I've been looking at unit fractions, and found a paper by Erdős ["Some properties of partial sums of the harmonic series"](http://www.ams.org/journals/bull/1946-52-04/S0002-9904-1946-08550-X/S0002-9904-1946-08550-X.pdf) that proves a few things, and gives a reference for the following theorem:


$$\sum\_{k=0}^n \frac{1}{m+kd}$$ is not an integer.


The source is:



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> Cf. T. Nagell, *Eine Eigenschaft gewissen Summen*, Skrifter Oslo, no. 13 (1923) pp. 10-15.
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Although I would like to find this source, I've checked with my university library and it seems pretty out of reach. What I'm really hoping for is a source that's more recent or even written in English.


Finding this specific source isn't everything, I'll be fine with pointers to places with similar results.



Unit fraction, equally spaced denominators not integer
You can cite H. Belbachir and A. Khelladi, On a sum involving powers of reciprocals of an arithmetic progression, Ann. Math. Inform. 34 (2007) 29-31, MR2385421, where a more general result is given. If you are OK with Russian, there is Z. D. Gorskaya, On an arithmetic property of a harmonic sum, Ukrain. Mat. Z. 6 (1954) 375-384, MR0069200 (16,998j).


Nagell wrote a nice introductory number theory textbook, which was republished by the American Math. Society. Maybe the result is in it.


EDIT: I have had a look at the Belbachir and Khelladi paper, at <http://www.kurims.kyoto-u.ac.jp/EMIS/journals/AMI/2007/ami2007-belbachir.pdf>, and I find that it rests heavily on the Shorey and Tijdeman paper cited in Gjergji Zaimi's answer.


FURTHER EDIT: I think that Erdős himself proves the result in the paper "Egy Kürschák-féle elemi számelméleti tétel általánosítása", [KöMaL](https://www.komal.hu/verseny/korabbi.e.shtml) 39 (1932) 17-24. This is freely available, with a summary in German at the end, at [http://www.renyi.hu/~p\_erdos/1932-02.pdf](http://www.renyi.hu/%7Ep_erdos/1932-02.pdf). It would seem that in 1932 Erdős was unaware of Nagell's work.


