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(*https://www.mersenneforum.org/forumdisplay.php?f=78*)

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(*https://www.mersenneforum.org/showthread.php?t=2397*)

number theory helpa homogeneous polynomial in 2 variables x,y, of degree 2, say, f(x,y)=ax^2+bxy+cy^2 with a,b,c, all integers is called a quadratic form over the integers. the discriminant of the above quadratic form is d=b^2-4ac. a change of variables u,v, is x=αu+βv, y=λu+δv, α, β, λ, δ are all integers, αδ-βλ= ±1. thus you get g(u,v)=f(x,y)=f(αu+βv, λu+δv).
--->show that if f(x,y) is a quadratic form with positive discriminant, then the equation f(x,y)=n may have infinitely many solutions by exhibiting an example. -thanks. |

[QUOTE=math]a homogeneous polynomial in 2 variables x,y, of degree 2, say, f(x,y)=ax^2+bxy+cy^2 with a,b,c, all integers is called a quadratic form over the integers. the discriminant of the above quadratic form is d=b^2-4ac. a change of variables u,v, is x=αu+βv, y=λu+δv, α, β, λ, δ are all integers, αδ-βλ= ±1. thus you get g(u,v)=f(x,y)=f(αu+βv, λu+δv).
--->show that if f(x,y) is a quadratic form with positive discriminant, then the equation f(x,y)=n may have infinitely many solutions by exhibiting an example. -thanks.[/QUOTE] isn't this also what is called an ellyptic curve? |

[QUOTE=juergen]isn't this also what is called an ellyptic curve?[/QUOTE]
Nope. An elliptic curve, E(x,y) is a polynomial which is cubic in x and quadratic in y. The formula given is indeed a quadratic form. The way in which it is phrased makes me almost certain that the original post was an attempt to get assistance with a homework problem. Paul |

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