## Corrections and comments on my 2002 paper

K.R. Matthews, The Diophantine equation ax2+bxy+cy2=N, D=b2-4ac > 0, Journal de Théorie des Nombres de Bordeaux, 14 (2002) 257-270. This gives an account of a neglected algorithm of Lagrange, generalising my earlier Expos. Math. paper on x2-Dy2=N, with an unexpected twist when D=5 and aN<0.
1. In the statement of Theorem 1(i), Page 263, line -2:
Replace "X/y is a convergent to ω"
by "X/y is a convergent Ai-1/Bi-1 to ω and Qi=(-1)i2N/|N|;"
2. In the statement of Theorem 1(ii)(a), Page 264, line 1:
Replace "X/y is a convergent to ω*;"
by "X/y is a convergent Ai-1Bi-1to ω* and Qi=(-1)i+12N/|N|;"
3. In the "conversely" part of statement of Theorem 1, Page 264, lines 11,12:
Replace "will be a solution to (4.1), possibly imprimitive"
by "will be a primitive solution to (4.1)."
4. I didn't emphasise that if D=5 and aN < 0, there is always an exceptional solution.
For, from page 269, we have QX2+(2P+1)Xy+Ry2=(-1)r-1
and from page 270, (-1)r-1Q < 0, ie. (-1)r-1a|N| < 0.
So if aN < 0, we have (-1)r-1N/|N| > 0 and consequently (-1)r-1=N/|N|. Hence QX2+(2P+1)Xy+Ry2=N/|N|, which implies that ax2+bxy+cy2=N, where x=y\theta+|N|X.
5. On page 260, on line -10, delete "if r > 0" and "s > 0 ".
6. On page 262, case (iii)(c), I noticed that ω = (αX + t)/(αy + (QX + Py)), where y > –(QX + Py) > 0. This observation led me to a conjecture which was solved by John Robertson. See Note. It implies that X = Ar – Ar-1 and y = Br – Br-1. Then on page 263, we also have X = As – As-1 and y = Bs – Bs-1.
7. On page 264, replace "possibly primitive" by "which is primitive".
8. On page 265, delete the sentence "If ω or ω* is purely periodic, we must examine Q2, which corresponds to the third period."
9. On page 268, on line 3, there is a missing partial quotient 1 before the 3.
10. On page 268, on line -2, delete "if r > 0" and "s > 0 ".
11. On page 270, there is confusion! However the argument on page 269 is correct and gives r – 1 ≡ s (mod 2).