known or being shown anywhere. This may seem intriguing, but it is only possible if the same number of last digits of the previous square is known. It is explained for the last five digits in the second note of the table above, but i will demonstrate it here for the seven squares of squares in the table below, in which even the last ten digits have been calculated separately. The calculation starts with 0000000004, that is, the number 4 with nine leading zeros, so that we have a ten-digit notation. When 0000000004 is squared we get 16 preceded by eighteen 0s, of which we only use the ten last digits, that is, 0000000016, which is the 10-digit radix-10 notation for (the) square (of squares) 1. The ten last digits of square 2 are obtained by squaring 0000000016, which is 256 preceded by seventeen 0s, of which the abbreviation is 0000000256. The number 0000000256 squared is 65536 preceded by fifteen 0s, but abbreviated to 000006553, the 10-digit radix-10 notation for square 3. Square 4 is the first square without leading zeros in its 10-digit tail, because 256^2=4,294,967,296, which is exactly ten digits long. Squaring 4294967296 yields 18,446,744,073,709,551,616 in a scientific calculator, which is the whole notation for square 5, of which the last ten digits are 3709551616. From now on the last ten digits will form only the end part of the number in radix-10 notation. At the same time, calculating the whole number by just squaring the previous square may not be possible anymore. Just try 18446744073709551616^2 and what you (may) get is 3.4028236692093846346337460743177e+38, which is an approximation, not the exact number! However, in this case we can combine this approximation with our precise knowledge of the last ten digits and so contruct the exact number. (See note 2 in the table below.) Squaring 3709551616 yields 13,760,773,191,768,211,456, of which the last ten digits are 1768211456, the 10-digit end of the notation for square 6. Squaring 1768211456 results in 3,126,571,753,129,639,936, of which the last ten digits are 3129639936, the end of square 7.

The 1- and 4-term one- and two-digit cycles

The long table above starts with the same eight squares, but it shows the final digit cycles as well. Since 6^2=36, which ends in 6 too, the first cycle is a one-term one-digit cycle, shown alternately in light yellow and light aqua or 'cyan'. The second set of vertically arranged numbers is {16,56,36,96}, which, apart from the sixes of the first cycle does not tell us anything. But once the first number reappears in the same column, we know that the whole set {16,56,36,96} must reappear, and that this set represents a four-term final two-digit cycle. It is shown alternately in light green or 'lime' and light purple.

A 20-term three-digit cycle

Four terms is a short cycle, but what do we do with the final 3-digit set {016,256,536,296,616,456,936} in the first seven squares of squares? So far, there is no return of 016 te be seen, but there is an occurrence of 616, precisely at the place where the previous, two-digit cycle restarts. Hence, as long as 016 does not show up, there is a good reason for paying special attention to the numbers where the previous cycle restarts. And, indeed, all numbers in those places end in 16: square 5 has 616 as we knew already, square 9 has 216, square 13 has 816 and square 17 has 416. Is there just a chance of 1 in 10 that 016 will turn up among 116, 216, 316, 416, and so on? The succession of numbers in {016,616,216,816,416} looks wholly arbitrary, and yet, on closer inspection there is a regularity, given that it is actually only the last three digits we are seeing. The difference 616-016=600 and 216-616=-400, which is of no help. However, if we take the 600 as a common difference in an arithmetic sequence, then what follows after 616 is 1216 without the first 1; and then 1816 without the first 1; and then 2416 without the first 2. If this is correct, what follows should be 3016 without the first 3, which is indeed the case, and the beginning of the second occurrence of the same 20-term final three-digit cycle, shown in light yellow and light aqua again. We are lucky that the three-digit cycle is only 20 terms long, because, all other things being equal, we had a 1 in 10 chance that the first digit was 0 and, in view of the two-digit cycle, a 1 in 4 chance that the second digit was 1, altogether a 1 in 40 chance that 01 followed by 6 would reappear.

A 100-term four-digit cycle

There is no regularity in the lengths of the first three cycles: they are, in the number of terms, 1, 4 and 20 (instead of, say, 16). Since 20=1.25*16, the fourth cycle threatens to be about 1.25*5*20=125 terms long; and what is the size of that square 125, or its neighbor, going to be, given that square 15 cannot even be approximated anymore by a scientific calculator because of overflow? Moreover, the arithmetic sequence found in {016,616,216,816,416} by allowing these terms to stand for numbers with an extra digit in front, is nowhere to be found in the set {16,56,36,96}, where 36 does not fit into any similar scheme. The column with the final four-digit numbers may be as unpredictable as the one with the final two-digit numbers and keep us waiting for more than 125 squares before 0016 occurs again. There is a 1 in 10 chance of 0 returning at the place of the first digit. After that, there is a 1 in 20 chance of 016 returning in the place of the second, third and fourth digits, in view of the three-digit cycle. Hence, the chance of 0016 returning is 1 in 200. Fortunately, the numbers at the beginning of each three-digit cycle tell us more: 0016 for square 1, 8016 for square 21, 6016 for square 41. Given that each quartet of digits may have a hidden digit in front, the set {0016,8016,6016} may be read as {0016,8016,16016}, which is the three-term beginning of the arithmetic sequence {0016,8016,16016,24016,32016,40016} with 8000 as the common difference. When we reduce the six-term beginning to six quartets again there emerges the set {0016,8016,6016,4016,2016,0016} which shows the return of 0016 after five 3-digit cycles of 20 terms, that is, one 4-digit cycle of 100 terms. The figures in the four-digit column of the table do indeed confirm this.

A 5-term five-digit (sub?)cycle starting from square 101

Is it possible to discern a five-digit cycle with the information in the table above? Let's assume that the regular pattern we have found in the three- and four-digit cycles persists, what data do we need then? We need at least the final five digits of square 101, because now it will be the squares 101, 201, 301, 401, and so on, which determine where to look for the return of 00016 in this five-digit cycle. The difference between square 1 and square 101 should become the addend or 'common difference' of an arithmetic sequence then, a sequence which can tell us in advance how long the five-digit cycle is going to be. In my table the last five digits of square 101 have been derived from the last five of square 100, and WolframAlpha at www.wolframalpha.com/, giving the last ten digits of the output for 2^(2^102), confirms that the last five are indeed 90016. This means that the addend might be 90,000 and the beginning of the sequence {00016, 90016, 80016, 70016, 60016} for the squares 1, 101, 201, 301 and 401 respectively. However, consulting WolframAlpha for the squares 201 to 401 i found the final-digit strings 30016 for square 201 and 10016 for square 401; only square 301 was 'predicted' correctly. Nonetheless, the crucial question which remained was still, Does the string 00016 return somewhere among the squares with order number 201, 301, 401, 501, and so on? I checked this until square 1101, a number as large as 10^(10^331.2), only to find out that 00016 does not return (within this range). And yet, my search was not fruitless, because i did find a 5-term 5-digit cycle (or subcycle?), but it does not begin at square 1; it begins at square 101! The final 5-digit strings for squares 101 to 1101 are: 90016, 30016, 70016, 10016, 50016, 90016, 30016, 70016, 10016 and 50016. Looking at them as numbers in there own right, we see the head-90016 addend-40000 sequence {90016, 130016, 170016, 210016, 250016, 290016, 330016, 370016, 410016, 450016}. Looking at them as strings again, and confining ourselves to the last five digits, we discover the consecutively recurring strings 90016, 30016, 70016, 10016 and 50016. For now, this leaves the squares 1 to 100 without any five-digit cycle, putting an end to my attempt to find the length of the final five-digit cycle in the radix-10 notation of the squares of squares. It does give a good impression, however, of what order of magnitude is involved here. (While the value of square 1101 is 10^(10^331.2), one googolplex is 'only' 10^(10^100)!)

THE DISTRIBUTION OF SQUARES OF SQUARES

The universalization of operators may be a big, perhaps, revolutionary contemporary step forward in arithmetic, an even bigger step was the introduction of multiplication in ancient times, because it was in this stage that a formalized operation of iteration was first introduced. (Doubtless, numerical repetition existed long before that stage already, but it was a revolution to treat it as a systematic process in itself.) With multiplication arithmetic moved from a noniterative chong-0 stage to an iterative chong-1 stage. It was also in this stage that a distinction was discovered between integers which can be created by means of multiplication and integers which cannot; or, conversely, larger composite numbers which are the product of two smaller integers, and those which are not. The latter ones were called "prime numbers" by ancient Greek mathematicians. They are in present terms positive integers which cannot be generated with a chong-1 operator other than on or by 1. These 'prime' numbers have fascinated number players and number theorists ever since. However little the prime numbers which cannot be generated on the first level of iteration and the squares of squares which can be generated on the third level of iteration have in common, there is one interesting feature which they share: they occur less and less frequently as they become larger, without disappearing entirely.

In the table below i show the distribution of the members of the chong-3 sequence of squares of squares by counting the number of integers left out between them from square 0 to square 7. It starts with a mere 11 integers between 4 (square 0) and 16 (square 1); it ends, that is, 'pauses', with approximately 1.1 times 10^77 integers between 340 282 366 920 938 463 463 374 607 431 768 211 456 (square 6) and 115 792 089 237 316 195 423 570 985 008 687 907 853 269 984 665 640 564 039 457 584 007 913 129 639 936 (square 7). In prime number theory there is a prime number theorem describing the asymptotic distribution of the prime numbers among the positive integers. There is already an analog for this theorem describing the 'distribution' of irreducible polynomials over a finite field. (See, e.g., en.wikipedia.org/[ ]wiki/Prime_number_theorem.) I do not know whether there will, similarly, ever be an analog for the sequence of squares of squares.

IN THE WAY OF THE SQUARES

Two-to-one and many-to-one correspondences

It is part of the way of the rut to think of a numeral system as a distinct system with one radix isolated from other numbers for being the most suitable or least unsuitable base or 'radix' there is. People who have thought about alternatives to the common base-10 numeral system, may swear by a new radix such as 12 or 16 —2 and 8 are for the computer— as 'the true one', as one radix of one system without any connection to other numeral systems. Now, 12 may be quite a distance from 144, and 256 an even greater distance from 16 when considering one's choice of radix, this type of relatively remote connection is not at all characteristic of what we encounter when we start at the beginning. At the very beginning there is 2, followed by 4 (2^2), 8 (2^3), 16 (itself 2^4), 32 (2^5), 64 (2^6) and 128 (2^7) before we get close to 144. And at the beginning there is 3, followed by 9 (3^2), 27 (3^3) and 81 (3^4). These numbers may seem of little interest, until it turns out that they represent fixed correspondences between number notations for varying radixes: two-to-one correspondences in which each couple of figures in the notations of the one numeral system can be directly replaced, that is, without any need for (further) calculation, with one figure in the other numeral system; three-to-one correspondences in which each triplet in the one system can be directly replaced with one figure in the other; four-to-one correspondences, and so on and so forth. Therefore, those who think they are choosing a radix and a single numeral system are actually choosing a numeral supersystem, for the right or for the wrong reason (or a mixture of the two).

Numeral supersystems

The concept of a numeral supersystem must play a central role in any adequate integrated linguistic-mathematical treatment of numbers, and i cannot fully discuss it in this article. May it suffice here to describe it in outline, while confining myself to positive integers. To start with, instead of just having a radix the supersystem has one superradix and an infinite number of subradixes which can be mathematically derived from it, the sole criterion being that a larger subradix will yield a shorter notation in a fixed two- or many-to-one correspondence between the figures in these notations. If the superradix is 2, the square of 2 and all squares of squares of 2 will be quadratic subradixes. Consider, for example, the number 14, of which the notation is 1110 with radix 2: it will be 32 with radix 2^2=4 after replacing the 11 pair with 3 (the notation for 3 in the radix-4 subsystem) and 10 with 2 (the notation for 2). Note that such a correspondence also applies to the subradix 16, where two two-to-one correspondences amount to one four-to-one correspondence between the radix-2 and -16 subsystems. It does not apply, however, to 8 as a subradix: the notation for the number 15, for instance, is 1111 with radix 2, 33 with radix 4, and 17 (1*8+7) with radix 8, which cannot be directly derived. Yet, 8 is a subradix too, as 8=2^3; and the subradix a cubic one. For cubic subradixes there is a three-to-one correspondence between their notation and the one with the superradix or previous subradix. Starting from 1111 for 15, we should first read this as 001111 with two triplets, 001, which is 1, and 111, which is 7 with radix 2; together 17 for 15 again. The following table shows the number of quadratic, cubic and other subradixes between 4 and 128 inclusive for the first eleven numeral supersystems. (The eleventh, the system with superradix 12, does not even have one subradix in this range!)

The (micro)macrobinary supersystem

There are at least three good reasons to opt for a superradix 2, with the radixes 4, 8 and 16 in its wake:

1. the number 2 is the elementary integer radix for a place-value system, which can even be used to deal with the true and the false
2. the radix-2 supersystem is the supersystem with by far the most subradixes of all of them, thus providing within its system the most two-to-one and many-to-one correspondences with other radixes
3. the radix-2, -4 and -16 supersystems are the only ones which cover all squares of squares as subradixes, altho they also cover squares which are subradixes for other reasons (such as 64=2^6=4^3=8^2)

Of course, the divisibility of a radix also plays a role: 6=2*3, 10=2*5, 15=3*5 and 30=2*3*5 all have something a radix derived from the superradix 2 does not have. In a final choice this too will have to be taken into account, but the answer is not as simplistic as tradionalists may expect, for the systematic selection of a radix must not depend on facts and feelings which are generated or engendered by the use of one or more different radixes in the past or at the same time. Take the subdivision of the naturally given whole day, for instance, which has, like the subdivision of the logically given full circle, grown into a radical hotchpotch. For an integrated linguistic-mathematical system there is also the indispensable criterion that a radix must not require more morphemes than are (made) available for the naming of numbers. Moreover, it should not be forgotten that one has to be able to more or less easily remember all these morphemes (and the tables of addition and multiplication which come with them)! Because of the high density of subradixes in the superradix-2 system especially these latter criterions can be satisfied better than in any other numeral supersystem. In this article i shall use a traditional 'scientific' terminology for its name and call it "the micromacrobinary supersystem" or, since i have only discussed integer radixes here, just "the macrobinary supersystem". It should come as no surprise now, that the squares of squares in particular are waiting for a macrobinary approach.

IN RADIX-16 NOTATION

A conversion which needs hardly any calculating

When converting a number from a radix-R to a radix-R² numeral system, each pair of digits in the radix-R system is being replaced, in the same order, with one in the radix-R² system. (If the number of digits is odd, you will have to add 0 in front.) Hence, 111001 [2] (111001 in the radix-2 system) will be 321 [4] (321 in the radix-4 system), with binary 11 read as "3", binary 10 as "2", and (every) 01 as "1". And, in turn, 321 [4], the same as 0321 [4], will be 39 [16], with 03 read as "3", and 21 as "9". (While this number may be 3*16+9*1=48+9=57 in denary terms, such a calculation is not by any manner of means part of the conversions.) Thus, when we see in the above table that Square Three is 1 0000 0000 0000 0000 in radix-4 notation (with a space between the segments instead of a comma for typographical reasons), we may conclude immediately that this will be 1 0000 0000 in radix-16 notation, because every couple 00 in the former notation corresponds with one 0 in the latter, and 1 or 01 refers to 1 in both numeral systems. However, in the scientific notation the exponent cannot be converted in the same direct way. It needs a little calculation, because, for x>0, 4^x has a value which differs from 16^x! Nonetheless, if x is the exponent in the radix-4 system and y the one in the radix-16 system, 4^x=16^y implies that 4^x=(4²)^y, from which it follows that x=2*y and y=x/2. Given that Square Three equals 1E+20 in radix-4 notation, it is therefore not 20 which must be converted, but a pair with half its value, that is, 10; and this 10 refers to 4 in the radix-4 system, while 4 refers to 4 in the radix-16 system. It follows that 1E+20 [4]=1E+4 [16]. In the case of Square Twenty-one it is the 200 which must first be reduced to 100 or 0100; the number of zeros following will also have to be reduced from 8 (two four-digit segments) to 4 (one four-digit segment). Since 0100 [4]=10 [16], the radix-16 equivalent of 1E+200 0000 0000 [4] will be 1E+10 0000 [16]. In other words, there is no need anymore to show radix 4 side by side with radix 16. Radix 16 suffices.

An unchanging head and ever-elongating tail

With the changeover from the radix-10 to the radix-16 notations for squares of squares their intriguing final-digit cycles disappear like snow in summer. From Square One on all squares of squares simply end in a zero, albeit in an increasing number of them: one zero in Square One (10=1E+1) to more than a million zeros in Square Twenty-one (1E+10,0000), as shown by the table above; and, for short- and long-scale base-10 'illionaires', to more than 340 uncillion or 'sextillion' zeros in Square One-hundred-and-twenty-nine (1E+1;0000,0000,0000,0000;0000,0000,0000,0000), as shown by the table below. Some will, perhaps, argue that there is a one-term final one-digit cycle not unlike the one in the radix-10 notation of the squares of squares, except that this one digit is 0 instead of 6. But if this is accepted as correct, it only is so in a very trivial sense, because, in the radix-10 notation, the squares never end in (6(6...6)6)66 —they end in 16, 56, 36 or 96— whereas, in the radix-16 notation, they all end in (0(0...0)0)0. Is there perhaps a nontrivial cycle at the beginning instead? Looking at the two macrobinary tables, there seems to be a four-term initial one-digit cycle in which the terms start with 1, 2, 4 and 8, after which the 1 returns in 10 (2*8=16 [10]=10 [16]). This, however, is a miserable illusion, for it may be a real cycle in the exponents of the shorthand representation of the numbers, it is not at all a cycle in the numbers themselves. At each step the string of zeros in a square becomes twice as long in the next square. (Compare, for example, 10,000² = 10,000*10,000 = (10^4)*(10^4) = 10^(2*4) = 10^8 =100,000,000.) While the notation of the squares of squares starts with 1, there follows first only 1 zero (in 10), then 2 zeros (in 100), 4 zeros (in 1,0000), 8 zeros (in 1,0000,0000, the number 1E+8 [16] refers to), 16 zeros (in the number 1E+10 [16] refers to), and so on and so forth. It is with a head remaining 1 forever, an ever-elongating tail of no more than zeros, and no body in between, that the broad or 'long' jumper in track-and-field mathematics attempts to reach the infinitely large, jump by jump.

The connection between the squares and morphemic economy

In the special paragraph about the micromacrobinary supersystem i have already touched upon the fact that in an integrated linguistic-mathematical system a radix must not require too many morphemes, a reason why a superradix larger than, say, twenty stands no, or little, chance of ever becoming the superradix of an integrated numeral system. Furthermore, it would be totally out of place to select, for exampple, a radix 16 and to continue to call numbers such as 13 [10]=D [16] and 31 [10]=1F [16] "thirteen" and "thirty-one", because their names of two and three morphemes could only acquire their meaning in a denary morphemic arithmetic. Another point is —and this depends, of course, on the natural or synthetic language used— that morphemes have to be assigned more carefully and more sparingly as fewer of them are or can be made available. Thus, in English there is a special morpheme for 1,000 (thousand), whereas there is no common morpheme (anymore) for 10,000. (Myriad is antiquated, while Chinese, for instance, has always had 万, wàn.) From the perspective of morphemic economy you would call 1,000 "ten hundred", unless you have plenty of morphemes left; and you would use a new morpheme for 10,000, because a duplication of the same morpheme on the same level, as in hundredhundred is undefined on morphemic-arithmetic grounds. (For the same reason 100 itself is not named "tenten".) In English there is a special morpheme for 1,000,000 too (million), but in a radix-16 system the special morpheme should go to 1,0000,0000, a 'hundred million' as far as the number of zeros is concerned. (In Chinese, the morpheme for this number is 亿, yì.) In general, you need either a new morpheme when you reach the next square of squares or you must have some type of formula which tells you which combination of at least one formulaic morpheme and one or more morphemes for smaller numbers refers to the larger number. In any event, keeping track of the chong-3 squares of squares will be indispensable.

CONCLUSION

Universalization opens up new vistas, by putting an end to outdated preconceptions and old prejudices; and by defying certain conventions. One such convention is right-associativity in mathematics. In the universalization of arithmetic operations the exponents of right-associativity never seriously analysed the pros and cons of the two mutually exclusive approaches possible: they kept their convention and were probably also pleased that it led to the biggest numbers in the fastest way. In this article i have discussed one of the most impressive features of the left-associative chong operators: the chong-3 sequence of squares of squares. Initially i stuck to radix 10 to show how extraordinary this sequence is, before changing over to radix 16, not as a paragon in itself, but as a subradix in a supersystem with the first radix of all. My radix-10 presentation led to numerous interesting observations; my radix-16 presentation proved how futile they were, however much similar observations may be useful in other cases. As far as i know, no-one before has shown any serious interest in an integrated linguistic-mathematical numeral system, if only because linguistics and mathematics do their work on branches of the tree of science very remote from each other. Yet, these two branches need to meet where it concerns the vocabulary of arithmetic, especially the words for numbers. I have not denied, let alone disproved, that right-associativity in arithmetic universalization serves the BIC best. Ironically, however, it is the entirely systematic approach of the squares of squares which discloses a numerical world in which googols and googolplexes (of size 10^(10^100) [10]) are mere peanuts in comparison with a number such as Square Four-hundred-and-one (of size 10^(10^120.49) [10]) and its successors. Nevertheless, no approach in arithmetic and in the history of counting is systematic that implicitly and unquestioningly takes the primacy of integers over portions (denigratingly called "fractions") for granted. In this article i have only confined myself temporarily to integers in order to make it easier to read for a contemporary audience of whom most may suffer from an (adult) integer bias; and of whom all lack the terminological apparatus to deal with portions on an equal footing. This may be an incitement rather than part of a conclusion: that the systematic number traveler goes up to the infinite and down to the infinitessimal in one move, without fear or favor.

APPENDIX: The 21st Square of Squares Written in Full

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Showing the exact radix-16 notation for Square Twenty-one

Finding and showing the exact radix-10 notation for Square Twenty-one

Calculation of the number of digits

The outcome of the Online Big Number Calculator consists of 1,683,483 characters (figures and spaces) from '20 650 635' (the beginning) to '394 198 016 ' (the end with space after '6'). First add one '0' in front, so that the notation starts with '020 650 '. Now the full number notation contains nothing but 4-character segments starting with three figures and ending in one space. Hence, there are (1683483+1)/4=420871 segments and the same number of spaces. Substract 420871 (spaces) from 1683484 and the next-to-final result is 1683484-420871=1262613. Finally delete the '0' before '20 ' at the beginning again, and the total number of digits turns out to be 1262613-1=1262612, which is also the number of decimal digits according to WolframAlpha.