1. Introduction. For an integer ν > 1, we define P (ν) to be the greatest prime factor of ν and we write P (1) = 1. Let m ≥ 0 and k ≥ 2 be integers.
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Further we apply the method of Halberstam and Roth as in [8] for deriving that there exists a subset S 3 of S 2 with |S 3 | ≥ u 4 k 1−ε1
(10) x i > k 2−τ1
In fact, (9) is valid with τ replaced by τ 0 = (1 + ε 0 /4)ν l −1 where ε 0 = (10 6 l 5 ) −1 , and we use this estimate for deriving (10). Put s 3 = |S 3 |. By per- muting the subscripts of d 1 , . . . , d t , there is no loss of generality in assuming that a 1 , a 2 , . . . , a s3
≤ u 5 log k k 1−ε1
a µ < 2u 5 log k k 1−ε1
k 1−ε1
Finally, we combine (12), (15) and (10) in order to derive that k (2−τ1
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