2007 Diophantine equation.....

Number Theory Level 5

Find all the positive integer solutions $(x;y)$ of the equation: $xy+x=2007y+2$ .

Call $S=x_1+y_1+x_2+y_2+.........+x_{n-1}+y_{n-1}+x_n+y_n$ which is the value of $S$ ?

1 solution

Ronald Overwater
Dec 27, 2014

$xy+x=x(y+1)=2007y+2$

$x=\dfrac{2007y+2}{y+1}=2007-\dfrac{2005}{y+1}$

Factorizing we get $2005=5 \times 401$

So using $y>0$ , we get three solutions for $(y+1): \{5, 401,2005\}$ , which gives the solutions: $(x,y)= \{(1606,4);(2002,400); (2006,2004)\}$

Summing all x and y coordinates we get a sum $S = 8022$

Damn it. I forgot that $y=0$ cannot be a positive solution -_-

Josh Banister - 6 years, 2 months ago

Similar to Ronald Overwater's Solution:

Alternatively, we can isolate y instead of x (for those who blindly isolate y from equations upon sight),

y = $\frac {x-2}{2007-x}$ = - $\frac{2-x}{2007-x}$ = $\frac {2005}{2007-x}$ - 1

Factoring 2005 yields:

(2007-x) ; {5, 401, 2005}

--> x = {2002, 1606, 2006}

--> y = {400, 4, 2004}

--> S = 8022

Mark Wo - 6 years, 1 month ago

Use Simon's Favourite Factoring Trick....

Anubhav Mahapatra - 3 years, 4 months ago