Smallest Positive Integer which is Sum of 2 Odd Primes in n Ways
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Theorem
The sequence of positive integers $n$ which are the smallest such that they are the sum of $2$ odd primes in $k$ different ways begins as follows:
$k$ $n$ $1$ $6$ $2$ $10$ $3$ $22$ $4$ $34$ $5$ $48$ $6$ $60$
This sequence is A001172 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
| \(\ds 6\) | \(=\) | \(\ds 3 + 3\) | $1$ way |
| \(\ds 8\) | \(=\) | \(\ds 5 + 3\) | $1$ way |
| \(\ds 10\) | \(=\) | \(\ds 7 + 3\) | $2$ ways | |||||||||||
| \(\ds \) | \(=\) | \(\ds 5 + 5\) |
| \(\ds 12\) | \(=\) | \(\ds 7 + 5\) | $1$ way |
| \(\ds 14\) | \(=\) | \(\ds 11 + 3\) | $2$ ways | |||||||||||
| \(\ds \) | \(=\) | \(\ds 7 + 7\) |
| \(\ds 16\) | \(=\) | \(\ds 13 + 3\) | $2$ ways | |||||||||||
| \(\ds \) | \(=\) | \(\ds 11 + 5\) |
| \(\ds 18\) | \(=\) | \(\ds 13 + 5\) | $2$ ways | |||||||||||
| \(\ds \) | \(=\) | \(\ds 11 + 7\) |
| \(\ds 20\) | \(=\) | \(\ds 17 + 3\) | $2$ ways | |||||||||||
| \(\ds \) | \(=\) | \(\ds 13 + 7\) |
| \(\ds 22\) | \(=\) | \(\ds 19 + 3\) | $3$ ways | |||||||||||
| \(\ds \) | \(=\) | \(\ds 17 + 5\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 11 + 11\) |
| \(\ds 24\) | \(=\) | \(\ds 19 + 5\) | $2$ ways | |||||||||||
| \(\ds \) | \(=\) | \(\ds 13 + 11\) |
| \(\ds 26\) | \(=\) | \(\ds 23 + 3\) | $3$ ways | |||||||||||
| \(\ds \) | \(=\) | \(\ds 19 + 7\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 13 + 13\) |
| \(\ds 28\) | \(=\) | \(\ds 23 + 5\) | $2$ ways | |||||||||||
| \(\ds \) | \(=\) | \(\ds 17 + 11\) |
| \(\ds 30\) | \(=\) | \(\ds 23 + 7\) | $3$ ways | |||||||||||
| \(\ds \) | \(=\) | \(\ds 19 + 11\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 17 + 13\) |
| \(\ds 32\) | \(=\) | \(\ds 29 + 3\) | $2$ ways | |||||||||||
| \(\ds \) | \(=\) | \(\ds 19 + 13\) |
| \(\ds 34\) | \(=\) | \(\ds 31 + 3\) | $4$ ways | |||||||||||
| \(\ds \) | \(=\) | \(\ds 29 + 5\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 23 + 11\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 17 + 17\) |
| \(\ds 36\) | \(=\) | \(\ds 31 + 5\) | $4$ ways | |||||||||||
| \(\ds \) | \(=\) | \(\ds 29 + 7\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 23 + 13\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 19 + 17\) |
| \(\ds 38\) | \(=\) | \(\ds 31 + 7\) | $2$ ways | |||||||||||
| \(\ds \) | \(=\) | \(\ds 19 + 19\) |
| \(\ds 40\) | \(=\) | \(\ds 37 + 3\) | $3$ ways | |||||||||||
| \(\ds \) | \(=\) | \(\ds 29 + 11\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 23 + 17\) |
| \(\ds 42\) | \(=\) | \(\ds 37 + 5\) | $4$ ways | |||||||||||
| \(\ds \) | \(=\) | \(\ds 31 + 11\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 29 + 13\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 23 + 19\) |
| \(\ds 44\) | \(=\) | \(\ds 41 + 3\) | $3$ ways | |||||||||||
| \(\ds \) | \(=\) | \(\ds 37 + 7\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 31 + 13\) |
| \(\ds 46\) | \(=\) | \(\ds 43 + 3\) | $4$ ways | |||||||||||
| \(\ds \) | \(=\) | \(\ds 41 + 5\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 29 + 17\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 23 + 23\) |
| \(\ds 48\) | \(=\) | \(\ds 43 + 5\) | $5$ ways | |||||||||||
| \(\ds \) | \(=\) | \(\ds 41 + 7\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 37 + 11\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 31 + 17\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 29 + 19\) |
From Smallest Positive Integer which is Sum of 2 Odd Primes in 6 Ways, the smallest positive integer which is the sum of $2$ odd primes in $6$ different ways is $60$.
$\blacksquare$
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $60$