Find the largest number that cannot be written as a sum of distinct primes of the form 6*n+1.

Well as it's entitled 'mission impossible' I assume there is no such largest number.

But is it true?

Consider this range:

1 331 331  
2 332 109+223  
3 333 13+43+277  
4 334 13+31+109+181  
5 335 13+19+31+61+211  
6 336 13+19+31+37+79+157  
7 337 337  
8 338 7+331  
9 339 7+332  
10 340 7+333  
11 341 7+334  
12 342 7+335  
13 343 7+336  
14 344 7+337  
15 345 13+332  
16 346 (19+7)+43+277  
17 347 (19+7)+31+109+181  
18 348 13+19+31+37+97+151
19 349 349  
20 350 19+331  etc.

We could have selected a higher starting number and left out both 7 and 13 to start with, or 7 and 13 and 19, etc. and still obtain a series of indefinite length that could be extended by adding 7,13,19,etc. to earlier solutions in the series that did not themselves contain those small primes.

 

 

Edited on August 27, 2013, 5:30 am

Posted by broll on 2013-08-27 05:28:06