Smallest Positive Integer which is Sum of 2 Odd Primes in 6 Ways
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Theorem
The smallest positive integer which is the sum of $2$ odd primes in $6$ different ways is $60$.
Proof
| \(\ds 60\) | \(=\) | \(\ds 7 + 53\) | $6$ ways | |||||||||||
| \(\ds \) | \(=\) | \(\ds 13 + 47\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 17 + 43\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 19 + 41\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 23 + 37\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 29 + 31\) |
It is determined that there are no smaller numbers with this property by inspection:
| \(\ds 58\) | \(=\) | \(\ds 53 + 5\) | $3$ ways | |||||||||||
| \(\ds \) | \(=\) | \(\ds 47 + 11\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 41 + 17\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 29 + 29\) |
| \(\ds 56\) | \(=\) | \(\ds 53 + 3\) | $3$ ways | |||||||||||
| \(\ds \) | \(=\) | \(\ds 43 + 13\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 37 + 19\) |
| \(\ds 54\) | \(=\) | \(\ds 47 + 7\) | $5$ ways | |||||||||||
| \(\ds \) | \(=\) | \(\ds 43 + 11\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 41 + 13\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 37 + 17\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 31 + 23\) |
| \(\ds 52\) | \(=\) | \(\ds 47 + 5\) | $3$ ways | |||||||||||
| \(\ds \) | \(=\) | \(\ds 41 + 11\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 29 + 23\) |
| \(\ds 50\) | \(=\) | \(\ds 47 + 3\) | $4$ ways | |||||||||||
| \(\ds \) | \(=\) | \(\ds 43 + 7\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 37 + 13\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 31 + 19\) |
From the workings of Smallest Positive Integer which is Sum of 2 Odd Primes in n Ways, $48$ is the smallest positive integer which is the sum of $2$ odd primes in $5$ different ways.
Thus there is no need to explore smaller numbers.
$\blacksquare$
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $60$