30034 625 14429 = Use Kraitchik's method to factor the number 20437.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Educational Content: Elementary Number Theory - Problems 5.4**
**1. Fermat’s Method for Factoring**
Use Fermat’s method to factor each of the following numbers:
(a) 2279
(b) 10541
(c) 340663 *[Hint: The smallest square just exceeding 340663 is 584²]*
**2. Prove a Perfect Square Pattern**
Prove that a perfect square must end in one of the following pairs of digits: 00, 01, 04, 09, 16, 21, 24, 25, 29, 36, 41, 44, 49, 56, 61, 64, 69, 76, 81, 84, 89, 96.
*[Hint: Because \( x^2 \equiv (50 + x)^2 \pmod{100} \) and \( x^2 \equiv (50 - x)^2 \pmod{100} \), it suffices to examine the final digits of \( x^2 \) for the 26 values \( x = 0, 1, 2, \ldots, 25 \).]*
**3. Fermat's Factorization of Powers of 2**
Factor the number \( 2^{11} - 1 \) using Fermat’s factorization method.
**4. Mersenne’s Observation (1647)**
In 1647, Mersenne noted that when a number can be written as a sum of two relative prime squares in two distinct ways, it is composite and can be factored as follows:
If \( n = a^2 + b^2 = c^2 + d^2 \), then
\[
n = \frac{(ac + bd)(ac - bd)}{(a + d)(a - d)}
\]
Use this result to factor the numbers:
\[
493 = 18^2 + 13^2 = 22^2 + 3^2
\]
\[
38025 = 168^2 + 99^2 = 156^2 + 117^2
\]
**5. Generalized Fermat Method**
Employ the generalized Fermat](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66c7b497-dac7-4855-b923-2e60bbc73063%2F5123ac2c-d5ae-43ed-ba3a-57f0003b4f1e%2Fv9h6b2.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Content: Elementary Number Theory - Problems 5.4**
**1. Fermat’s Method for Factoring**
Use Fermat’s method to factor each of the following numbers:
(a) 2279
(b) 10541
(c) 340663 *[Hint: The smallest square just exceeding 340663 is 584²]*
**2. Prove a Perfect Square Pattern**
Prove that a perfect square must end in one of the following pairs of digits: 00, 01, 04, 09, 16, 21, 24, 25, 29, 36, 41, 44, 49, 56, 61, 64, 69, 76, 81, 84, 89, 96.
*[Hint: Because \( x^2 \equiv (50 + x)^2 \pmod{100} \) and \( x^2 \equiv (50 - x)^2 \pmod{100} \), it suffices to examine the final digits of \( x^2 \) for the 26 values \( x = 0, 1, 2, \ldots, 25 \).]*
**3. Fermat's Factorization of Powers of 2**
Factor the number \( 2^{11} - 1 \) using Fermat’s factorization method.
**4. Mersenne’s Observation (1647)**
In 1647, Mersenne noted that when a number can be written as a sum of two relative prime squares in two distinct ways, it is composite and can be factored as follows:
If \( n = a^2 + b^2 = c^2 + d^2 \), then
\[
n = \frac{(ac + bd)(ac - bd)}{(a + d)(a - d)}
\]
Use this result to factor the numbers:
\[
493 = 18^2 + 13^2 = 22^2 + 3^2
\]
\[
38025 = 168^2 + 99^2 = 156^2 + 117^2
\]
**5. Generalized Fermat Method**
Employ the generalized Fermat
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