Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 344 — #352
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Chapter 9
Number Theory
implies r 1 D r 2 .
(c) Show that multiplicative inverses are unique, that is, show that
r ˝ r 1 D 1
r ˝ r 2 D 1
implies r 1 D r 2 .
Problem 9.33.
This problem will use elementary properties of congruences to prove that every
positive integer divides infinitely many Fibonacci numbers.
A function f W N ! N that satisifies
f .n/ D c 1 f .n 1/ C c 2 f .n 2/ C C c d f .n d/ (9.26)
for some c i 2 N and all n d is called a degree d linear-recursive.
A function f W N ! N has a degree d repeat modulo m at n and k when it
satisfies the following repeat congruences:
f .n/ f .k/ .mod m/;
f .n 1/ f .k 1/ .mod m/;
:
f .n .d 1// f .k .d 1// .mod m/:
for k > n d 1.
For the rest of this problem, assume linear-recursive functions and repeats are
degree d > 0.
(a) Prove that if a linear-recursive function has a repeat modulo m at n and k, then
it has one at n C 1 and k C 1.
(b) Prove that for all m > 1, every linear-recursive function repeats modulo m at
n and k for some n; k 2 Œd 1; d C m d /.
(c) A linear-recursive function is reverse-linear if its dth coefficient c d D ˙1.
Prove that if a reverse-linear function repeats modulo m at n and k for some n d,
then it repeats modulo m at n 1 and k 1.
(d) Conclude that every reverse-linear function must repeat modulo m at d 1
and .d 1/ C j for some j > 0.