Arithmetic Progressions on Pell Equations

IF Introduction sn IWWW fremner ‘I“ ™onsidered —rithmeti™ progressions on ellipti™ ™urvesF fremner ™onstru™ted ellipti™ ™urves with —rithmeti™ progressions of length UD iFeF r—tion—l points @X; Y A whose XE ™oordin—tes —re in —rithmeti™ progressionF sn — following p—per fremnerD ƒilverm—n —nd „z—n—kis ‘P“ showed th—t — su˜group of the ellipti™ ™urve E@QA with E X Y 2 a X@X 2 n 2 A of r—nk I does not h—ve nonEtrivi—l integr—l —rithmeti™ progressionsD provided n ! IF gontr—ry to the results of fremnerD ƒilverm—n —nd „z—n—kis ‘P“D g—mp˜ell ‘Q“ found —n innite f—mily of ellipti™ ™urves with W integr—l points in —rithmeti™ progressionsF „his result w—s improved ˜y …l—s ‘IP“D where —n innite f—mily w—s found with —n —rithmeti™ progression ™onsisting of IP integr—l pointsF sn this p—per we ™onsider ™urves of genus HD in p—rti™ul—r hyper˜ol—D with integr—l —rithmeti™ progressionsF snspired ˜y the results of fremner ‘I“D fremnerD ƒilverm—n —nd „z—n—kis ‘P“D g—mpE ˜ell ‘Q“ —nd …l—s ‘IP“ the —im of this p—per is to prove the following theoremsF Theorem 1. Let H < d P Z, d not a square and H T a m P Z. If there are three solutions