Richert Theorem

In 1852, Chebyshev proved the Bertrand’s postulate:

For any integer $n>1$, there always exists at least one prime number $p$ with $n < p < 2n$.

You can find Erdős’ elementary proof here. In this post I would like to discuss an application of this fantastic result, discovered by Hans-Egon Richert in 1948:

Every integer $ n\geq 7$ can be expressed as a sum of distinct primes.

There are several proofs available in literature, but we will follow the short proof given by Richert himself (english translation has been taken from here and here):

Consider the set of prime integers $ {p_1, p_2,p_3,\ldots}$ where $ p_1=2, p_2=3, p_3=5,\ldots $. By Bertrand’s postulate we know that $ \boxed{p_i < p_{i+1} < 2p_i }$.

Next, we observe that, any integer between 7 and 19 can be written as a sum of distinct first 5 prime integers $ {2,3,5,7,11}$:

7 = 5+2; 8 = 5+3; 9 = 7+2; 10 = 5+3+2; 11 = 11; 12 = 7+5; 13 = 11+2; 14 = 7+5+2; 15 = 7+5+3; 16 = 11+5; 17 = 7+5+3+2; 18 = 11+7; 19 = 11+5+3

Hence we fix $ a=7-1=6$, $ b=19-a=13$, and $ k=5$ to conclude that $ \boxed{b \geq p_{k+1}}$.

Let, $ S_i = {a+1,a+2,\ldots, a+p_{i+1}}={7,8,\ldots, 6+p_{i+1}}$. Then by the above observation we know that the elements of $ S_k = S_5 = {7,8,\ldots, 19}$ are the sum of distinct first $ k=5$ prime integers. Moreover, if the elements of $ S_i$ can be written as the sum of distinct first $ i$ prime integers, then the elements of $ S_{i+1}$ can also be written as the sum of distinct first $ i+1$ prime integers since

$ S_{i+1}\subset S_i \cup \{m + p_{i+1}: m\in S_{i}\}$

as a consequence of $ 2p_{i+1} \geq p_{i+2}$.

Hence inductively the result follows by considering $ \bigcup_{i\ge k} S_i$, which contains all integers greater than $ a=6$, and contains only elements which are distinct sums of primes.

Exercise: Use Bertrand’s postulate to generalize the statement proved earlier: If $ n>1$ and $ k$ are natural numbers , then the sum

$ \displaystyle{\frac{1}{n}+ \frac{1}{n+1} + \ldots + \frac{1}{n+k}}$

cannot be an integer.

[HINT: Look at the comment by Dan in the earlier post.]

References:

[0] Turner, C. (2015) A theorem of Richert. Math.SE profile: https://math.stackexchange.com/users/37268/sharkos

[1] Richert, H. E. (1950). Über Zerfällungen in ungleiche Primzahlen. (German). Mathematische Zeitschrift 52, pp. 342-343. https://doi.org/10.1007/BF02230699

[2] Sierpiński,W. (1988). Elementary theory of numbers. North-Holland Mathematical Library 31, pp. 137-153.