Re: Is it possible to have several solutions in this wa y to this equation using Pari/ɢᴘ ?

Ok.

though I admit I used this as part of the Kunerth’s algorithm…

and after the following code from the thread last year :

nfr=nfroots(,xx);

print("nfr=",nfr);

{

foreach(nfr,X,

Y=Mod(alpha*X+beta,b);

if(lift(Y^2)==c,

print("X=",X);

print("Y=",Mod(lift(Y),semiprime));

print("-Y=",Mod(-lift(Y),semiprime));

print("Y^2=",Mod(lift(Y)^2,semiprime) )));

}

the end result is the same square root than with the original nfroot
(when beta=0). I suppose then when modular inverse are exluded that
it means it’s impossible to use the algorithm to get more
than 1 possible square root ?

Cordialement,

Le 25/01/2025 à 12:26, Bill Allombert a écrit :

On Sat, Jan 25, 2025 at 11:34:34AM +0100,
Laël Cellier wrote:

Oui,

Though I’d like to find other nfroots values given such initial
inputs…

You can use the function I provided.

fun(V,W,b,c,f)=

{

my(A= (-4*W^2*b*f+4*W^4*c), B = (-4*V*W*b*f+8*V*W^3*c)/2, C =
b^2*f^2+4*V^2*W^2*c);

my(M=[A, B, 0; B, C, 0; 0, 0, -1]);

my(S=qfsolve(M));

my(beta =S[1]/S[2]);

my(alpha= W*(V+W*beta));

my(X=nfroots(,alpha^2*x^2+(2*alpha*beta-f*b)*x+(beta^2-c)));

[alpha,beta,X];

}

Then one get

[alpha, beta,X] = fun(V,W,b,c,f);

? alpha

%12 =
-4180572546532158292278454142928666898146832680251024517983166235549996866457463625641813260586575779196664910573656996328431370516487393422559951029010876416586647166710244772350178099099618887593929936011318605706333374790648799565747747747561947250858017026766816415725437778202649010183207102576212750748135057263603141004437390038337911000152260109200201058346971040459320496802802301131571820036166693387212506103166741562118958988034328700974427922415964682112189387592813583381733167516458481948652743252684612101923172981549774492120807150063338763749210936339908775337732314066796727025818787181846214184221923610775161865542769994173571543693201424049580560482622721565114959862604847790194010517550442220655090091817011313539885397132610770833607380656525772796225889856348233175585529525872015832633185556927483750069081231848276531038569943395243141588370577330303878534658346065270579066123937284487619885028558513613850687309241680694943195370410619768514844951355759869316467650570612713838336564935958755899222980761665982884365037955185720294076412106450194271889/25

? beta

%13 =
-672225628067123147116558149881995710798276995011447987050592328976012714865450985916447995662328919412047497279423045117446686729257128275949500746711043539189609725096112283848011986679159419553919508258509140939280500687680096737399531725624206572451821801127194059612210107252008495654751179194156314087179604128231565028033956354284599827623105506753176633926911713425185713779171315276262107447531504708473972729375105320057836498161030991644012650913993856801096153646999371792888290083986221329352989125748813940274976373895395812973914959320432965507828213792019827258803534256546308413647072827145489240887133613964672582561783150543532894610277494904922885667981168421365684201818690509788680019717332275925513478852164737660971344539051206444984192470274604219167707476386543960680385929829179479402230347862231026387872217799903692210388580045747193829740200677896152713759174914475170921132879055170900968153557141/621900203143507540299291235115041475720508671983046645926801396672339663042813373324041813111683010979449390037318880936304666091135011937935140396872157900

Though beta is not an integer, one can find other solutions using
qfparam.

Cheers,

Bill.