3. Prove that if 2 |un, then 4 | (u2 4. For the Fibonacci sequence, establish the following: 19ª is always a perfect square. 1-u1); and similarly, if 3 | un, then 9| (u?1 - u²_1). n+1 (a) (mod
3. Prove that if 2 |un, then 4 | (u2 4. For the Fibonacci sequence, establish the following: 19ª is always a perfect square. 1-u1); and similarly, if 3 | un, then 9| (u?1 - u²_1). n+1 (a) (mod
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
3
![sparse; only 42 of thêm are
Ts are so
pre
preselmuy khown, the largest being the 126377-digit u604711-
Let us present one more proof of the infinitude of primes, this one involv-
Fibonacci numbers. Suppose that there are only finitely many primes, say!
ing
primes 2, 3, 5, . Pr, arranged in ascending order. Next, consider the correspond-
Fibonacci numbers u2, u3, U5,...,
relatively prime in pairs. Exclude u2 = 1. Each of the remaining r – 1 numbers is
prime
divisible by a single prime with the possible exception that one of them has two
factors (there being only r primes in all). A contradiction occurs because
Up,. According to Theorem 14.3, these are
|3|
-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 up+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(-1)" is always a perfect square.
3. Prove that if 2 | un, then 4 | (u2
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, u 10, 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
%3D
+1-u1); and similarly, if 3| un, then 9| (u²1- u_1).
n+1
are all even integers.
= unun+1
n++En+ n + n
[Hint: For n 2, u = unun+1 - Unun-1.]
6. Utilize the identity in Problem 5 to prove that for n > 3
.2
%3D
u+3u-1 +2(12-2 + u-3 +. +u+ 내)
2
0. Find the Fibonacci numbers that divide both u24 and u36.
the fact that um |u, if and only if m | n 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.
1, prove that u mu, divides umn for all m, n 1.
%3D
number or um
is a Fibonacci number. Give examples illustrating both cases.
numbers. Find them.
I 1, prove that 23-1u, =n (mod 5).
%3D
a Fibonacci number
15.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66c7b497-dac7-4855-b923-2e60bbc73063%2F92cb33ff-ff88-4ba7-888a-98191cded7f0%2Flf0ggrd.jpeg&w=3840&q=75)
Transcribed Image Text:sparse; only 42 of thêm are
Ts are so
pre
preselmuy khown, the largest being the 126377-digit u604711-
Let us present one more proof of the infinitude of primes, this one involv-
Fibonacci numbers. Suppose that there are only finitely many primes, say!
ing
primes 2, 3, 5, . Pr, arranged in ascending order. Next, consider the correspond-
Fibonacci numbers u2, u3, U5,...,
relatively prime in pairs. Exclude u2 = 1. Each of the remaining r – 1 numbers is
prime
divisible by a single prime with the possible exception that one of them has two
factors (there being only r primes in all). A contradiction occurs because
Up,. According to Theorem 14.3, these are
|3|
-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 up+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(-1)" is always a perfect square.
3. Prove that if 2 | un, then 4 | (u2
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, u 10, 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
%3D
+1-u1); and similarly, if 3| un, then 9| (u²1- u_1).
n+1
are all even integers.
= unun+1
n++En+ n + n
[Hint: For n 2, u = unun+1 - Unun-1.]
6. Utilize the identity in Problem 5 to prove that for n > 3
.2
%3D
u+3u-1 +2(12-2 + u-3 +. +u+ 내)
2
0. Find the Fibonacci numbers that divide both u24 and u36.
the fact that um |u, if and only if m | n 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.
1, prove that u mu, divides umn for all m, n 1.
%3D
number or um
is a Fibonacci number. Give examples illustrating both cases.
numbers. Find them.
I 1, prove that 23-1u, =n (mod 5).
%3D
a Fibonacci number
15.
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