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Integers represented as a sum of primes and powers of two
 Asian J. Math
"... It was shown by Linnik [10] that there is an absolute constant K such that every sufficiently large even integer can be written as a sum of two primes and at most ..."
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It was shown by Linnik [10] that there is an absolute constant K such that every sufficiently large even integer can be written as a sum of two primes and at most
A Numerical Bound For Baker's Constant  Some Explicit Estimates For Small Prime Solutions Of Linear Equations
"... INTRODUCTION Let a 1 ; a 2 ; a 3 be any nonzero integers such that gcd(a 1 ; a 2 ; a 3 ) = 1: (1.1) Let b be any integer satisfying b j a 1 + a 2 + a 3 (mod2) and gcd(b; a i ; a j ) = 1 for 1 i ! j 3: (1.2) Write A := maxf3; ja 1 j; ja 2 j; ja 3 jg. In [11], M.C. Liu and K. M. Tsang studied a pr ..."
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INTRODUCTION Let a 1 ; a 2 ; a 3 be any nonzero integers such that gcd(a 1 ; a 2 ; a 3 ) = 1: (1.1) Let b be any integer satisfying b j a 1 + a 2 + a 3 (mod2) and gcd(b; a i ; a j ) = 1 for 1 i ! j 3: (1.2) Write A := maxf3; ja 1 j; ja 2 j; ja 3 jg. In [11], M.C. Liu and K. M. Tsang studied a problem of A. Baker [1], namely, the solubility and the size of small prime solutions p 1 ; p 2 ; p 3 of the linear equation a 1 p 1<
THE PARITY PROBLEM FOR REDUCIBLE CUBIC FORMS
, 2005
"... Let f ∈ Z[x, y] be a reducible homogeneous polynomial of degree 3. We show that f(x, y) has an even number of prime factors as often as an odd number of prime factors. ..."
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Let f ∈ Z[x, y] be a reducible homogeneous polynomial of degree 3. We show that f(x, y) has an even number of prime factors as often as an odd number of prime factors.
Let SUMS OF THREE OR MORE PRIMES
"... Abstract. It has long been known that, under the assumption of the Riemann Hypothesis, one can give upper and lower bounds for the error � p≤x log p − x in the Prime Number Theorem, such bounds being within a factor of (log x) 2 of each other and this fact being equivalent to the Riemann Hypothesis. ..."
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Abstract. It has long been known that, under the assumption of the Riemann Hypothesis, one can give upper and lower bounds for the error � p≤x log p − x in the Prime Number Theorem, such bounds being within a factor of (log x) 2 of each other and this fact being equivalent to the Riemann Hypothesis. In this paper we show that, provided “Riemann Hypothesis ” is replaced by “Generalized Riemann Hypothesis”, results of similar (often greater) precision hold in the case of the corresponding formula for the representation of an integer as the sum of k primes for k ≥ 4, and, in a mean square sense, for k ≥ 3. We also sharpen, in most cases to best possible form, the original estimates of Hardy and Littlewood which were based on the assumption of a “QuasiRiemann Hypothesis”. We incidentally give a slight sharpening to a wellknown exponential sum estimate of VinogradovVaughan. r(n)=rk(n)= 1.
On multiplicative congruences
, 807
"... Let ε be a fixed positive quantity, m be a large integer, xj denote integer variables. We prove that for any positive integers N1,N2,N3 with N1N2N3> m 1+ε, the set {x1x2x3 (mod m) : xj ∈ [1,Nj]} contains almost all the residue classes modulo m (i.e., its cardinality is equal to m+o(m)). We further s ..."
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Let ε be a fixed positive quantity, m be a large integer, xj denote integer variables. We prove that for any positive integers N1,N2,N3 with N1N2N3> m 1+ε, the set {x1x2x3 (mod m) : xj ∈ [1,Nj]} contains almost all the residue classes modulo m (i.e., its cardinality is equal to m+o(m)). We further show that if m is cubefree, then for any positive integers N1,N2,N3,N4 with N1N2N3N4> m 1+ε, the set {x1x2x3x4 (mod m) : xj ∈ [1,Nj]} also contains almost all the residue classes modulo m. Let p be a large prime parameter and let p> N> p 63/76+ε. We prove that for any nonzero integer constant k and any integer λ ̸ ≡ 0 (mod p) the congruence p1p2(p3 + k) ≡ λ (mod p) admits (1 + o(1))π(N) 3 /p solutions in prime numbers p1,p2,p3 ≤ N. 2000 Mathematics Subject Classification: 11L40 1 1