Linear Equations with Small Prime and Almost Prime Solutions
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AbstractLet b1, b2 be any integers such that gcd(b1, b2) = 1 and c1|b1| < |b2| ≤ c2|b1|, where c1, c2 are any given positive constants. Let n be any integer satisfying gcd(n, bi) = 1, i = 1, 2. Let Pk denote any integer with no more than k prime factors, counted according to multiplicity. In this paper, for almost all b2, we prove (i) a sharp lower bound for n such that the equation b1p + b2m = n is solvable in prime p and almost prime m = Pk, k ≥ 3 whenever both bi are positive, and (ii) a sharp upper bound for the least solutions p, m of the above equation whenever bi are not of the same sign, where p is a prime and m = Pk, k ≥ 3.
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2005 ◽
Vol 70
(10)
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pp. 1193-1197
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2011 ◽
Vol 54
(3)
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pp. 685-693
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Keyword(s):
1991 ◽
Vol 44
(1)
◽
pp. 54-74
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Keyword(s):