+

+=

++=

+=

++++=

+===+=

++++++=

+=

++=

+=+===+=+=

++++++++++=

+===+=

Explain the pattern and find the next five rows.

*Hint: the symbols I chose have nothing to do with the pattern.*

For every integer n, there is an integer f which is the product of the prime factors of n. The nth element is f symbols long. The kth symbol in an element is '+' if the gcd of f and k is 1, otherwise the symbol is '='.

Example 1: n=12

12 = 2*2*3, then f = 2*3 = 6

gcd(6,1) = 1 '+'

gcd(6,2) = 2 '='

gcd(6,3) = 3 '='

gcd(6,4) = 2 '='

gcd(6,5) = 1 '+'

gcd(6,6) = 6 '='

6: +===+=

Example 2: n=10

10 = 2*5, then f = 2*5 = 10

gcd(10,1) = 1 '+'

gcd(10,2) = 2 '='

gcd(10,3) = 1 '+'

gcd(10,4) = 2 '='

gcd(10,5) = 5 '='

gcd(10,6) = 2 '='

gcd(10,7) = 1 '+'

gcd(10,8) = 2 '='

gcd(10,9) = 1 '+'

gcd(10,10) = 10 '='

10: +=+===+=+=

Continuing this algorithm for n=13 to 20 yeilds:

13: ++++++++++++=

14: +=+=+===+=+=+=

15: ++=+==++==+=++=

16: +=

17: ++++++++++++++++=

18: +===+=

19: ++++++++++++++++++=

20: +=+===+=+=

Brian Wainscott asked what would happen for n=30: +=====+===+=+===+=+===+=====+=