Mind Over Mathematics:
Higher Arithmetic --
Residues of Powers

by Bruce Director

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Pedagogical Discussion

Beyond Counting -- A Preparatory Experiment

by Bruce Director

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        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. 
        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.

Beyond Counting -- Part II

by Bruce Director

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        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;

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
        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.

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The preceding article is a rough version of the article that appeared in The American Almanac. It is made available here with the permission of The New Federalist Newspaper. Any use of, or quotations from, this article must attribute them to The New Federalist, and The American Almanac.

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