text: integral solutions for 1/x+1/y=1/99..i need all possible solutions..negative nd positive....?   the more general problem is 1/x + 1/y = 1/m for some m.  it turns out that the way to find all solutions for integers is to find all factorizations:      m^2 = a*b  (a,b!=-m)  then the solution is x=a+m and y=b+m.  it's easy to show that this works:     1/x + 1/y = (x+y)/xy = (a+b+2m)/(a+m)(b+m)  but (a+m)(b+m) = ab + (a+b)m + m^2      = (a+b)m + 2m^2  (since ab = m^2)     = m(a+b+2m)  so 1/x + 1/y = 1/m  on the other hand, if 1/x + 1/y = 1/m, then xy = m(x+y).  subtract m(x+y) from both sides, then add m^2 to both sides, and you get:    (x-m)(y-m) = m^2  so, in the case of m=99, we are looking to factor 99^2 = 3^4 * 11^2 = 9801.  a=1, b=9801, x = 100, y=9900 a=3, b=3267, x = 102, y=3366 a=9, b=1089, x = 108, y=1188 a=11, b=891, x = 110, y=990 a=27, b=363, x = 126, y=462 a=33, b=297, x = 132, y=396 a=81, b=121, x = 180, y=220 a=99, b=99, x = 198, y=198 (case where a>b just switches x and y, so we skip those.) a=-1, b=-9801, x=98, y=-9702 a=-3, b=-3267, x=96, y=-3168 a=-9, b=-1089, x=90, y=-990 a=-27, b=-363, x=72, y=-264 a=-33, b=-297, x=66, y=-198 a=-81, b=-121, x=18, y=-22 a=-99, b=-99, x=0, y=0 (obviously not a solution)  again for |a|>|b|, we just switch the x and y.  so, that's all your solutions.