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## Pattern Play - Act 17

Pick a natural number n

Find the product n(n+1)

Now add 17 to that product

What do you notice?

To make the pattern more noticeable, test the first ten natural numbers using this rule:

• n = 1 leads to 19
• n = 2 leads to 23
• n = 3 leads to 29
• n = 4 leads to 37
• n = 5 leads to 47
• n = 6 leads to 59
• n = 7 leads to 73
• n = 8 leads to 89
• n = 9 leads to 107
• n = 10 leads to 127
Now, what do you notice?

Will this pattern always work?

And what happens if you try a different prime other than 17. Use a spreadsheet, as it is easy to set up a grid that will print out a table of results for any prime number.

What happens if you try adding a composite number?

Source: J.W. (Math teacher)

Hint: For the number 17 and the given table, do you notice that all of the final sums are prime. Now, extend the table for n = 11, 12, 13, ....

Solution Commentary: You should find that this nice "prime" pattern breaks down when n = 16 or n = 17. Use algebra and the distributive law on the expression n(n+1)+17 to see why this happens...and this can be generalized into a proof that the pattern will always break down for any number other than 17 as well.

A new question: Why does this pattern seem to work for 17 but seems to rapidly fail for other primes such as 11, 13, or 19, or even 37?

The teacher (J.W.) who gave me this problem added this note: "Patterns in numbers have always interested me. Maybe it's because you don't need to have a lot of mathematical background to notice patterns. Numbers are very fascinating in that there is no end to what you can find working with them. The variety of paths a student can take with this problem is the other reason I chose it...The students who are really interested can pursue the problem further using non-primes...they could also study the primes not generated and the differences between the primes generated. This could keep a young body productively busy for a while..."