Determine the possible nonzero units digits of a duodecimal positive integer n such that:

Each of n and n+2 is a prime number, and:

n+2 is expressible as the sum of squares of two positive itegers.

Any twin primes will be of the form (6x-1, 6x+1) except for (3, 5). Otherwise, one of them will be divisible by 2 or 3. All squares are of the form 4x or 4x+1. Therefore, the sum of two squares is of the form 4x, 4x+1, or 4x+2. Since n+2 is prime, it is of the form 4x+1. Therefore, n+2 is 1 more than a multiple of 4 and 6, so it is of the form 12x+1. Then, n is of the form 12x-1, so it ends in the digit for 11 in base 12. The answer is B or 11.

 

Posted by Math Man on 2011-09-22 21:50:22