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a≤4D ∑ b,c,f∈N 2f<b<c bc−f2≤ 4D a b≡4c+2 1 − ∑ a∈4N a≤4D ∑ b,c∈N b<c< 4D ab + b 4 b≡4c≡40 1 + ∑ a∈4N a≤4D ∑ b,c∈N b<c< 4D ab + b 4 b≡4c≡43 1− ∑ a∈2N\4N a≤4D ∑ b,c∈N b<c< 4D ab + b 4 0≡4b≡4c+2 1 + ∑ a∈2N\4N [...] a∈2N\4N a≤4D ∑ b,c∈N b<c< 4D ab + b 4 3≡4b≡4c+2 1(65) + ∑ a∈4N a≤4D ∑ b,c∈N b<c< 4D ab + b 4 bc− 4D a >1 b≡4c O(1) + ∑ a∈2N\4N a≤4D ∑ b,c∈N b<c< 4D ab + b 4 bc− 4D a >1 b≡4c+2 O(1) ) .(66) We claim that the [...] a∈N a≤4D a≡4x ∑ b∈N b<4 √ D 3a b≡4y ∑ c∈N b<c< 4D ab + b 4 c≡4z 1 = ∑ a∈N a≤4D a≡4x ∑ b∈N b<4 √ D 3a b≡4y ( 1 4 ∑ c∈N b<c< 4D ab + b 4 1 +O(1) ) = 1 4 ∑ a∈N a≤4D a≡4x ∑ c∈N c< 16D 3a ∑ b∈N b<c< 4D ab + …

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