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Three Consecutive Odd Numbers

Age 11 to 16
Challenge Level

 

$3, 5, 7$ are a set of three consecutive odd numbers which are all prime.

$109, 111$ and $113$ are a set of three consecutive odd numbers which are not all prime $(111=37\times3)$.

Can you find another set of three consecutive odd numbers which are all prime?

If not, might it be impossible?

Mathematicians aren't usually satisfied with testing a few examples to convince themselves that something is always true, and look to proofs to provide rigorous and convincing arguments and justifications.

Can you prove that there is only one set of three consecutive odd numbers which are all prime?

Below is a proof that has been scrambled up.
Can you rearrange it into its original order?
 

If you can find a proof which is different to the one in our proof sorter, then please do let us know by submitting it as a solution.
 

Extension:

Take a look at Take Three from Five which requires similar reasoning to this problem.

We are very grateful to the Heilbronn Institute for Mathematical Research for their generous support for the development of this resource.