# What is the biggest positive integer within 100 that can be/can not be written as the differences between two positive primes?

What is the biggest positive integer within 100 that can be/can not be written as the differences between two positive primes?

Can someone answer the cannot part?

There are two parts in the question.

Thanks in advance!

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## 1 Answer

Differences of primes are generally even (except when $2$ is involved).

$99$ can be written as $101 - 2$. Therefore a lower bound for the largest number which cannot be written as a difference of primes is $97$. It remains to check that $100$ and $98$ actually can be written as a difference of primes. Some simple trial and error will reveal that $$101 - 3 = 98$$ $$103 - 3 = 100$$

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What about the can not be part? –  Victor Oct 16 '12 at 21:38
That's where $97$ comes in. –  EuYu Oct 16 '12 at 21:44
Eu Yu - Thanks you very much. –  Victor Oct 16 '12 at 21:45