This was a practice problem for a class. The class is over now and I never solved it, so I thought I'd ask here.
Let's ignored the fact that adding extra security to single textbook RSA is unnecessary. As hilariously stated here:
Think about it this way, if it is estimated to take 500 years for a prisoner to chew through the bars on [their] prison cell to escape, is the public any safer if we add a second set of bars so that it will take 1000 years to chew through the two sets before the prisoner can escape? Not really.
The proposed question:
Does encrypting data twice with RSA and different keys increase security over encrypting data once with RSA?
Thinking about it now, I feel it would because you're introducing a second factorization problem that the adversary must compute.
Could someone please explain, mathematically, why I am correct/incorrect?
Note: For notation and algorithms, I am more familiar with that used in the original paper, particularly the use of Euler's totient function, not Carmichael's totient function.