Expressing Integers As A Difference Of Squares

If a number can be expressed as $a^2-b^2$ , then it can be factored as $(a+b)(a-b)$ . Let $a+b=m$ and $a-b=n$ .

Then $a=\frac{m+n}{2}$ and $b=\frac{m-n}{2}$ . Therefore, in order for a number to be expressed as $a^2-b^2$ , it must be factorable into two odd numbers or two even numbers.

For odd non-perfect squares, the number of ways to express as $a^2-b^2$ is equal to the number of factor pairs. We can find the number of factor pairs by dividing the number of factors by $2$ . For example, $63$ 's prime factorization is $3^2\times7^1$ . The number of factors of $63$ is $(2+1)(1+1)=6$ . The number of factor pairs will be $6\div2=3$ . Therefore, it can be expressed as $a^2-b^2$ in 3 different ways: $12^2-9^2$ , $32^2-31^2$ , and $8^2-1^2$ .

For even non-perfect squares, the number of ways to express as $a^2-b^2$ is equal to the number of even factor pairs. For example, $24$ has prime factorization $2^3\times3^1$ . To find the number of even factor pairs, follow the process to find the number of factors except subtract $1$ from $2$ 's exponent instead of adding $1$ : $(3-1)(1+1)=4$ . The number of even factor pairs will be: $4\div2=2$ . Therefore, $24$ can be expressed as $a^2-b^2$ in 2 different ways: $7^2-5^2$ and $5^2-1^2$ .

For perfect squares, the process of finding factor pairs is similar, but you must not include the factor pair in which the two factors are equal.

The process of finding $g(n)$ for any positive integer involves some trial and error. I used the smallest possible primes to find numbers that have as many factor pairs (or even factor pairs in the case of even numbers) as the $n$ that I am trying to find $g(n)$ for. This yielded:

$g(1)=3^1=3$

$g(2)=3^1\times5^1=15$

$g(3)=3^2\times5^1=45$

$g(4)=2^5\times3^1=96$

$g(5)=2^6\times3^1=192$

The least common multiple of these numbers is $M=2^6\times3^2\times5^1=2880$ . The number of even factor pairs of this number is $(6-1)(2+1)(1+1)\div2=15$ . Therefore $f(M)=\boxed{15}$ .