Number as Sum of Distinct Primes greater than 11

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Theorem

Every number greater than $45$ can be expressed as the sum of distinct primes greater than $11$.


Proof

Let $S = \set {s_n}_{n \mathop \in N}$ be the set of primes greater than $11$ ordered by size.

Then $S = \set {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, \dots}$.


By Bertrand-Chebyshev Theorem:

$s_{n + 1} \le 2 s_n$ for all $n \in \N$.

We observe that every integer $n$ where $45 < n \le 45 + s_{11} = 92$ can be expressed as a sum of distinct elements in $\set {s_1, \dots, s_{10}} = \set {11, 13, 17, 19, 23, 29, 31, 37, 41, 43}$.

Hence the result by Richert's Theorem.

$\Box$


Here is a demonstration of our claim:

\(\ds 46\) \(=\) \(\ds 17 + 29\)
\(\ds 47\) \(=\) \(\ds 11 + 17 + 19\)
\(\ds 48\) \(=\) \(\ds 19 + 29\)
\(\ds 49\) \(=\) \(\ds 13 + 17 + 19\)
\(\ds 50\) \(=\) \(\ds 19 + 31\)
\(\ds 51\) \(=\) \(\ds 11 + 17 + 23\)
\(\ds 52\) \(=\) \(\ds 23 + 29\)
\(\ds 53\) \(=\) \(\ds 13 + 17 + 23\)
\(\ds 54\) \(=\) \(\ds 23 + 31\)
\(\ds 55\) \(=\) \(\ds 13 + 19 + 23\)
\(\ds 56\) \(=\) \(\ds 19 + 37\)
\(\ds 57\) \(=\) \(\ds 11 + 17 + 29\)
\(\ds 58\) \(=\) \(\ds 17 + 41\)
\(\ds 59\) \(=\) \(\ds 13 + 17 + 29\)
\(\ds 60\) \(=\) \(\ds 19 + 41\)
\(\ds 61\) \(=\) \(\ds 13 + 19 + 29\)
\(\ds 62\) \(=\) \(\ds 19 + 43\)
\(\ds 63\) \(=\) \(\ds 13 + 19 + 31\)
\(\ds 64\) \(=\) \(\ds 23 + 41\)
\(\ds 65\) \(=\) \(\ds 11 + 17 + 37\)
\(\ds 66\) \(=\) \(\ds 23 + 43\)
\(\ds 67\) \(=\) \(\ds 13 + 17 + 37\)
\(\ds 68\) \(=\) \(\ds 31 + 37\)
\(\ds 69\) \(=\) \(\ds 13 + 19 + 37\)
\(\ds 70\) \(=\) \(\ds 11 + 17 + 19 + 23\)
\(\ds 71\) \(=\) \(\ds 17 + 23 + 31\)
\(\ds 72\) \(=\) \(\ds 13 + 17 + 19 + 23\)
\(\ds 73\) \(=\) \(\ds 19 + 23 + 31\)
\(\ds 74\) \(=\) \(\ds 11 + 13 + 19 + 31\)
\(\ds 75\) \(=\) \(\ds 13 + 19 + 43\)
\(\ds 76\) \(=\) \(\ds 11 + 17 + 19 + 29\)
\(\ds 77\) \(=\) \(\ds 17 + 19 + 41\)
\(\ds 78\) \(=\) \(\ds 11 + 17 + 19 + 31\)
\(\ds 79\) \(=\) \(\ds 17 + 19 + 43\)
\(\ds 80\) \(=\) \(\ds 13 + 17 + 19 + 31\)
\(\ds 81\) \(=\) \(\ds 17 + 23 + 41\)
\(\ds 82\) \(=\) \(\ds 11 + 17 + 23 + 31\)
\(\ds 83\) \(=\) \(\ds 19 + 23 + 41\)
\(\ds 84\) \(=\) \(\ds 13 + 17 + 23 + 31\)
\(\ds 85\) \(=\) \(\ds 19 + 23 + 43\)
\(\ds 86\) \(=\) \(\ds 13 + 19 + 23 + 31\)
\(\ds 87\) \(=\) \(\ds 17 + 29 + 41\)
\(\ds 88\) \(=\) \(\ds 11 + 17 + 29 + 31\)
\(\ds 89\) \(=\) \(\ds 19 + 29 + 41\)
\(\ds 90\) \(=\) \(\ds 13 + 17 + 29 + 31\)
\(\ds 91\) \(=\) \(\ds 19 + 31 + 41\)
\(\ds 92\) \(=\) \(\ds 13 + 19 + 29 + 31\)

$\blacksquare$


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