I am trying to find the five smallest pairs of positive integers p,q that satisfy the unusual Pell equation
$$p^2− \frac{251934169}{4} q^2 = 1.$$
One obvious trivial solution is p=1 and q=0 , but this does not count. I am trying to find the five smallest pairs of positive integers p,q that satisfy this equation. Note that the coefficient of $$q^2$$ is not an integer but is a rational number. I would appreciate your help.
Best Answer
If $p^2− \frac{251934169}{4} q^2 = 1 $, then $4p^2− 251934169q^2 = 4 $, so that $q$ has to be even.
Let $q = 2y$ so that $4p^2− 251934169\cdot 4 y^2 = 4 $ or $p^2− 251934169 y^2 = 1 $ and this becomes a standard Pell equation.
After searching for a Pell equation solver that could solve this, I found http://www.numbertheory.org/php/pell.php which said that
the least solution of $u^2-251934169·v^2=1$ is (u,v)=(225271244990700943315267956280411800555634015051308090277884868604104962338540162508662780103851110134288923470733214492576705304921942447496877316432148755394149624610709778342253829532034182955585281785605125774062067042871533474548723189550080879461940755054931210785029037747926428769992333668074349218880978527807663394127680486193384843986997533304388501754725074957250497216933172269065473852746178843181608306140365685996039286536330244666823292214037380277106713895640581283369187712083172695225344399463019401537455882605714942067846468852520297222251755337316170041845823661624202782666250115734237819683042082985171526282800382222714576727626094980629885366352000131303132135359463108635521565410488112287912844686192672738472001947756812093648970418119040213989933215258586913997217342108856777957037627456499897383934170286644285878283503578561992303795946148570976156177701503287373115452930781765236053893727748799204417831966968237024161827983124601075826155762117660452380906950456102876389134502447701968296040064628681054852876640098371536019198418255253941528944307814328148190184143000799951889089084659644347638577183884192073467082108506523506372732340274528141456740473415969963650656773018158163222322270988303396424521240862140458260591486474043045092368034899520901292279521856198411310822116084681636806884280270173841934612885750400068863907732865349185857915555235934810570449154233256465716941270067608222665579808387449666859364948244303214692248884665160509209379252943392494272634976297214741913847533965479527154125323880743892254267831724482576916275662958095933387932932551059660582766222292883165493697748307219389789209133143531345455092189866850439804333199651752360242139683699342368412986247802850384829304599110060419091883992716219173015297451039403047743238488985860729267543533352329687314970438461416693387772967943131761604987413334536162985092790422123282237028624645785633489664726218957148609919685769428122222084852809562569214413965720771408729185099918424343296677568166613090433613515483031252901694323178595995462642028964386862010093420726769418355184234720946181775332737659395967587225532076407623909565596790910765885818561865412162538848363553668612714198691288861437047601157353591221118724976585902490459382998907534939868158790879001671152433044317432949761648923963614896693699457510715718361695625294634696966284878410574160929772400728092044256419270960522476326245255572559146175279700400659950509793788476378848175390369825427301273114735692514379330175924474553668351745584388982458068437797359188706503857643164095213477929089334214332342050356909188421399238606889543471922745845239071010651328302511512759263707483733323614395083737283232899508265127479172494312201913477981379407799842709252906180434716507141311051322982240657546127971114278462713116439134410377103157523161671298844486230796509326848933853071382190119442436807702852665446333225245223437120117269735297360569365484765265678599514845945070544634609637487148190908846353641866101791106162516313305822511866796089071796216876389782118935966244678804350508265429071175447438279751936690584484769952697448915336468581146844571603113602643709943866971797062017021207642033921312513697587999212884164536775768402206018046772509817554058217211657510926455226154119243190033360301765607288639334708843614852895130144204537103834220400026878753494472540566640552609407477709815083333874222841700725622211428144359211679202120655592804708419101143418091589880834113404679889876903100324371120800767782414611156383670868080765233711399358523011346075275028463738338742036397612554950891163082740807747720798395812396497715250419270439996118401688429363409408587403868552632449,
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