Vol. 7,1921 MATHEMATICS: H. F. BLICHFELDT 
317 
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 w+1 integers 1 x h x 2 , . . ., x n , w 
exist which satisfy approximately the n equations 2 
%x — aiW — j8i = 0, . . . , X n — a n w — $ n = 0, (1) 
where a h . . . , a n , ft, . . . , /3 n 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 equa- 
tions (i.e., the "errors") can simultaneously be made smaller than e. 
The condition (C) imposed upon a h a 2 , .... a n is this : they are to be 
irrational numbers of such a nature that a linear expression of the form 
a 0 + ai<xi + . . . + a n a n , with integral coefficients a 0 , ai, . . . , a n , can 
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 e h . . . , e n . 
2. In the two special cases: (I) n = 1, or (II) ft = j3 2 = ■ . • = fi n — 0, 
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 
€i < e and at the same time that ei < 1/(4 | w |) ; in case (II) we can demand 
that ei, . . ., e n be each < e and at the same time < k/( \ w \ ) m , where k 
= n/(n + 1), and m = — in — l)/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 
{%', w') f {%", w"), .... exist which, when substituted in turn in % — aw — 
j8 = 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 as- 
sumed, 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 (x h x% . . . , 
. . . ., if we demand the following degree of accuracy for each solution: 
ei < /( I w | ), . . ., e n < /( | w | ). In fact, we may even substitute the 
weaker demand. 
