Question

I've been looking at unit fractions, and found a paper by Erdos "Some Properties Of Partial Sums Of The Harmonic Series" 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:

Cf. T. Nagell, Eine Eigenschaft gewissen Summen, Skrifter Oslo, no. 13 (1923) pp. 10-15.

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Question

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.

Answer 1

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, MR 2009d:11018, 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, MR 16, 998j.

Nagell wrote a nice intro 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 Erdos himself proves the result in a paper of 1932 (but it's in Hungarian), Egy Kurschak-fele elemi szamelmeleti tetel altalanositasa, Lapok 39 (1932) 17-24 (my apologies for omitting the numerous diacritical marks). This is freely available, with a summary in German at the end, at http://www.renyi.hu/~p_erdos/1932-02.pdf It would seem that in 1932 Erdos was unaware of Nagell's work.

Answer 2

If you are willing to use some heavier results, the fact that $$\sum_{k=0}^{n}\frac{1}{m+kd}$$ is never an integer follows from the following theorem by Shorey and Tijdeman (which refines a theorem of Sylvester):

The greatest prime factor of the product $m(m+d)\cdots (m+nd)$ is greater than $n+1$ unless $(m,d,n)=(2,7,2).$

This is proven in "On the greatest prime factor of an arithmetical progression", and refinements are given in subsequent papers of the authors, finding these references shouldn't be that hard. It implies your result because it shows that one of the fractions has a denominator divisible by a prime $p$ which doesn't appear in any other denominator.