Even Integers not Expressible as Sum of 3, 5 or 7 with Prime
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Theorem
The even integers that cannot be expressed as the sum of $2$ prime numbers where one of those primes is $3$, $5$ or $7$ begins:
- $98, 122, 124, 126, 128, 148, 150, \ldots$
This sequence is A283555 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
These are the primes which coincide with the upper end of a prime gap greater than $6$.
These can be found at:
and so on.
We have that:
\(\ds 98\) | \(=\) | \(\ds 19 + 79\) | ||||||||||||
\(\ds 122\) | \(=\) | \(\ds 13 + 109\) | ||||||||||||
\(\ds 124\) | \(=\) | \(\ds 11 + 113\) | ||||||||||||
\(\ds 126\) | \(=\) | \(\ds 13 + 113\) | ||||||||||||
\(\ds 128\) | \(=\) | \(\ds 19 + 109\) |
and so on.
$\blacksquare$
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $98$