Prime Factorization of 49 + 39 - MathFest 2004

[devious_]

Is there a method one can use to obtain the prime factorization of a certain number?

For example:
Find the prime factorization of 4⁹ + 3⁹. [MathFest 2004]

I realize that I can re-write the expression as 2⁹.2⁹+3⁹, but that's about as far as I can go.

 

[shmoe]

For this specific number, notice it's a sum of cubes, $$4^9+3^9=(4^3)^3+(3^3)^3$$ and use the general formula for the sum of cubes. This breaks it into 2 factors nicely, both are much easier to deal with then the orignial. Just use trial division on these 2 factors to furthur break them down.

 

[devious_]

I did use the sum of two cubes formula. This is what I got:
64³+27³=(91)(64²-64.27+27²)=(7)(13)(4096-1728+729)=7.13.19.163

That's the correct answer, but I had to multiply the second bracket out to get it. I was just wondering if there was some other way I could use that wouldn't involve this level of multiplication.