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Find Integer Solutions Challenge

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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,689
Find all pairs $(p, q)$ of integers such that $1+1996p+1998q=pq$.
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,708
Hint:
What is $(p-1)(q+1)$?

Further hint:
$1997$ is a prime number.
 

kaliprasad

Well-known member
Mar 31, 2013
1,309
I shall proceed differently from Opalg

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