interesting math problem

[gumpzilla]

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The prime decompositions of different integers m and n involve the same primes. The integers m+1 and n+1 also have this property. Is the number of such pairs (m,n) finite or infinite?

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Here's a related question. What are all the pairs of numbers (k,k+1) such that one is a power of 2 and the other is a power of 3.

Examples are (1,2), (2,3), (3,4), (8,9). Are there any others. Anyone know? Does anyone know of techniques to answer questions like this? Is it some standard number theory, or is it really difficult?

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There's a trick I remember for proving that in base 10, all possible natural numbers will eventually occur as the leading digits in some power of 2. If I remember correctly, the trick involved looking at the circle map that you get by considering the fractional part of the log base 10 of successive powers of 2. Since this is like adding log_10 of 2 over and over again, and that number is irrational, you can make an argument about the distribution of points being dense in the circle and then you're done.

I thought for a minute a similar trick might work for this. But now I realize it probably won't, because for each iteration around the circle, you need a closer and closer approach of the two points in order for them to be different by 1; it's not a fixed epsilon that stays constant no matter how many times you traverse the circle. So so much for that.