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:
\(\ds 46\) | \(=\) | \(\ds 17 + 29\) | ||||||||||||
\(\ds 47\) | \(=\) | \(\ds 11 + 17 + 19\) | ||||||||||||
\(\ds 48\) | \(=\) | \(\ds 19 + 29\) | ||||||||||||
\(\ds 49\) | \(=\) | \(\ds 13 + 17 + 19\) | ||||||||||||
\(\ds 50\) | \(=\) | \(\ds 19 + 31\) | ||||||||||||
\(\ds 51\) | \(=\) | \(\ds 11 + 17 + 23\) | ||||||||||||
\(\ds 52\) | \(=\) | \(\ds 23 + 29\) | ||||||||||||
\(\ds 53\) | \(=\) | \(\ds 13 + 17 + 23\) | ||||||||||||
\(\ds 54\) | \(=\) | \(\ds 23 + 31\) | ||||||||||||
\(\ds 55\) | \(=\) | \(\ds 13 + 19 + 23\) | ||||||||||||
\(\ds 56\) | \(=\) | \(\ds 19 + 37\) | ||||||||||||
\(\ds 57\) | \(=\) | \(\ds 11 + 17 + 29\) | ||||||||||||
\(\ds 58\) | \(=\) | \(\ds 17 + 41\) | ||||||||||||
\(\ds 59\) | \(=\) | \(\ds 13 + 17 + 29\) | ||||||||||||
\(\ds 60\) | \(=\) | \(\ds 19 + 41\) | ||||||||||||
\(\ds 61\) | \(=\) | \(\ds 13 + 19 + 29\) | ||||||||||||
\(\ds 62\) | \(=\) | \(\ds 19 + 43\) | ||||||||||||
\(\ds 63\) | \(=\) | \(\ds 13 + 19 + 31\) | ||||||||||||
\(\ds 64\) | \(=\) | \(\ds 23 + 41\) | ||||||||||||
\(\ds 65\) | \(=\) | \(\ds 11 + 17 + 37\) | ||||||||||||
\(\ds 66\) | \(=\) | \(\ds 23 + 43\) | ||||||||||||
\(\ds 67\) | \(=\) | \(\ds 13 + 17 + 37\) | ||||||||||||
\(\ds 68\) | \(=\) | \(\ds 31 + 37\) | ||||||||||||
\(\ds 69\) | \(=\) | \(\ds 13 + 19 + 37\) | ||||||||||||
\(\ds 70\) | \(=\) | \(\ds 11 + 17 + 19 + 23\) | ||||||||||||
\(\ds 71\) | \(=\) | \(\ds 17 + 23 + 31\) | ||||||||||||
\(\ds 72\) | \(=\) | \(\ds 13 + 17 + 19 + 23\) | ||||||||||||
\(\ds 73\) | \(=\) | \(\ds 19 + 23 + 31\) | ||||||||||||
\(\ds 74\) | \(=\) | \(\ds 11 + 13 + 19 + 31\) | ||||||||||||
\(\ds 75\) | \(=\) | \(\ds 13 + 19 + 43\) | ||||||||||||
\(\ds 76\) | \(=\) | \(\ds 11 + 17 + 19 + 29\) | ||||||||||||
\(\ds 77\) | \(=\) | \(\ds 17 + 19 + 41\) | ||||||||||||
\(\ds 78\) | \(=\) | \(\ds 11 + 17 + 19 + 31\) | ||||||||||||
\(\ds 79\) | \(=\) | \(\ds 17 + 19 + 43\) | ||||||||||||
\(\ds 80\) | \(=\) | \(\ds 13 + 17 + 19 + 31\) | ||||||||||||
\(\ds 81\) | \(=\) | \(\ds 17 + 23 + 41\) | ||||||||||||
\(\ds 82\) | \(=\) | \(\ds 11 + 17 + 23 + 31\) | ||||||||||||
\(\ds 83\) | \(=\) | \(\ds 19 + 23 + 41\) | ||||||||||||
\(\ds 84\) | \(=\) | \(\ds 13 + 17 + 23 + 31\) | ||||||||||||
\(\ds 85\) | \(=\) | \(\ds 19 + 23 + 43\) | ||||||||||||
\(\ds 86\) | \(=\) | \(\ds 13 + 19 + 23 + 31\) | ||||||||||||
\(\ds 87\) | \(=\) | \(\ds 17 + 29 + 41\) | ||||||||||||
\(\ds 88\) | \(=\) | \(\ds 11 + 17 + 29 + 31\) | ||||||||||||
\(\ds 89\) | \(=\) | \(\ds 19 + 29 + 41\) | ||||||||||||
\(\ds 90\) | \(=\) | \(\ds 13 + 17 + 29 + 31\) | ||||||||||||
\(\ds 91\) | \(=\) | \(\ds 19 + 31 + 41\) | ||||||||||||
\(\ds 92\) | \(=\) | \(\ds 13 + 19 + 29 + 31\) |
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $45$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $45$