On some diophantine inequalities involving primes

x) When writing the original draft of this paper I had not seen the recent work of Wolfgang Schwarz [ü b er 
die Lösbarkeit gewisser Ungleichungen durch Primzahlen, J. Reine Angew. Math. *212 (1963), 150— 157]. Here a 
theorem involving prime powers is proved which includes an inequality of the type (3) as a special case. Schwarz's 
work suggests that analogous generalisations can be established for the result of the present paper.
(i)
(2)
(3)
I * i P i  4
h
P
i4- K P z  I <

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B a k er, On some diophantine inequalities involving primes
167
many primes P i , p 2, p$ satisfying the inequality
(4) 
I 
XiPi
+
hPi
+
h
1
< (log
where p denotes the maximum of p Lyp 2iP^^
In th e course of the proof of this result we are led to consider equations of the type
(5) 
b1p 1 + b2p 2 + bzp z = m,
where bx, b 2, bz and m denote integers. Under suitable conditions, we establish the 
existence of a solution of such an equation w ith primes p l y p 2,Ps below a specified 
bound. This extends some previous work of Richert (see [4]). From the special case m — 0 
we see th a t, unlike (2), the condition th a t one a t least of the ratios XJXj is irrational is 
necessary in th e enunciation of the above theorem ; for if 615 ¿>2, bz were relatively prime 
b u t 6j, b2 were both divisible by th e same composite num ber then clearly (5) would not 
be soluble. On th e other hand, we shall prove th a t, provided only bx, b 2, bz are relatively 
prime and do not all have the same sign, then (5) always possesses a solution w ith either 
m = I or m — 2.
Finally we rem ark th a t th e case A* = <%, A2 = A3 = ± 1 of our theorem implies 
th a t for any real num ber ¿x, not rational, and any positive integer n there exist infinitely 
m any rationals q/p, with p prime, such th a t
1
< P(log 
'
The result can be obtained by more direct means (for example, by methods similar to 
those indicated in the notes on p. 180 of [5]) and it would be of interest to ascertain 
w hether any sharper inequality of this type is v a lid 2).
2. Notation
We suppose throughout th a t a 1 < 0 , a2 > 0, az > 0 and th a t is irrational. 
Also we suppose th a t | a x |, u2, cr3 lie between 1 and A  > 1 (inclusive). We denote by 
62, bz relatively prime integers such th a t bx < 0, b2 > 0, bz > 0 and we suppose th a t 
1 
I? ^
2
? bz are each less th an B.
Let N  be any integer > 1 0 0 and, corresponding to N,  define 
'-■/ ' 
• • • 
. r ■
■
. 
■
. ' 
• -./ 
v
S(oc) = X e(txp), 
1(a) = X e(am)l\og m,
m = 2
V pr:#ne
where e(x) denotes e2jttx. Let H  be an integer > 4 . We use 9Jia i to denote th e “ m ajor arc” 
consisting of all real a in the interval [0, 1] such th a t relatively prime integers a, q exist 
w ith 
; 
,
. , - 1 ^ a q ^ (log N ) H and | a — a\q | < (log N ) HIN.
The usual convention applies in which the right hand half of 
is supposed translated
to the left by an am ount 1. The complement m of th e union 9JI of all the m ajor arcs in 
[0, 1] comprises the “ m inor arcs” .