Opteron Cpu

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What are the practical uses of large integers (> 10^90)?

My honors calc professor (who went to Princeton and is a genius) told me that there isn't much practical use for numbers greater than 10^90 (which is the number of particles in the universe). Don't financial institutions use large prime numbers for encryption? Are there any other practical purposes?

I'm not sure of purposes besides prime number encryption. The idea behind prime numbers is actually using composite numbers created from large primes and then relying on breaking them down into their prime factorization. If you think about how you go about doing this, it's really just a guess and check sort of process, so when the primes become large, since there isn't any formulaic way to decompose the numbers, it takes enormous amounts of time.

For example, factoring the number:
123018668453011775513049495838496272077
285356959533479219732245215172640050726
365751874520219978646938995647494277406
384592519255732630345373154826850791702
612214213461670429214311602221240479274
737794080665351419597459856902143413
(RSA-768)

"took almost 2000 2.2GHz-Opteron-CPU years according to the submitters, just short of 3 years of calendar time."

The factors are the primes:
334780716989568987860441698482126908177
047949837137685689124313889828837938780
02287614711652531743087737814467999489

and

36746043666799590428244633799627952632
27915816434308764267603228381573966651
12792333734171433968102700927987
36308917

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