tut02sols - 2 The University of Sydney MATH2068 Number...

This preview shows pages 1–2. Sign up to view the full content.

The University of Sydney MATH2068 Number Theory and Cryptography (http://www.maths.usyd.edu.au/u/UG/IM/MATH2068/) Semester 2, 2008 Lecturer: R. Howlett Tutorial 2 1. For each natural number n , let a n be the residue of 2 n modulo 13. Observe that a k +1 2 a k (mod 13) for each k N , and thus the a k are easy to compute recursively, starting at a 0 = 1 and repeatedly doubling and reducing mod 13 to get successive terms of the sequence. Compute the ﬁrst 15 or 20 terms, and then compute a 2008 . Solution. 1, 2, 4, 8, 3, 6, 12, 11, 9, 5, 10, 7, 1 – and then it repeats. In other words, a 0 = a 12 = a 24 = ··· , and a 1 = a 13 = a 25 = ··· , etc. In general, a i = a j if and only if i j (mod 12). (In the terminology introduced in lectures, ord 13 (2) = 12.) Now 2004 4 (mod 12); so a 2008 = a 4 = 3. 2. Repeat Question 1 with 13 replaced by 3, 5, 7, 11, 31, 47. Solution. For 3 the sequence is just 1, 2, 1, 2, . . . ; that is, a i = 1 if i is even and a i = 2 if i is odd (and ord 3 (2) = 2). So a 2008 = 1. Using 5 we get 1, 2, 4, 3, 1, . . . . So ord 5 (2) = 4, and a 2008 = a 0 = 1 (since 2008 0 (mod 4)). Using 7 we get 1, 2, 4, 1, . . . . So ord 7 (2) = 3, and a 2008 = a 1 = 2 (since 2008 1 (mod 3)). Using 11 we get 1, 2, 4, 8, 5, 10, 9, 7, 3, 6, 1, . . . . So ord 11 (2) = 10, and a 2008 = a 8 = 3 (since 2008 8 (mod 10)). Using 31 we get 1, 2, 4, 8, 16, 1, . . . . So ord 31 (2) = 5, and a 2008 = a 3 = 8 (since 2008 3 (mod 5)). Using 47 we get 1, 2, 4, 8, 16, 32, 17, 34, 21, 42, 37, 27, 7, 14, 28, 9, 18, 36, 25, 3, 6, 12, 24, 1, . . . . So ord 47 (2) = 23, and a 2008 = a 7 = 34 (since 2008 7 (mod 23)). Observe that in all these case, ord

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 2

tut02sols - 2 The University of Sydney MATH2068 Number...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online