Lagrange’s algorithm revisited: solving at 2 +btu+cu 2 =n in the case of negative discriminant. We make more accessible a neglected continued fraction algorithm of Lagrange for solving the equation at 2 +btu+cu 2 =n in relatively prime integers t,u, where a>0,gcd(a,n)=1, and D=b 2 -4ac<0. The cases D=-4 and D=-3 present a consecutive convergents phenomenon which aids the search for solutions. Webservice: Finding primitive solutions of the diophantine equation ax2+bxy+cy2=n, where b2-4ac < 0, gcd(a,b,c)=1 and a > 0, n > 0.

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  1. Matthews, Keith R.: Lagrange’s algorithm revisited: solving (at^2 + btu + cu^2 = n) in the case of negative discriminant (2014)