I have some large integers that I want to add. The problem is, I can't even store them in a usual way. I store their prime decompositions, since they are so large. Here is one example of a product I want:
$2^{100}\cdot97^{300}+13^{50}\cdot17^{200}$
One thing I know, they have no common factors. How do I add them, and save the result as a prime decomposition again? I need an algorithm, that can avoid expanding the integers into 10 radix form. To make it a math problem (and not an algorithm problem), does anyone know how to calculate the factors in the decomposition directly from the factors in the sum? Also, if this is the wrong forum, please let me know in the comments the appropriate forum.
This number has $627$ digits, which is small for an arbitrary precision program. A vanishingly small proportion of numbers like this do not have a factor of at least, say, $100$ digits. Your sum is probably in that class. Sums of small powers are a poor way to store large numbers because most large numbers cannot be stored that way.
I don't think there is any way to calculate the factor decomposition of the sum from what you have. It might even be prime. Any arbitrary precision program can handle small numbers like this. If I feed it to Alpha I get the fact that the number has $627$ digits and that a partial factorization is $$5^3×17729×21737×50221×9079945801×16441924501×269980701319081×1397886802598297673234594043519579352028658054947394866487636502506015815941812887266642090877085788431701291456010743473720785420046423436856120466425336965441750777517400799989328659843101703902332792720478140833462239869241240106006785458142390107793347098685251282065940839772545979646478677050049002607236158820818815906611239067665805873384741916310281844310798477993420772443148384106393452706286551901829163928497142254006017819919953919702562895678891182858220972336093353922116710032345881403814710715099124523636128153352090629908397114483871566968260206555482086069$$ (7 distinct prime factors, 1 distinct composite factor) but it admits that the last factor is composite. Alpha will happily give me all the digits if I want to store it that way.