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Diophantine Equations Day3

The document provides an introduction to Diophantine equations, focusing on divisibility techniques and presenting various examples and practice problems. It includes theorems and examples that illustrate how to find integers satisfying specific divisibility conditions. Additionally, it offers a series of practice problems for further exploration of the topic.

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วีรวัฒน์ พรเรืองทรัพย์
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33 views3 pages

Diophantine Equations Day3

The document provides an introduction to Diophantine equations, focusing on divisibility techniques and presenting various examples and practice problems. It includes theorems and examples that illustrate how to find integers satisfying specific divisibility conditions. Additionally, it offers a series of practice problems for further exploration of the topic.

Uploaded by

วีรวัฒน์ พรเรืองทรัพย์
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Weerawat P.

— 24 April 2024 A Brief Introduction to Diophantine Equations

§5 Usage of Divisibility Techniques

Example 5.1
(i) Find all positive integers n such that n divides n + 3.
(ii) Find all positive integers n such that 4n + 2 divides 6n + 5.

Solution.

8
Weerawat P. — 24 April 2024 A Brief Introduction to Diophantine Equations

Theorem 5.2
Let P(x) be a polynomial with integer coefficients with x, k 2 Z such that

x k|P(x) for some integer x,

then x k|P(k).

Proof.

Example 5.3
Find the greatest integer n such that n + 10|n3 + 100.

Solution.

9
Weerawat P. — 24 April 2024 A Brief Introduction to Diophantine Equations

§6 Practice Problems
[1?] (POSN 2566) Find all possible values of a 2 Z such that all roots of the equation

ax2 (3a 4)x + 2a 4=0

are integers.

[2?] Find the sum of all natural number n such that

n3 n2 5n + 32
n 2
is a positive integer.

[2?] Find the sum of all natural number x such that there exists an integer N which

x3 x + 120
N= .
x2 1
[3?] Find the elements of the set
⇢
x3 3x + 2
S= x2Z 2Z .
2x + 1

n5 + 3 n5 + 3
[3?] (OBEC R1 66) Let n be a positive integer such that is an integer. If  1000,
n+3 n+3
n5 + 3
find the maximum value of .
n+3
[4?] (IWYMIC 2016) What is the largest integer n < 999 such that (n 1)2 divides n2016 1?
x y z
[2?] (IWYMIC 2014) If x, y and z are three consecutive positive integers such that + + +
y z x
y x z
+ + is an integer, what is the value of x + y + z?
x z y
1260
[2?] (CHINA 2003) Given that 2 is a positive integer, where a is a positive inte-
a +a 6
ger. Find the value of a.

[1?] (POSN 2561) Let n 2 N such that 1  n  200 which 5 divides n2 + 9. Find the num-
ber of positive integer n possible.

[2?] (CHINA 2001) Find the number of positive integer solutions to the equation
x 14
+ = 3.
3 y

[20?] (IMO 1994) Determine all ordered pairs (m, n) of positive integers such that

n3 + 1
mn 1
is an integer.

10

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