▁the ▁case , ▁it ▁implies ▁that ▁the ▁second ▁Hard y – L ittle wood ▁conject ure , ▁in ▁contrast , ▁is ▁false . ▁ ▁Prime ▁arithmetic ▁progress ions ▁ ▁A ▁prime ▁k - tuple ▁of ▁the ▁form ▁( 0 , ▁n , ▁ 2 n , ▁...) ▁is ▁said ▁to ▁be ▁a ▁prime ▁arithmetic ▁pro gression . ▁ ▁In ▁order ▁for ▁such ▁a ▁k - tuple ▁to ▁meet ▁the ▁adm iss ibility ▁test , ▁n ▁must ▁be ▁a ▁multiple ▁of ▁the ▁prim orial ▁of ▁k . ▁ ▁S ke w es ▁numbers ▁The ▁S ke w es ▁numbers ▁for ▁prime ▁k - tu ples ▁are ▁an ▁extension ▁of ▁the ▁definition ▁of ▁S ke w es ' ▁number ▁to ▁prime ▁k - tu ples ▁based ▁on ▁the ▁first ▁Hard y - L ittle wood ▁conject ure ▁( ). ▁Let ▁ ▁denote ▁a ▁prime ▁k - tuple , ▁ ▁the ▁number ▁of ▁pr imes ▁ ▁below ▁ ▁such ▁that ▁ ▁are ▁all ▁prime , ▁let ▁ ▁and ▁let ▁ ▁denote ▁its ▁Hard y - L ittle wood ▁constant ▁( see ▁first ▁Hard y - L ittle wood ▁conject ure ). ▁Then ▁the ▁first ▁prime ▁ ▁that ▁viol ates ▁the ▁Hard y - L ittle wood ▁inequality ▁for ▁the ▁k - tuple ▁, ▁i . e ., ▁such ▁that ▁▁ ▁( if ▁such ▁a ▁prime ▁exists ) ▁is ▁the ▁S ke w es ▁number ▁for ▁. ▁ ▁The ▁table ▁below ▁shows ▁the ▁currently ▁known ▁S ke w es ▁numbers ▁for ▁prime ▁k - tu ples :