The difference is $a^2 - b^2 = (a - b).(a + b)$
But what about when I have $a^{25} + 1$ ? According to wolfram alpha, the alternate form is:
- $(a+1) (a^4 -a^3 + a^2 -a + 1)( a^{20} - a^{15} + a^{10} -a^5 +1)$
However, the square root of 25 is a rational number 5.
But If I had 50, where the square root of 50 is irrational ?
- $(a^2 + 1) (a^8 -a^6 +a^4 -a^2 +1) (a^{40} -a^{30} +a^{20} -a^{10} +1 )$
In fact, I'm just wondering and trying to find out patterns, I'm very curious about that, since I haven't found anything related, only the alternate forms generated by wolfram. My question is how and what mathematical algorithm they used to find $k$ forms for $a^n + 1$ ?
Thanks in advance.