bertrand’s postulate | Gaurish4Math

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.

If you enjoyed reading Lewis Carroll’s Alice’s Adventures in Wonderland, George Gamow’s Mr Tompkins, Abbott’s Flatland, Malba Tahan’s The Man Who Counted, Imre Lakatos’s Proofs and Refutations or Tarasov’s Calculus, then you will enjoy reading Enzensberger’s The Number Devil. But that is not an if and only if statement.

Originally written in German and published as Der Zahlenteufel, so far it has been translated into 26 languages (as per the back cover).

After reading this book one will have some knowledge of infinity, infinitesimal, zero, decimal number system, prime numbers (sieve of eratothenes, Bertrand’s postulate, Goldbach conjecture), rational numbers (0.999… = 1.0, fractions with 7 in denominator), irrational numbers (√2 = 1.4142…, uncountable), triangular numbers, square numbers, Fibonacci numbers, Pascal’s triangle (glimpse of Sierpinski triangle in it), combinatorics (permutations and combinations, role of Pascal’s triangle), cardinality of sets (countable sets like even numbers, prime numbers,…), infinite series (geometric series, harmonic series), golden ratio (recursive relations, continued fractions..), Euler characteristic (polyhedra and planar graphs), how to prove (11111111111^2 does not give numerical palindrome, Principia Mathematica), travelling salesman problem, Klein bottle, types of infinities (Cantor’s work), Euler product formula, imaginary numbers (Gaussian integer), Pythagoras theorem, lack of women mathematicians and pi.

Since this is a translation of original work into English, you might not be happy with the language. Though the author is not a mathematician, he is a well-known and respected European intellectual and author with wide-ranging interests. He gave a speech on mathematics and culture, “Zugbrücke außer Betrieb, oder die Mathematik im Jenseits der Kultur—eine Außenansicht” (“Drawbridge out of order, or mathematics outside of culture—a view from the outside”), in the program for the general public at the International Congress of Mathematicians in Berlin in 1998. The speech was published under the joint sponsorship of the American Mathematical Society and the Deutsche Mathematiker Vereinigung as a pamphlet in German with facing English translation under the title Drawbridge Up: Mathematics—A Cultural Anathema, with an introduction by David Mumford.

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Number Devil