Smallest Integer not Sum of Two Ulam Numbers
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Theorem
The smallest integer greater than $1$ which is not the sum of two Ulam numbers is $23$.
Proof
Recall the Ulam numbers:
The sequence of Ulam numbers begins as follows:
- $1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, 38, 47, 48, 53, 57, 62, 69, \ldots{}$
We have:
| \(\ds 2\) | \(=\) | \(\ds 1 + 1\) | ||||||||||||
| \(\ds 3\) | \(=\) | \(\ds 2 + 1\) | ||||||||||||
| \(\ds 4\) | \(=\) | \(\ds 3 + 1\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 2 + 2\) | ||||||||||||
| \(\ds 5\) | \(=\) | \(\ds 4 + 1\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 3 + 2\) | ||||||||||||
| \(\ds 6\) | \(=\) | \(\ds 4 + 2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 3 + 3\) | ||||||||||||
| \(\ds 7\) | \(=\) | \(\ds 6 + 1\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 4 + 3\) | ||||||||||||
| \(\ds 8\) | \(=\) | \(\ds 6 + 2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 4 + 4\) | ||||||||||||
| \(\ds 9\) | \(=\) | \(\ds 8 + 1\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 6 + 3\) | ||||||||||||
| \(\ds 10\) | \(=\) | \(\ds 8 + 2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 6 + 4\) | ||||||||||||
| \(\ds 11\) | \(=\) | \(\ds 8 + 3\) | ||||||||||||
| \(\ds 12\) | \(=\) | \(\ds 11 + 1\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 8 + 4\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 6 + 6\) | ||||||||||||
| \(\ds 13\) | \(=\) | \(\ds 11 + 2\) | ||||||||||||
| \(\ds 14\) | \(=\) | \(\ds 13 + 1\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 11 + 3\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 8 + 6\) | ||||||||||||
| \(\ds 15\) | \(=\) | \(\ds 13 + 2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 11 + 4\) | ||||||||||||
| \(\ds 16\) | \(=\) | \(\ds 13 + 3\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 8 + 8\) | ||||||||||||
| \(\ds 17\) | \(=\) | \(\ds 16 + 1\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 13 + 4\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 11 + 6\) | ||||||||||||
| \(\ds 18\) | \(=\) | \(\ds 16 + 2\) | ||||||||||||
| \(\ds 19\) | \(=\) | \(\ds 18 + 1\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 16 + 3\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 13 + 6\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 11 + 8\) | ||||||||||||
| \(\ds 20\) | \(=\) | \(\ds 18 + 2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 16 + 4\) | ||||||||||||
| \(\ds 21\) | \(=\) | \(\ds 18 + 3\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 13 + 8\) | ||||||||||||
| \(\ds 22\) | \(=\) | \(\ds 18 + 4\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 16 + 6\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 11 + 11\) |
Now consider the the difference between $23$ and successive Ulam numbers:
| \(\ds 23 - 18\) | \(=\) | \(\ds 5\) | not a Ulam number | |||||||||||
| \(\ds 23 - 16\) | \(=\) | \(\ds 7\) | not a Ulam number | |||||||||||
| \(\ds 23 - 13\) | \(=\) | \(\ds 10\) | not a Ulam number | |||||||||||
| \(\ds 23 - 11\) | \(=\) | \(\ds 12\) | not a Ulam number |
and it is not necessary to go further back.
Hence the result.
$\blacksquare$
Sources
- 1994: Richard K. Guy: Unsolved Problems in Number Theory (2nd ed.)
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $23$