Thread by @TamasGorbe

Harmonic numbers
Hₖ = 1 + 1/2 + ... + 1/k
are never integers (except for H₁=1 of course).

Here is a proof from THE BOOK. (1/4)

In fact, I'll show you the proof of a stronger statement:

The reciprocals of two or more consecutive positive integers never add up to an integer.

In other words, the difference of distinct harmonic numbers is not an integer. (2/4)

First, we prove a simple yet powerful lemma:

Only one number among n+1, n+2, ..., n+k is divisible by a largest power of 2.

Proof. If n+i < n+j are multiples of 2ᵐ with maximal m, then they're at least 2ᵐ apart and (n+i)+2ᵐ is divisible by 2ᵐ⁺¹, so m isn't maximal. (3/4)

Proof from THE BOOK. Let N be the unique number among n+1,n+2,...,n+k with N=a×2ᵐ (a odd, m maximal). The sum of reciprocals without 1/N is of the form b/(c×2ᵈ) with b,c odd and d<m. If we add 1/N to this, we get (ab2ᵐ⁻ᵈ + c)/ac2ᵐ = odd/even, a non-integer (4/4)

This proof is by the Hungarian mathematician József Kürschák.
mathshistory.st-andrews.ac.uk/Biographies/Ku…

It was published in 1918 in a journal called "Mathematikai és Physikai Lapok" real-j.mtak.hu/7278

This fact combined with the divergence of the harmonic series implies that the sequence of harmonic numbers jumps over every positive integer missing each and every one of them (except 1).

It also means that no harmonic bridge has the length that's an integer multiple of the length of its building blocks

Some harmonic numbers that come right after an integer:
2 ≲ H₄ ,
3 ≲ H₁₁ ,
4 ≲ H₃₁ ,
5 ≲ H₈₃ ,
6 ≲ H₂₂₇ ,
7 ≲ H₆₁₆ ,
8 ≲ H₁₆₇₄ ,
9 ≲ H₄₅₅₀ ,
10 ≲ H₁₂₃₆₇

100 ≲ H₁₅₀₉₂₆₈₈₆₂₂₁₁₃₇₈₈₃₂₃₆₉₃₅₆₃₂₆₄₅₃₈₁₀₁₄₄₉₈₅₉₄₉₇

so yeah, it grows very slowly

In fact, the slow logarithmic growth of harmonic numbers is easy to show by comparing areas.

ln(k+1) < Hₖ < ln(k)+1