ON THE APPROXIMATE SOLUTIONS IN INTEGERS OF A SET OF LINEAR EQUATIONS

BY H. F. BLICHFELDT

DEPARTMENT of Mathematics, Stanford UNIVERSITY, CALIFORNIA

Read before the Academy, April 26, 1921

1. It has been proved that a set of n+1 integers1 x1, x2, exist which satisfy approximately the n equations2

where α1, ..., αn, B1,

(1)

ẞ, are 2n given real numbers, the first n of which are subject to a condition (C) stated below. The degree of approximation is as follows: having specified a positive number e as small as we please, then the numerical values of the left-hand members of the above equations (i.e., the "errors") can simultaneously be made smaller than e. The condition (C) imposed upon a1, a2, .... an is this: they are to be irrational numbers of such a nature that a linear expression of the form a。 + a1α1 + ... + anan, with integral coefficients a, a, vanish exactly only when these coefficients are all = 0. For a given approximate solution of the equations (1) the errors shall be designated, respectively, by e1, ..., .

En.

=

βε

=

... an, can

=

βη

=

0,

2. In the two special cases: (I) n = 1, or (II) B1 we can make additional demands upon the errors, namely (and this is of importance in applications) we can cause them to be smaller than certain corresponding functions of w. Thus, in the case (I), we can demand that ← < € and at the same time that < 1/(4 | w |); in case (II) we can demand ......., eŋ be each <e and at the same time < k/(w), where k n/(n + 1), and m = -(n-1)/n; furthermore, the condition (C) is not necessary in case (II).3 These results may be stated in another way: in case (I), a being irrational, an infinite number of sets of integers (x', w'), (x", w"), .... exist which, when substituted in turn in x

that €1,

=

β

απ

B = 0, will produce errors e', e", .... of such magnitudes that e' < 1/(4 | w' | ), e′′ < 1/(4 | w"), etc. Similarly, in case (II) we have an infinite number of solutions satisfying corresponding inequalities.

3. In the general case the errors cannot be made to satisfy such extra demands. Stated more precisely: no matter what function f(w) be assumed, if only it be subjected to the conditions (D) below, we can always construct a set of equations (1) with attendant condition (C), which do not admit an infinite number of approximate solutions in integers (x1, x2, ..., x),...., if we demand the following degree of accuracy for each solution: €1 < ƒ( | w | ), ..., en < ƒ( | w | ). In fact, we may even substitute the

weaker demand.

+ €162. . En _1 < ƒ( | w |).

The conditions (D) referred to are: for every positive integer w, f(w) is a positive number which decreases when w increases, and approaches 0 as a limit when w tends to ∞.

4. We proceed to give an outline of the proof, limiting ourselves to the case n = 3, which is sufficiently illustrative. The function f(w) being given subject to (D), we shall take as our equations the following set (under an obvious change of notation):

Here r, s, t represent three arbitrary, but different integers, while a, ß, y are defined as the limits approached when j tends to∞ by three series of generalized (Jacobian) continued fractions x;/w;, y;/wj, zj/w;; j = 1, 2, .... We take at the outset any four sets of non-negative integers (x1, .., wi), ., (X4, w1), such that the determinant (x1223W) = 1; succeeding sets (xj, .., w;; j>4) are constructed from previous sets by the rule:

.....

Xj = Sj−1Xj−1 + Xj− 4, Yj = Sj−1Y;−1 + y;-4, etc., involving a series of positive integers S4, S5, S6, which, in turn, are expressed in terms of another series designated by [3], [4], [5], ..; namely, we put s; = 2 [5] − [j − 1] ̧ The members of the new series are any set of positive integers satisfying the following conditions:

5. For convenience in the further development we make use of the letters M, m, μ to represent certain non-negative coefficients (constants or variables), the actual values of which are not required: M for a number having a lower bound > 0, m for one having an upper bound, and μ for one having both a lower bound > 0 and an upper bound.

We note the following preliminary relations:

Xj μ.25-11, y; = μ.2-11, etc.;

=

§;=x; — aw; = ±m/sj, nj=y;— ßw; = ±m/Sj, $j=%; —YW; =±m/sj; = ±m. 2−1]-[i]+[i-1] when j > i, Y;w; XiWj xjW;= (x;w;) ji, — YjW; = (y;W;) = = etc., etc. Moreover, (x;w;), etc., do not vanish when j is taken sufficiently high.

(The condition (C), §1, is satisfied by a, B, y. For, an equation a。 +a1α +α2ß+αзy = 0 would imply aw; +ax; +a1⁄2yj+а3%; = 0 for every subscript j above a certain number. But this requires a。=A1=A2=A3=0, since the determinant (x,y;+1;+2Wj+3)= 1.)

6. If x, y, z, w represent any four integers, then at least two of the numbers

A =

XWj

=

x; (w+r), B = yw; Vj (w+s), C ZWj Zj (w+1) are of magnitude M.2-11-2 - 21, when j exceeds a certain number. For selecting any pair, say A, B, we have

(Aq − Bp) w;−1 + (r − s)pq=0

(mod. w;),

Aq

=

where p (x;w;−1), q=(y;w;−1). But this congruence is found to imply wj-1>wj for sufficiently high values of j. Hence, either Aq|or| Bp or both are > w;/w;-1. 17 for

7. Reverting now to the equations (3), we take any approximate solution in integers, say x, y, z, w. Assuming for our purpose that w w≥ 2[4]-[3], we determine the positive integer j such that

We have

=

2-11--21 | w | < 2-1-11.
2] ≤ 1]

WjE1 |A+(w+r)§j], wj6= |B+(w+s)n;\, w;e3= |C+(w+1)$;], and it follows from the results above that at least one of the three products €12, 62€3, €€ is of magnitude M.2-4-2) for sufficiently large values of j. But, under the same condition,

M. 2-4[j-2] > 2-6 (j − 2] > ƒ(2lj − 1] - [j − 2]) ≥ ƒ( [w] ).

Hence, when j exceeds a certain number, the condition (2) is not fulfilled. That is, the values of w satisfying this condition are limited.

1 Throughout this paper the term "integer" means "a positive or negative integer or zero," when it is not specially defined.

Kronecker, L., Monatsberichte Kgl. Preuss. Aka. Wiss., 1884 (1179ff., 1271ff.); Werke, 31, Leipzig, 1899 (49-109).

'See Dickson, L. E., History of the Theory of Numbers, 2; Carnegie Institution, Wash., 1920 (93-99), for references.

A PHYSICAL BASIS FOR EPIDEMIOLOGY

BY SIMON FLEXNER AND HAROLD L. AMOSS

LABORATORIES OF THE ROCKEFELLER INSTItute for MedICAL RESEARCH

Read before the Academy, April 26, 1921

In the autumn of 1918 there swept through one of the mouse breeding rooms of the Rockefeller Institute a destructive epidemic of mouse typhoid-an infection of mice with a bacillus of the enteritidis group of organisms to which the name of Bacillus typhi murium has been given. The history of the epidemic is instructive: the original mice of the population, numbering about 3000 at the time of the epidemic, came from a breeder in Massachusetts and had been purchased some time before and moved en masse to the Rockefeller Institute. In the meantime many new mice had been born of this stock, and many of the original mice had died or been employed for experiment, so that only a small residue of the original population remained. In other words, the epidemic of mouse typhoid arose among chiefly a new stock of mice, the offspring of an old stock believed on good grounds to have passed through previous outbreaks of the disease.

There is still another reason for supposing that the epidemic arose from within and was not imported from without this stock. Besides the breed

ing room for the population accruing from the purchased mice, a second breeding room not far away is maintained for mice bred at the Institute from perfectly healthy stock. This second breeding station has been developed from small beginnings and now has an average mouse population of 3000. The personnel in charge of each breeding room is distinct and does not mingle. During the period of many months in which the epidemic continued, rose and fell in the one room, no death from mouse typhoid occurred in the other.

Our attention has long been directed to the phenomena of the epidemic prevalence of disease, and since the early years of the founding of the Rockefeller Institute opportunity was presented to observe in succession epidemics of meningitis, poliomyelitis, influenza, and latterly of encephalitis. The waves of these diseases, which have swept over the world since 1904, have aroused profound interest in the facts of epidemiology. We seized, therefore, upon this outbreak of mouse typhoid in order to study experimentally an epidemic disease among small laboratory animals which could be assembled and observed in fairly large numbers. The epidemic referred to has supplied the cultures of the Bacillus typhi murium for the experiment and certain data of a statistical nature with which to compare our experimental results. The healthy stock of mice also referred to provided an unexceptionable material with which to attempt the production of an intentional epidemic.

The conditions of the experiment were simple. A kind of mouse village was set up and an isolated room away from all other animals was selected in which shelves were erected and the cages placed. The latter may be taken for streets and houses in an ordinary village. Each cage contained five healthy mice. The original population, later increased from time to time by the introduction of fresh increments, was 100. The epidemic was started by feeding with a virulent culture of Bacillus typhi murium ten mice placed in two cages midway of the other cages. The incidental contact between the culture fed mice and the others was secured by the attendant who fed the animals and periodically cleaned the cages. A spot map was kept in order to follow the events.

The preliminary feeding experiments led to the death from mouse typhoid of eight of the ten hand-fed and seven contact mice distributed in as many cages. No epidemic in the real sense ensued. The fatal outbreak of mouse typhoid which arose partook of the nature of sporadic instances of the disease.

When the conditions as regards death from the infection had become stationary (at about the end of 30 days), a fresh addition of 200 mice contained in cages of five each was brought into the village. The effect of this addition was striking: after a lapse of about five days the new

mice began to die of mouse typhoid, and five to ten days later the old mice (original population) also began to succumb. The total losses in this outbreak were 70 per cent, and the number of cages involved all, or 100 per cent. The epidemic endured about four weeks and then subsided, although an occasional death still occurred at intervals.

When a second equilibrium had been established, another addition of normal or healthy stock was introduced into the village. The succession of events was about the same as that just described. However, the peak of the epidemic was less high, the total percentage of deaths being 5 per cent, although 100 per cent of the cages was again attacked. After an interval, another equilibrium was reached, when still another addition of normal mice brought about a recrudescence, passing as in the other examples from the healthy new population to the exposed old population, with however a total percentage of deaths below the others.

These fluctuations or movements of the epidemic consisting of a series of waves have been repeated now about ten times. The general character or type of the wave or curve is always the same, although the height or peak as determined by the percentage of mortality varies. In every instance the mortality rate among the old mice became finally about the same as among the new.

The experiments are followed in two main ways: according to the mortality, controlled of course by bacteriological examination of the animals succumbing, and according to the living carriers of the Bacillus typhi murium produced. The feces and the urine of living mice are tested for the bacillus, and in general it may be stated that the percentage of carriers stands in inverse ratio to the mortality. In other words, high death rate means a low carrier rate and vice versa. An obvious, although by no means safe, deduction from this observation would be that the carriers have all recovered from non-lethal infection and hence are rendered resistant by an acquired immunity, and that the production of carriers is nature's way of limiting the epidemic and of bringing about the state of equilibrium described.

The main results of the study presented may be viewed as the mere beginnings of an undertaking to follow artificially reproduced epidemics among animals, which not only should simulate the epidemics naturally or spontaneously arising, as we say, among men and animals, but which being controllable may be made to yield up some of the underlying conditions affecting the movements among those naturally occurring epidemic outbreaks which we represent graphically in the form of curves, but which in relation to cause and effect we still are so largely ignorant. The impulse to use epidemic diseases among laboratory animals to help solve the problem of epidemics among man has been felt by other experimenters. In particular Topley1 in London has independently made use