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 Three forever (Posted on 2010-09-30)
Choose a prime number greater than 3.
Multiply it by itself and add 14.
Divide by 12 and write down the remainder.
It will always be 3.

WHY?

 No Solution Yet Submitted by Ady TZIDON Rating: 2.7500 (4 votes)

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 Solution | Comment 3 of 5 |
All primes greater than 3 are not divisible by 2 so the primes must be odd
All primes greater than 3 are not divisible by 3 so primes after this are of the form 6n+1 or 6n-1

(6n+1)²= 36n²+12n+1
add 14 to this to get 36n²+12n+15

Now dividing by 12: the first two terms give no remainder but the last gives a remainder of 3.

6n-1 works the same way.

Incidentally this proof shows the trick isn't so much about primes as about numbers divisible by neither 2 nor 3.  You can check it works with the composites such as 25, 35, 49, etc.

 Posted by Jer on 2010-09-30 15:49:41
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