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.

I A prime of the form 4k+1 must be of the form 12k+1, 12k+5, or 12k+9; but for 12k+9 we have 3m, for some m.

II A prime of the form 4k+3 must be of the form 12k+7, 12k+11, or 12k+3, but for 12k+3 we have 3m, for some m.

III If n+2 is expressible as the sum of 2 squares, it is of the form 4k+1. So its counterpart n must be of the form 4k+3. From I and II, this can only happen if the smaller prime is of the form 12k+11 and the larger of the form 12k+1 (there is a corresponding set of numbers where the 4k+1 prime is of the form 12k+5 and the 4k+3 prime is of the form 12+7)

IV The only exception is where 12k+3=3, i.e. k=0; so unless the smaller prime is 3, the units digit of n_12 is B.

Posted by broll on 2011-09-22 21:21:03