Editing Harmonic number (section)

== Arithmetic properties ==
The harmonic numbers have several interesting arithmetic properties. It is well-known that <math display="inline">H_n</math> is an integer [[if and only if]] <math display="inline">n=1</math>, a result often attributed to Taeisinger.<ref>{{Cite book|title=CRC Concise Encyclopedia of Mathematics|last=Weisstein|first=Eric W.|publisher=Chapman & Hall/CRC|year=2003|isbn=978-1-58488-347-0|location=Boca Raton, FL|pages=3115}}</ref> Indeed, using [[P-adic valuation|2-adic valuation]], it is not difficult to prove that for <math display="inline">n \ge 2</math> the numerator of <math display="inline">H_n</math> is an odd number while the denominator of <math display="inline">H_n</math> is an even number. More precisely,
<math display="block">H_n=\frac{1}{2^{\lfloor\log_2(n)\rfloor}}\frac{a_n}{b_n}</math>
with some odd integers <math display="inline">a_n</math> and <math display="inline">b_n</math>.

As a consequence of [[Wolstenholme's theorem]], for any prime number <math>p \ge 5</math> the numerator of <math>H_{p-1}</math> is divisible by <math display="inline">p^2</math>. Furthermore, Eisenstein<ref>{{Cite journal|last=Eisenstein|first=Ferdinand Gotthold Max|year=1850|title=Eine neue Gattung zahlentheoretischer Funktionen, welche von zwei Elementen ahhängen und durch gewisse lineare Funktional-Gleichungen definirt werden|journal=Berichte Königl. Preuβ. Akad. Wiss. Berlin|volume=15|pages=36–42}}</ref> proved that for all odd prime number <math display="inline">p</math> it holds
<math display="block">H_{(p-1)/2} \equiv -2q_p(2) \pmod p</math>
where <math display="inline">q_p(2) = (2^{p-1}  -1)/p</math> is a [[Fermat quotient]], with the consequence that <math display="inline">p</math> divides the numerator of <math>H_{(p-1)/2}</math> if and only if <math display="inline">p</math> is a [[Wieferich prime]].

In 1991, Eswarathasan and Levine<ref>{{Cite journal|last1=Eswarathasan|first1=Arulappah|last2=Levine|first2=Eugene|year=1991|title=p-integral harmonic sums|journal=Discrete Mathematics|volume=91|issue=3|pages=249–257|doi=10.1016/0012-365X(90)90234-9|doi-access=free}}</ref> defined <math>J_p</math> as the set of all positive integers <math>n</math> such that the numerator of <math>H_n</math> is divisible by a prime number <math>p.</math> They proved that
<math display="block">\{p-1,p^2-p,p^2-1\}\subseteq J_p</math>
for all prime numbers <math>p \ge 5,</math> and they defined ''harmonic primes'' to be the primes <math display="inline">p</math> such that <math>J_p</math> has exactly 3 elements.

Eswarathasan and Levine also conjectured that <math>J_p</math> is a [[finite set]] for all primes <math>p,</math> and that there are infinitely many harmonic primes. Boyd<ref>{{Cite journal|last=Boyd|first=David W.|year=1994|title=A p-adic study of the partial sums of the harmonic series|url=http://projecteuclid.org/euclid.em/1048515811|journal=Experimental Mathematics|volume=3|issue=4|pages=287–302|doi=10.1080/10586458.1994.10504298|citeseerx=10.1.1.56.7026}}</ref> verified that <math>J_p</math> is finite for all prime numbers up to <math>p = 547</math> except 83, 127, and 397; and he gave a heuristic suggesting that the [[Natural density|density]] of the harmonic primes in the set of all primes should be <math>1/e</math>. Sanna<ref>{{Cite journal|last=Sanna|first=Carlo|year=2016|title=On the p-adic valuation of harmonic numbers|journal=Journal of Number Theory|volume=166|pages=41–46|doi=10.1016/j.jnt.2016.02.020|hdl=2318/1622121|url=https://iris.unito.it/bitstream/2318/1622121/1/padicharm.pdf|doi-access=free}}</ref> showed that <math>J_p</math> has zero [[Natural density|asymptotic density]], while Bing-Ling Wu and Yong-Gao Chen<ref>{{Cite journal|last1=Chen|first1=Yong-Gao|last2=Wu|first2=Bing-Ling|year=2017|title=On certain properties of harmonic numbers|journal=Journal of Number Theory|volume=175|pages=66–86|doi=10.1016/j.jnt.2016.11.027|doi-access=}}</ref> proved that the number of elements of <math>J_p</math> not exceeding <math>x</math> is at most <math>3x^{\frac{2}{3}+\frac1{25 \log p}}</math>, for all <math>x \geq 1</math>.