A FEW CLASSIC UNKNOWNS IN MATHEMATICS
BY PROFESSOR G. A. MILLER
UNIVERSITY OF ILLINOIS
KING HIERO is said to have remarked, in view of the marvelous
mechanical devices of Archimedes, that he would henceforth
doubt nothing that had been asserted by Archimedes. This spirit of
unbounded confidence in those who have exhibited unusual mathematical
ability is still extant. Even our large city papers sometimes speak of a
mathematical genius who could solve every mathematical problem that
was proposed to him. The numerous unexpected and far-reaching results
contained in the elementary mathematical text-books, and the ease
with which the skilful mathematics teachers often cleared away what
appeared to be great difficulties to the students have filled many with a
kind of awe for unusual mathematical ability.
In recent years the unbounded confidence in mathematical results
has been somewhat shaken by a wave of mathematical skepticism which
gained momentum through some of the popular writings of H. Poincaré
and Bertrand Russell. As instances of expressions which might at first
tend to diminish such confidence we may refer to Poincaré's contention
that geometrical axioms are conventions guided by experimental facts
and limited by the necessity to avoid all contradictions, and to Russell's
statement that "mathematics may be defined as the subject in which we
never know what we are talking about nor whether what we are saying
is true.''
The mathematical skepticism which such statements may awaken is
usually mitigated by reflection, since it soon appears that philosophical
difficulties abound in all domains of knowledge, and that mathematical
results continue to inspire relatively the highest degrees of confidence.
The unknowns in mathematics to which we aim to direct attention here
are not of this philosophical type but relate to questions of the most
simple nature. It is perhaps unfortunate that in the teaching of elementary
mathematics the unknowns receive so little attention. In fact,
it seems to be customary to direct no attention whatever to the unsolved
mathematical difficulties until the students begin to specialize in
mathematics in the colleges or universities.
One of the earliest opportunities to impress on the student the fact
that mathematical knowledge is very limited in certain directions presents
itself in connection with the study of prime numbers. Among the
small prime numbers there appear many which differ only by 2. For
instance, 3 and 5, 5 and 7, 11 and 13, 17 and 19, 29 and 31, constitute
such pairs of prime numbers. The question arises whether there is a
limit to such pairs of primes, or whether beyond each such pair of prime
numbers there must exist another such pair.
This question can be understood by all and might at first appear to
be easy to answer, yet no one has succeeded up to the present time in
finding which of the two possible answers is correct. It is interesting to
note that in 1911 E. Poincaré transmitted a note written by M. Merlin
to the Paris Academy of Sciences in which a theorem was announced
from which its author deduced that there actually is an infinite number
of such prime number pairs, but this result has not been accepted because
no definite proof of the theorem in question was produced.
Another unanswered question which can be understood by all is
whether every even number is the sum of two prime numbers. It is very
easy to verify that each one of the small even numbers is the sum of a
pair of prime numbers, if we include unity among the prime numbers;
and, in 1742, C. Goldbach expressed the theorem, without proof, that
every possible even number is actually the sum of at least one pair of
prime numbers. Hence this theorem is known as Goldbach's theorem,
but no one has as yet succeeded in either proving or disproving it.
Although the proof or the disproof of such theorems may not appear
to be of great consequence, yet the interdependence of mathematical
theorems is most marvelous, and the mathematical investigator is attracted
by such difficulties of long standing. These particular difficulties
are mentioned here mainly because they seem to be among the simplest
illustrations of the fact that mathematics is teeming with classic
unknowns as well as with knowns. By classic unknowns we mean here
those things which are not yet known to any one, but which have been
objects of study on the part of mathematicians for some time. As our
elementary mathematical text-books usually confine themselves to an
exposition of what has been fully established, and hence is known, the
average educated man is led to believe too frequently that modern
mathematical investigations relate entirely to things which lie far beyond
his training.
It seems very unfortunate that there should be, on the part of educated
people, a feeling of total isolation from the investigations in any
important field of knowledge. The modern mathematical investigator
seems to be in special danger of isolation, and this may be unavoidable
in many cases, but it can be materially lessened by directing attention
to some of the unsolved mathematical problems which can be most
easily understood. Moreover, these unsolved problems should have an
educational value since they serve to exhibit boundaries of modern
scientific achievements, and hence they throw some light on the extent
of these achievements in certain directions.
Both of the given instances of unanswered classic questions relate to
prime numbers. As an instance of one which does not relate to prime
numbers we may refer to the question whether there exists an odd perfect
number. A perfect number is a natural number which is equal to
the sum of its aliquot parts. Thus 6 is perfect because it is equal to
1 + 2 + 3, and 28 is perfect because it is equal to 1 + 2 + 4 + 7 + 14.
Euclid stated a formula which gives all the even perfect numbers, but no
one has ever succeeded in proving either the existence or the non-existence
of an odd perfect number. A considerable number of properties
of odd perfect numbers are known in case such numbers exist.
In fact, a very noted professor in Berlin University developed a
series of properties of odd perfect numbers in his lectures on the theory
of numbers, and then followed these developments with the statement
that it is not known whether any such numbers exist. This raises the
interesting philosophical question whether one can know things about
what is not known to exist; but the main interest from our present point
of view relates to the fact that the meaning of odd perfect number is so
very elementary that all can easily grasp it, and yet no one has ever
succeeded in proving either the existence or the non-existence of such
numbers.
It would not be difficult to increase greatly the number of the given
illustrations of unsolved questions relating directly to the natural numbers.
In fact, the well-known greater Fermat theorem is a question of
this type, which does not appear more important intrinsically than many
others but has received unusual attention in recent years on account of
a very large prize offered for its solution. In view of the fact that those
who have become interested in this theorem often experience difficulty
in finding the desired information in any English publication, we proceed
to give some details about this theorem and the offered prize. The
following is a free translation of a part of the announcement made in
regard to this prize by the Königliche Gesellschaft der Wissenschaften,
Göttingen, Germany:
On the basis of the bequest left to us by the deceased Dr. Paul
Wolskehl, of Darmstadt, a prize of 100,000 mk., in words, one hundred
thousand marks, is hereby offered to the one who will first succeed to
produce a proof of the great Fermat theorem. Dr. Wolfskehl remarks
in his will that Fermat had maintained that the equation
xλ + y
λ = zλ
could not be satisfied by integers whenever λ is an odd prime number.
This Fermat theorem is to be proved either generally in the sense of
Fermat, or, in supplementing the investigations by Kummer, published
in
Crelle's Journal, volume 40, it is to be proved for all values
of λ for which it is actually true. For further literature consult
Hilbert's report
on the theory of algebraic number realms, published in volume 4 of the
Jahreshericht der Deutschen Mathernatiker-Vereinigung, and volume 1
of the Encyklopädie der mathematischen Wissenschaften.
The prize is offered under the following more particular conditions
The Königliche Gesellschaft der Wissenschaften in
Göttingen decides independently on the question to whom the prize
shall be awarded. Manuscripts intended to compete for the prize will not
be received, but, in awarding the prize only such mathematical papers
will be considered as have appeared either in the regular periodicals or
have been published in the form of monographs or books which were for
sale in the book-stores. The Gesellschaft leaves it to the option of the
author of such a paper to send to it about five printed copies.
Among the additional stipulations it may be of interest to note that
the prize will not be awarded before at least two years have elapsed since
the first publication of the paper which is adjudged as worthy of the
prize. In the meantime the mathematicians of various countries are
invited to express their opinion as regards the correctness of this paper.
The secretary of the Gesellschaft will write to the person to whom the
prize is awarded and will also publish in various places the fact that the
award has been made. If the prize has not been awarded before
September 13, 2007, no further applications will be considered.
While this prize is open to the people of all countries it has become
especially well known in Germany, and hundreds of Germans from a
very noted university professor of mathematics to engineers, pastors,
teachers, students, bankers, officers, etc., have published supposed proofs.
These publications are frequently very brief, covering only a few pages,
and usually they disclose the fact that the author had no idea in regard
to the real nature of the problem or the meaning of a mathematical
proof. In a few cases the authors were fully aware of the requirements
but were misled by errors in their work. Although the prize
was formally announced more than seven years ago no paper has as
yet been adjudged as fulfilling the conditions.
It may be of interest to note in this connection that a
mathematical proof implies a marshalling of mathematical results, or
accepted assumptions, in such a manner that the thing to be proved is a
necessary consequence. The non-mathematician is often inclined
to think that if he makes statements which can not be successfully
refuted he has carried his point. In mathematics such statements have no
real significance in an attempted proof. Unknowns must be labeled as
such and must retain these labels until they become knowns in view of
the conditions which they can be proved to satisfy. The pure
mathematician accepts only necessary conclusions with the exception that
basal postulates have to be assumed by common agreement.
The mathematical subject in which the student usually has to contend
most frequently with unknowns at the beginning of his studies is
the history of mathematics. The ancient Greeks had already attempted
to trace the development of every known concept, but the work along
this line appears still in its infancy. Even the development of our
common numerals is surrounded with many perplexing questions, as may
be seen by consulting the little volume entitled "The Hindu-Arabic
Numerals,'' by D. E. Smith and L. C. Karpinski.
The few mathematical unknowns explicitly noted above may suffice
to illustrate the fact that the path of the mathematical student often
leads around difficulties which are left behind. Sometimes the later
developments have enabled the mathematicians to overcome some of
these difficulties which had stood in the way for more than a thousand
years. This was done, for instance, by Gauss when he found a necessary
and sufficient condition that a regular polygon of a prime number of
sides can be constructed by elementary methods. It was also done by
Hermite, Lindemann and others by proving that e and π are
transcendental numbers. While such obstructions are thus being gradually
removed some of the most ancient ones still remain, and new ones are
rising rapidly in view of modern developments along the lines of
least resistance.
These obstructions have different effects on different people. Some
fix their attention almost wholly on them and are thus impressed by the
lack of progress in mathematics, while others overlook them almost
entirely and fix their attention on the routes into new fields which avoid
these difficulties. A correct view of mathematics seems to be the one
which looks at both, receiving inspiration from the real advances but
not forgetting the desirability of making the developments as continuous
as possible. At any rate the average educated man ought to know
that there is no mathematician who is able to solve all the mathematical
questions which could be proposed even by those having only slight
attainments along this line.