All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Just Math
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)

Comments: ( Back to comment list | You must be logged in to post comments.)
Possible solution | Comment 3 of 5 |

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
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (4)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2018 by Animus Pactum Consulting. All rights reserved. Privacy Information