- Thread starter
- Admin
- #1

- Feb 14, 2012

- 3,909

Find all pairs $(p, q)$ of integers such that $1+1996p+1998q=pq$.

- Thread starter anemone
- Start date

- Thread starter
- Admin
- #1

- Feb 14, 2012

- 3,909

Find all pairs $(p, q)$ of integers such that $1+1996p+1998q=pq$.

- Moderator
- #2

- Feb 7, 2012

- 2,785

Hint:

Further hint:

What is $(p-1)(q+1)$?

Further hint:

$1997$ is a prime number.

- Mar 31, 2013

- 1,346

Pq – 1996p – 1998q = 1

Or (p-1998)(q-1996) – 1998 * 1996 = 1

Or (p-1998)(q-1996) = 1998 * 1996 + 1 = 1997^2

We get all the solution set for (p-1998, q- 1996) to be ( 1, 1997^2), (1997,1997), ( 1997^2, 1)

(-1, - 1997^2), (-1997,- 1997), (- 1997^2, - 1) as 1997 is prime