Using 567 pb^{-1} of data collected with the BESIII detector at a center-of-mass energy of sqrt[s]=4.599 GeV, near the Λ_{c}^{+}Λ[over ¯]_{c}^{-} threshold, we study the singly Cabibbo-suppressed decays Λ_{c}^{+}→pπ^{+}π^{-} and Λ_{c}^{+}→pK^{+}K^{-}. By normalizing with respect to the Cabibbo-favored decay Λ_{c}^{+}→pK^{-}π^{+}, we obtain ratios of branching fractions: [B(Λ_{c}^{+}→pπ^{+}π^{-})/B(Λ_{c}^{+}→pK^{-}π^{+})]=(6.70±0.48±0.25)%, [B(Λ_{c}^{+}→pϕ)/B(Λ_{c}^{+}→pK^{-}π^{+})]=(1.81±0.33±0.13)%, and [B(Λ_{c}^{+}→pK^{+}K_{non-ϕ}^{-})/B(Λ_{c}^{+}→pK^{-}π^{+})]=(9.36±2.22±0.71)×10^{-3}, where the uncertainties are statistical and systematic, respectively. The absolute branching fractions are also presented. Among these measurements, the decay Λ_{c}^{+}→pπ^{+}π^{-} is observed for the first time, and the precision of the branching fraction for Λ_{c}^{+}→pK^{+}K_{non-ϕ}^{-} and Λ_{c}^{+}→pϕ is significantly improved.