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A note on the diophantine equation x 2 + b Y = c z

Maohua Le (2006)

Czechoslovak Mathematical Journal

Let a , b , c , r be positive integers such that a 2 + b 2 = c r , min ( a , b , c , r ) > 1 , gcd ( a , b ) = 1 , a is even and r is odd. In this paper we prove that if b 3 ( m o d 4 ) and either b or c is an odd prime power, then the equation x 2 + b y = c z has only the positive integer solution ( x , y , z ) = ( a , 2 , r ) with min ( y , z ) > 1 .

A note on the distribution of angles associated to indefinite integral binary quadratic forms

Dragan Đokić (2019)

Czechoslovak Mathematical Journal

To each indefinite integral binary quadratic form Q , we may associate the geodesic in through the roots of quadratic equation Q ( x , 1 ) . In this paper we study the asymptotic distribution (as discriminant tends to infinity) of the angles between these geodesics and one fixed vertical geodesic which intersects all of them.

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