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# Three Consecutive Odd Numbers

*We are very grateful to the Heilbronn Institute for Mathematical Research for their generous support for the development of this resource.*## You may also like

### Adding All Nine

### Doodles

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Age 11 to 16

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- Problem
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$3, 5, 7$ 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?

**Extension:**

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

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some other possibilities for yourself!

Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?