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Duodecimal Digits (Posted on 2011-09-22) Difficulty: 2 of 5
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.

See The Solution Submitted by K Sengupta    
Rating: 4.5000 (2 votes)

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Solution Answer and proof Comment 5 of 5 |

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
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