Simon's Favorite Factoring Trick
Contents
The General Statement
Simon's Favorite Factoring Trick (SFFT) is often used in a Diophantine equation where factoring is needed. The most common form it appears is when there is a constant on one side of the equation and a product of variables with each of those variables in a linear term on the other side. A extortive example would be: where
is the constant term,
is the product of the variables,
and
are the variables in linear terms.
Let's put it in general terms. We have an equation , where
,
, and
are integral constants. According to Simon's Favourite Factoring Trick, this equation can be transformed into:
Using the previous example,
is the same as:
If this is confusing or you would like to know the thought process behind SFFT, see this eight-minute video by Richard Rusczyk from AoPS: https://www.youtube.com/watch?v=0nN3H7w2LnI. For the thought process, start from https://youtu.be/0nN3H7w2LnI?t=366
Applications
This factorization frequently shows up on contest problems, especially those heavy on algebraic manipulation. Usually and
are variables and
are known constants. Also, it is typically necessary to add the
term to both sides to perform the factorization.
Fun Practice Problems
Introductory
- Two different prime numbers between
and
are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?
(Source)
Intermediate
are integers such that
. Find
.
(Source)
Olympiad
- The integer
is positive. There are exactly 2005 ordered pairs
of positive integers satisfying:
Prove that is a perfect square.
Source: (British Mathematical Olympiad Round 3, 2005)