This is a followup to
Divisibility to square.
Each of x and y is a positive integer such that x^2 + y^2 + x is divisible by xy.
1: Prove that there is an infinite number of (x,y) which make the quotient equal to 3.
2: Prove or disprove 3 is the only integer quotient possible.
Assuming the quotient, q, is positive (if the quotient is negative, x can never be positive for the equality x^2 + nxy + y^2 + x = 0 to hold):
Let x^2  xy + y^2 + x = 0, an ellipse with integer solutions (x,y}={1,1},{0,0},{1,0}
Let x^2  2xy + y^2 + x = 0, a parabola with integer solutions {x,y} ={u^2, (u^2u)}
Let x^23xy+y^2+x= 0, a hyperbola with integer solutions {x,y} = {0,0}, {1,0}, {1,1}, {1,2}, {4,2}; plainly there are infinitely many such solutions of the form:
Xn+1 = P Xn + Q Yn + K
Yn+1 = R Xn + S Yn + L
with characteristics
P = 3
Q = 1
K = 1
R = 1
S = 0
L = 1
or
P = 0
Q = 1
K = 1
R = 1
S = 3
L = 2
If the quotient is greater than 3, then the characteristics are of the form:
P = (q^21)
Q = q
K = 1
R = q
S = 1
L = 0
or
P = 1
Q = q
K = 1
R = q
S = (q^21)
L = q
Since the terms in P and Q, and in R and S, are of opposite sign, either x or y is never positive, contrary to the stipulations of the problem. So 3 is the only integer quotient possible, as was to be proved.
Edited on August 8, 2016, 7:12 am

Posted by broll
on 20160808 00:30:54 