Googling the problem leads us to a [similar Quora post](https://www.quora.com/How-do-you-find-the-positive-integer-solutions-to-frac-x-y+z-+-frac-y-z+x-+-frac-z-x+y-4/answer/Alon-Amit) on this challenge.
Modifying the script in the Quora post for N=10, we can solve for a,b, and c.To make all numbers positive, we can brute force a multiple of P until all 3 numbers are positive.
Here's the script in Sage:```def orig(P,N):    x=P[0]    y=P[1]    a=(8*(N+3)-x+y)/(2*(N+3)*(4-x))    b=(8*(N+3)-x-y)/(2*(N+3)*(4-x))    c=(-4*(N+3)-(N+2)*x)/((N+3)*(4-x))    da=denominator(a)    db=denominator(b)    dc=denominator(c)    l=lcm(da,lcm(db,dc))    return [a*l,b*l,c*l]
ee = EllipticCurve([0,517,0,416,0])P = ee(-416,4160)
for i in range(1,1000):    u = orig(P*i,10)    (a,b,c)=(u[0],u[1],u[2])    if a > 0 and b > 0 and c > 0:        print(a)        print(b)        print(c)        break```