Olympiad number theory problem with primes

Asked Nov 18 '23 at 22:52

Active Nov 26 '23 at 17:05

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Find all pairs of $x, y \in \mathbb Z$ that satisfy the following equation: $$1 + 1996x+1998y=xy$$ (Irish Mathematical Olympiad 1997)

I am stuck and I have not been able to make even a start to a solution. Any help would be greatly appreciated. I presume 1997 being a prime and being omitted in the question has something to do with the solution but I can't figure out what.

edited Nov 26 '23 at 17:05

asked Nov 18 '23 at 22:52

Ilikemath

Are $x,y\in\Bbb N$? – pancini Nov 18 '23 at 22:54
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Hint: Are you familiar with Simon's Favorite Factoring Trick? – Calvin Lin Nov 18 '23 at 22:55
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Complete the product as in the linked dupe. – Bill Dubuque Nov 18 '23 at 23:07
@BillDubuque Hello, after applying SFFT/Completing the product I am left with $(x-1998)(y-1996)=1+1998*1996=3988009$ and I am wondering how one would efficiently find the factor pairs especially in an Olympiad setting where time would be an issue. – Ilikemath Nov 19 '23 at 09:23
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$1+(n+1)(n-1)$ easily factors. $\ \ $ – Bill Dubuque Nov 19 '23 at 17:36
Thank you, I had not taught of that. – Ilikemath Nov 19 '23 at 19:18

Olympiad number theory problem with primes

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