Assuming is the nth prime number, establish that the sum is never an integer.

I don't know how to go about showing this. Any help is greatly appreciated, thanks!

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- Sep 6th 2010, 04:14 PMkimberuSum of inverses of primes
Assuming is the nth prime number, establish that the sum is never an integer.

I don't know how to go about showing this. Any help is greatly appreciated, thanks! - Sep 6th 2010, 06:17 PMmelese
Let . This leads to .

Assume to the contrary that is an integer; then divides . Try to find a contradiction...

By the way, using this idea can show that is not an integer if for , i.e., the are relatively prime in pairs. - Sep 7th 2010, 01:16 AMmeleseA different way, simpler maybe.
I didn't want to resist the following solution. Again let ; suppose is an integer.

Each term in expasion of is an integer except for , but then cannot be an integer. - Sep 7th 2010, 03:33 AMtonio