What is the smallest number that can be shown as a difference between two squares in 5 different ways?
Barring an oversight, the answer is 192.
The number must have at least 10 factors to fit 5 instances of (a+b)(ab). Also, (a+b) and (ab) will necessarily have the same parity as otherwise adding them leads to a contradiction.
Looking at increasingly larger numbers with at least 10 factors:
48 has 4 pairs of factors with same parity
80 has only 3
96 and 160 have 4
192 has 5, (96,2),(48,4),(32,6),(24,8),(16,12).
Setting a=1/2 the sum and b=1/2 the difference of the factors gives solutions (49,47),(26,22),(19,13),(16,8),(14,2).

Posted by xdog
on 20180707 13:10:32 