A bound for Wilson' s general theorem

Given any setK of positive integers and positive integer λ, letc(K,λ) denote the smallest integer such that v ∈B(K, λ) for every integerv≥c(K, λ) that satisfies the congruencesλv(v-1) ≡ 0 (modβ(K)) and λ (v-1)=≡ (modα(K)). LetK 0 be an equivalent set ofK, k andk* be the smallest and the largest integers inK 0. We prove that c(K, λ) ≤exp expQ 0 where $$Q_0 = \max \left\{ {2^{(2p(K_0 )^2 - k + k^2 \log _4 k)p(K_0 )^4 } ,(k^{k^2 } 4^{2y - k - 2} )^{(\begin{array}{*{20}c} y \\ 2 \\ \end{array} )} } \right\},$$ $p(K_0 ) = \prod\nolimits_{l \in K_0 } l $ andy=k*+k(k?1)+1.

Given any set K of positive integers and positive integer λ, let c( K, λ) denote the smallest integer such that v∈B( K, λ) for every integer v>>c( K, λ) that satisfies the congruences λv( v-1)≡0 (modβ (K)) and λ( v- 1)≡0 (modα(K)). Let K0 be an equivalent set of K, k and k* be the smallest and the largest integers in K0. We prove that c( K,λ)≤exp