Advanced Engineering Mathematics
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ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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14.2 question 2
![UMBERS
mewhat sparse; only 42 of them are
nown, the largest being the 126377-digit uGOAT4
correspond-
.u p.- According to Theorem 14.3, these are
primes 2, 3, 5,
ing Fibonacci numbers u2, u3, U5,
relatively prime in pairs. Exclude u2 =
divisible by a single prime with the possibie exception that one of the bers is
%3D
H37 =73 149-2221 has three prime factors.
%3D
PROBLEMS 14.2
1 Given any prime p 5, it is known that either u p-1 or u p+1 is divisible by p. Confirm
this in the cases of the primes 7, 11, 13, and 17.
2. For n = 1, 2, .. ., 10, show that 5u + 4(-1y" is always a perfect square.
3. Prove that if 2|un, then 4 | (u?
4. For the Fibonacci sequence, establish the following:
(a) un+3 =Un (mod 2), hence u3, u6, ug,
(b) un+5 = 3un (mod 5), hence u5, u10, u15, ... are all divisible by 5.
5. Show that the sum of the squares of the first n Fibonacci numbers is given by the formula
n+1
1--1); and similarly, if 3 | un, then 9 (u?. – u² )
%3D
are all even integers.
%3D
Itun"n = "n+ . +En+n +
[Hint: For n 2 2, u =unun+1 - UnUn-1.]
6. Utilize the identity in Problem 5 to prove that for n > 3
ut1=u + 3u-1 + 2(u-2 + u-3 때)
Evaluate gcd(u9, u12), gcd(u15, u 20), and gcd(u24, U36).
* Find the Fibonacci numbers that divide both u24 and U 36 -
he fact that um lu, if and only if m ln to verify each of the assertions below:
(a) 2|un if and only if 3 n.
(b) 3 un if and only if 4|n.
(c) 5|un if and only if 5 n.
(d)8unif and only if 6 n.
%3D
number or um
10 Fibonacci number. Give examples illustrating both cascs.
numbers. Find them.
15. For n> 1. prove that 25
\Hint: Use induction an
un =n (mod 5).
14. If un](https://content.bartleby.com/qna-images/question/66c7b497-dac7-4855-b923-2e60bbc73063/3e052b25-7ffa-43d6-9d30-747c00b48b2d/d65o7mf.jpeg)
Transcribed Image Text:UMBERS
mewhat sparse; only 42 of them are
nown, the largest being the 126377-digit uGOAT4
correspond-
.u p.- According to Theorem 14.3, these are
primes 2, 3, 5,
ing Fibonacci numbers u2, u3, U5,
relatively prime in pairs. Exclude u2 =
divisible by a single prime with the possibie exception that one of the bers is
%3D
H37 =73 149-2221 has three prime factors.
%3D
PROBLEMS 14.2
1 Given any prime p 5, it is known that either u p-1 or u p+1 is divisible by p. Confirm
this in the cases of the primes 7, 11, 13, and 17.
2. For n = 1, 2, .. ., 10, show that 5u + 4(-1y" is always a perfect square.
3. Prove that if 2|un, then 4 | (u?
4. For the Fibonacci sequence, establish the following:
(a) un+3 =Un (mod 2), hence u3, u6, ug,
(b) un+5 = 3un (mod 5), hence u5, u10, u15, ... are all divisible by 5.
5. Show that the sum of the squares of the first n Fibonacci numbers is given by the formula
n+1
1--1); and similarly, if 3 | un, then 9 (u?. – u² )
%3D
are all even integers.
%3D
Itun"n = "n+ . +En+n +
[Hint: For n 2 2, u =unun+1 - UnUn-1.]
6. Utilize the identity in Problem 5 to prove that for n > 3
ut1=u + 3u-1 + 2(u-2 + u-3 때)
Evaluate gcd(u9, u12), gcd(u15, u 20), and gcd(u24, U36).
* Find the Fibonacci numbers that divide both u24 and U 36 -
he fact that um lu, if and only if m ln to verify each of the assertions below:
(a) 2|un if and only if 3 n.
(b) 3 un if and only if 4|n.
(c) 5|un if and only if 5 n.
(d)8unif and only if 6 n.
%3D
number or um
10 Fibonacci number. Give examples illustrating both cascs.
numbers. Find them.
15. For n> 1. prove that 25
\Hint: Use induction an
un =n (mod 5).
14. If un
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