by Bruce Director

In previous pedagogical discussions on Higher Arithmetic, we investigated the ordering of numbers with respect to arithmetic (rectilinear) progressions, (as in the case of linear and polygonal numbers) and geometric (rotational) progressions, as in the case of geometric numbers, and prime numbers. (See Doc.#'s 97267bmd01; 97316bmd001; 97326bmd001;)

The deeper implications of these investigations, which form the basis of Gauss' re-working of Greek classical geometry, reveal themselves, only if we rise above intuition, and investigate the nature of numbers with the mind only.

In the coming weeks, we will begin to further investigate these principles. But, it will be much more efficient, if the reader first performs the following experiments:

As discovered in an earlier pedagogical discussion, the geometric progression is constructed by beginning with a square, whose sides are a unit length, and whose area is a unit area. We then add 2, 3, 4, or more squares forming a rectangle. We then double, triple, quadruple, this rectangle forming a new square, and so forth. With each successive action, the area of the corresponding square or rectangle increases, but the type of action, doubling, tripling, quadrupling, etc. doesn't change. A different type of number, incommensurable with rectilinear numbers, is discovered by this process. (This construction is discussed by Plato in the beginning of the Theatetus dialogue.)

In contradistinction, the rectilinear (polygonal) numbers, form a series in which each number associated with a given polygon, increases by an increasing amount, but the differences between the differences remains the same. And so, under Gauss' concept of congruence, all polygons of the same type can be brought into a One, because the differences are all congruent, relative to a modulus which is the number of sides minus 2. The totality of all polygons, can be thought of as a series of series, ordered by successively increasing moduli.

For geometric numbers, however, there is no simple modulus, under which the individual members of any given geometric progression can be made congruent. Or, put another way, the change from rectilinear (1 dimension) to rotational (2 dimension) changes the ordering principle. We must shift tactics. The old rules, don't apply. We must discover a new, higher type of congruence. This new higher type of congruence, opens the door to whole new domain.

To discover the nature of this domain, it is most efficient to follow in Gauss' footsteps, and first discover these orderings experimentally, and then investigate the deeper implications, which underlie these orderings.

Each different geometric progression can by also thought of as a series of numbers associated with the underlying action, in order of increasing actions. For example, 2 for doubling. The first number in the series is a unit area, which has undergone no doubling, i.e., 2^0 or 1. The second number is the first doubling, or 2^1 or 2. The third number is the second doubling or 2^2 or 4. The third number is the third doubling, or 2^3 or 8, etc. This forms the geometric progression, 1, 2, 4, 8, 16, ....

Another example, 3 for tripling. The first number is the unit area which has undergone no tripling or, 3^0 or 1. Then the first tripling, 3^1 or 3; the second tripling 3^2 or 9; the third tripling 3^3 or 27. This forms the series 1, 3, 9, 27, ....

Now investigate the congruences of these series with respect to odd prime numbers as moduli. Begin with modulus 3. Calculate the least positive residues of the numbers of the geometric progression based on 2 with respect to 3 as a modulus. Then take 5 as a modulus. Calculate the least positive residues of the numbers of the geometric progressions based on 2, 3, 4, with respect to 5 as a modulus. Then take 7 as a modulus. Calculate the least positive residues of the numbers of the geometric progressions based on 2, 3, 4, 5, 6 with respect to modulus 7.

What new type of orderings emerge? What's going on here? We will begin to investigate these questions, next week.

by Bruce Director

If you carried out the experiment in last week's discussion, you would have discovered the reflection of an ordering principle with respect to the residues of geometric progression. The experiment should have yielded the following result.

With respect to modulus 5, the residues of the geometric progressions based on the numbers 2-5 yield the following results: (The Powers are in the first row; the residues resulting from a specific geometric progression are in the rows which follow. The base is the type of action from which the geometric progression is generated -- 2 for doubling; 3 for tripling; etc.).

Powers: 0 1 2 3 4 5 6 7 8 9 10
Base 2: 1 2 4 3 1 2 4 3 1 2 4 etc.
Base 3: 1 3 4 2 1 3 4 2 1 3 4 etc.
Base 4: 1 4 1 4 1 4 1 4 1 4 1 etc.

For Modulus 7:
Powers: 0 1 2 3 4 5 6 7 8 9 10
Base 2: 1 2 4 1 2 4 1 2 4 1 2 etc. 
Base 3: 1 3 2 6 4 5 1 3 2 6 4 etc.
Base 4: 1 4 2 1 4 2 1 4 2 1 4 etc.
Base 5: 1 5 4 6 2 3 1 5 4 6 2 etc.
Base 6: 1 6 1 6 1 6 1 6 1 6 1 etc.

This is a surprising result. The unbounded, ever increasing geometric progression, is brought into a simple periodic ordering with respect a prime number modulus. No matter which type of change (base) of the geometric progression, a periodic cycle emerges with respect to a prime number modulus. Each period, begins with unity, making a sort of wave pattern. While the "wavelength" may change with the base, the "wavelength" is always either the modulus minus 1 (m-1) or a factor of m-1. No other "wavelengths" are possible. The bases whose "wavelengths" are m-1 are called "primitive roots." (In the examples above, 2 and 3 are primitive roots of 5; 3 and 5 are primitive roots of 7.)

These orderings were investigated by Fermat and Leibniz, and, according to Gauss, Leibniz' investigations of these orderings, were a subject of the oligarchical slave Euler's attack on Leibniz, played out in the famous fight between Koenig and Maupertuis. In his Disquisitiones Arithmeticae and the two Treatises on Biquadratic residues, Gauss unfolds even deeper implications of these orderings, which will be discussed in future pedagogical discussions. For now, it is sufficient to reflect on the subjective questions presented by the phenomena.

In order to even begin to discover what's going on here, you must think in an entirely different way about numbers. What accounts for these orderings? The answer will elude you, if you cannot free yourself from a conception of number associated with mere quantity of objects. Just as the discovery of valid physical principles, such as the orbit of the asteroid Ceres, will elude you, if you cannot free your mind from fixating on the mere observations. The answer lies outside the orderings themselves, and can only be reconstructed inside the mind, by reflecting on the paradoxes presented.

Instead of thinking of each number individually, think instead of a series from 1 to m-1, associated with a unique principle of generation, that contains each number. Each principle of generation is characterized by a distinct type of curvature. One principle of generation, is the principle of adding one (rectilinear). Another principle of generation is the principle of adding areas (sprial action). A third principle of generation, is the principle of congruence (circular rotation). A fourth principle of generation is the principle of prime numbers. The combination of all four characterizes a hypergeometry, the unfolding of which, generates the periodic orderings reflected in the residues of powers.

The subjective challenge, is to be able to conceive in your mind of the interconnection of these generating principles as a One, when that One cannot be expressed as a mathematical function. The functional relationship exists only in the mind. Just as the One of a musical composition exists not in the notes, or the physical characteristics of the well-tempered system, but in the Idea of the composition, which is transfinite with respect to the unfolding of the composition.

In the interest of not diverting attention from concentrating on these subjective questions, the reader is advised to continue these experiments with respect to the prime numbers 11, 13, 17, and 19. In future discussions, we will rediscover Gauss' application of this principle in his re-working and superseding classical geometry.