Numbers are a fundamental concept in mathematics. We learn of them in the early stages of our math education (in our elementary classes), but not many of us can explain the nature of numbers succinctly. In school, we were wrongly introduced to numbers as abstract symbols that are used in arithmetic operations.

If we want to go far in our journey of understanding mathematics, it is imperative that we have a clear understanding of the different kinds of numbers.

We define numbers as a proportion with respect to an arbitrarily chosen unit of measure.

Unit – In counting separate objects or in measuring magnitudes, the standards by which we count or measure are called unit.

Number- Repetitions of the unit are expressed by numbers.

The theory of numbers is a subset of mathematics that deals with different kinds of numbers. It is sometimes referred to as the queen of mathematics because of its foundational place in the discipline.

In his expository book which is entitled ‘A Scrap Book of Elementary Mathematics’, Professor William F. White explains the theory of numbers in an elegant manner.

In this article, we will mention some theorems in number theory as elaborated by Professor William F. White. He writes:

The theory of numbers has been called a “neglected but singularly fascinating subject.” “Magic charm” is the quality ascribed to it by the foremost mathematician of the nineteenth century (Gauss).

Gauss is quoted to have said ‘The most beautiful theorems of higher arithmetic (theory of numbers) have this peculiarity- They are easily discovered by using mathematical induction, but can be hard to prove demonstrably and can be ferreted out only by very searching investigations. It is precisely this which gives to higher arithmetic that magic charm which has made it the favorite science of leading mathematicians’. Gauss is also quoted as saying ‘the theory of numbers has an inexhaustible richness, wherein it so far excels all other parts of pure mathematics’.

Gauss said also: “Mathematics is the queen of the sciences, and arithmetic [i. e., theory of numbers] the crown of mathematics.” And he was master of the sciences of his time. “While it requires some facility in abstract reasoning, it may be taken up with practically no technical mathematics, is easily amenable to numerical exemplifications, and leads readily to the frontier. It is perhaps the only branch of mathematics where there is any possibility that new and valuable discoveries might be made without an extensive acquaintance with technical mathematics.”

An interesting exercise in higher arithmetic is to investigate theorems and the established properties of particular numbers to determine which have their origin in the nature of number itself and which are due to the decimal scale in which the numbers are expressed.

Fermat’s last theorem- Of the many theorems in numbers discovered by Fermat, nearly all have since been proved. A well-known exception is sometimes called his “last theorem.”

It “is to the effect that no integral values of x, y, z can be found to satisfy the equation xn + yn = zn, if n is an integer greater than 2. This proposition has acquired extraordinary celebrity from the fact that no general demonstration of it has been given, but there is no reason to doubt that it is true.”

It has been proved for special cases, and proved generally if certain assumptions be granted. Fermat asserted that he had a valid proof. That may yet be rediscovered; or, more likely, a new proof will be found by some new method of attack. “Interest in problems connected with the theory of numbers seems recently to have flagged, and possibly it may be found hereafter that the subject is approached better on other lines.”

Another popular theorem that acquired celebrity is the Wilson’s theorem. Of Wilson’s theorem, William F. White writes:

Wilson’s theorem may be stated as follows: If p is a prime, 1+ (p – 1)! is a multiple of p. This well-known proposition was enunciated by Wilson, first published by Waring in his Meditations Algebraic, and first proved by Lagrange in 1771.

Formulas for prime numbers. “It is easily demonstrated that no rational algebraic formula can always, give primes. Several remarkable expressions have been found, however, which give a large number of primes for consecutive values of x. Legendre gave 2x2+29, which gives primes for x = 0 to 28, or for 29 values of x. Euler found x2 + x + 41, which gives primes for x = 0 to 39, i. e., 40 values of x. I have found 6x2+ 6x +31, giving primes for 29 values of x; and 3x2+ 3x + 23, giving Primes for 22 values of x. These expressions give different primes. We can transform them so that they will give primes for more values of x, but not different primes. For instance, in Euler’s formula if we replace x by x – 40, we get x2 – 79x + 1601, which gives primes for 80 consecutive values of x.”

A Chinese criterion for prime numbers- With reference to the so called criterion, that a number p is prime when the condition, that 2p – 1 – 1 be divisible by p, is satisfied, Mr. Escott makes the following interesting comment:

“This is a well-known property of prime numbers (Fermat’s Theorem) but it is not sufficient. My attention was drawn to the problem by a question in L’Intermédiaire des Mathematicians, which led to a little article by me in the Messenger of Mathematics. As the smallest number which satisfies the condition and which is not prime is 341, and to verify it by ordinary arithmetic (not having the resources of the Theory of Numbers) would require the division of 2340 – 1 by 341, it is probable that the Chinese obtained the test by a mere induction.”

Additional reading * Are there more than one set of prime factors of a number? Most text-books answer no; and this answer is correct if only arithmetic numbers are considered. But when the conception of number is extended to include complex numbers, the proposition, that a number can be factored into prime factors in only one way, ceases to hold.

1. g., 26 = 2 *13 = (5 + )(5 – ). *

Asymptotic laws- This happily chosen name describes “laws which approximate more closely to accuracy as the numbers concerned become larger.”*Legendre is among the best-known names here. One of the most celebrated of the original researches of Dirichlet, in the middle of the last century, was on this branch of the theory of numbers.

Growth of the concept of number, from the arithmetic integers of the Greeks, through the rational fractions of Diophantus, ratios and irrationals recognized as numbers in the sixteenth century, negative versus positive numbers fully grasped by Girard and Descartes, imaginary and complex by Argand, Wessel, Euler and Gauss, has proceeded in recent times to new theories of irrationals and the establishing of the continuity of numbers without borrowing it from space.

Tables- Many computations would not be possible without the aid of tables. Some of them are monuments to the patient application of their makers. Once made, they are a permanent possession. The time saved to the computer who uses the table is the one item taken into account in judging of the value of a table. It is difficult to appreciate the variety and extent of the work that has been done in constructing tables. For this purpose an examination of Professor Glaisher’s article “Tables” in the Encyclopedia Britannica is instructive.

Anything that facilitates the use of a book of tables is important. Spacing, marginal tabs (in-cuts), projecting tabs—all such devices economize a little time at each handling of the book; and in the aggregate this economy is no trifle. Among American collections of tables for use in elementary mathematics the best example of convenience of arrangement for ready reference is doubtless Taylor’s Five-place Logarithmic and Trigonometric Tables (1905). Dietrichkeit’s Siebenstellige Logarithmen und Antilogarithmen (1903) is a model of convenience. When logarithms to many places are needed, they can be readily calculated by means of tables made for the purpose, such as Gray’s for carrying them to 24 places (London, 1876).

Factor tables have been printed which enable one to resolve into prime factors any composite number as far as the 10th million. They were computed by different calculators. “Prof. D. N. Lehmer, of the University of California, is now at work on factor tables which will extend to the 12th million. When completed they will be published by the Carnegie Institution, Washington, D.C. According to Petzval, tables giving the smallest prime factors of numbers as far as 100,000,000 have been calculated by Kulik, but have remained in manuscript in the possession of the Vienna Academy. . . Lebesgue’s Table des Diviseurs des Numbers goes as far as 115500 and is very compact, occupying only 20 pages.”

Some long numbers. The computation of the value of to 707 decimal places by Shanks* and of e to 346 places by Boorman are famous feats of calculation.

“Paradoxes of calculation sometimes appear as illustrations of the value of a new method. In 1863, Mr. G. Suffield, M.A., and Mr. J. R. Lunn, M.A., of Clare College and of St. John’s College, Cambridge, published the whole quotient of 10000 . . . divided by 7699, throughout the whole of one of the recurring periods, having 7698 digits. This was done in illustration of Mr. Suffield’s method of synthetic division.”

Exceptions have been found to Fermat’s theorem on binary powers (which he was careful to say he had not proved). The theorem is, that all numbers of the form 22n + 1 are prime. Euler showed, in 1732, that if n = 5, the formula gives 4,294,967,297, which = 641 x 6,700,417. “During the last thirty years it has been shown that the resulting numbers are composite when n = 6, 9, 11, 12, 18, 23, 36, and 38; the two last numbers contain many thousands of millions of digits.”§ To these values of n for which 22n + 1 is composite, must now be added the value n = 73. “Dr. J. C. Morehead has proved this year [1907] that this number is divisible by the prime number 275 . 5 + 1. This last number contains 24 digits and is probably the largest prime number discovered up to the present.”* If the number + 1 it were written in the ordinary notation without exponents, and if it were desired to print the number in figures the size of those on this page, how many volumes like this would be required? They would make a library many millions of times as large as the Library of Congress.

Additional Reading * Some results of permutation problems: The formulas for the number of permutations, and the number of combinations, of n dissimilar things taken r at a time is given in every higher algebra. The most important may be condensed into one equality:

nPr = n(n – 1)(n – 2) …. (n – r + 1) = = nCr *r!

There are 3,979,614,965,760 ways of arranging a set of 28 dominoes (i. e., a set from double zero to double six) in a line, with like numbers in contact.* “Suppose the letters of the alphabet to be wrote so small that no one of them shall take up more space than the hundredth part of a square inch: to find how many square yards it would require to write all the permutations of the 24 letters in that size.” Dr. Hooper computes that “it would require a surface 18620 times as large as that of the earth to write all the permutations of the 24 letters in the size above mentioned.”

Fear has been expressed that if the epidemic of organizing societies should persist, the combinations and permutations of initial letters might become exhausted. We have F.A.A.M., I.O.O.F., K.M.B., K.P.,

I.O.G.T., W.C.T.U., Y.M.C.A., Y.W.C.A., A.B.A., A.B.S., A.C.M.S., etc., An almanac names more than a hundred as “prominent in

New York City and its list is exclusive of fraternal organizations, of which the number is known to be vast. Already there are cases of two societies having names with the same initial letters. But by judicious choice this can long be avoided. Hooper’s calculation supposed the entire alphabet to be employed in every combination. Societies usually employ only 2, 3 or 4 letters. And a letter may repeat, as the A in the title of the A.L.A. or of the A.A.A. The present problem is therefore different from that above. The number of permutations of 26 letters taken two at a time, the two being not necessarily dissimilar, is 262; three at a time, 263; etc. As there is occasionally a society known by one letter and occasionally one known by five, we have

261 + 262 + 263 + 264 + 265 = 12,356,630.

This total of possible permutations is easily beyond immediate needs. By lengthening the names of societies (as seems to have already begun) the total can be made much larger. Since the time when Hooper’s calculations were made; two letters have been added to the alphabet. When the number of societies reaches about the twelve million mark, it would be well to agitate for a further extension of the alphabet. With these possibilities one may be assured, on the authority of exact science, that there is no cause for immediate alarm. The author hastens to allay the apprehensions of prospective organizers. *

*The content in italics is the added text*