Chapter 23 More conjectures I: Assorted

What follows are more conjectures, some of which will require one of the three methods mentioned in this chapter, other which are false, and others which can be proved using another method from this textbook.

They are not ordered by difficulty, so if you get stuck, you can move onto the next one without worrying that it will be even harder!

I have grouped some similar theorems together. However, this doesn’t mean you will prove (or disprove) them in a similar way, though they may rely on each other.

Rational and irrational numbers: A rational number is a number that can be expressed as the ratio of two integers.
For example $4$ is rational because $4$ can be written as $\frac{4}{1}$ or $\frac{8}{2}$ or $\frac{100}{25}$ etc.
$-3.5$ is rational because it can be written as $\frac{-7}{2}$ or $\frac{-35}{10}$ etc.
$\frac{1}{9} = 0.1111...$ and so $0.1111...$ is a rational number.
$0.142857142857142857...$ is rational because it is $1$ divided by $7$.
$\pi$ is not rational because it can’t be written as a fraction of integers.

Conjecture 23.1 : There is no rational number $a$ for which $a^3 + a + 1 = 0$.

Conjecture 23.2 : The ratio of two rational numbers is rational.

Conjecture 23.3 : The sum of two rational numbers is rational.

Conjecture 23.4 : The product of two rational numbers is rational.

Conjecture 23.5 : No two numbers in a Pythagorean triple are the same.

Conjecture 23.6 : If $a$ is an integer not divisible by $5$, then $a^2$ leaves a remainder of $1$ or $4$ when it is divided by $5$.

Conjecture 23.7 : For any number $a$, $a^2 \geq a$.

Conjecture 23.8 : If $a + b = 500$ and both $a$ and $b$ are integers greater than 0, the greatest common divisor (gcd) of $a$ and $b$ is not 7.

Conjecture 23.9 : If $a^3 - 1$ is even and $a$ is an integer, then $a$ is odd.

Conjecture 23.10 : If $a|bc$, then $a |$ both $b$ and $c$.

Conjecture 23.11 : If $a|bc$, then $a|$ either $b$ or $c$ or possibly both.

Conjecture 23.12 : For any numbers $a$ and $b$, if $a^2 | b^3$ then $a | b$.

Conjecture 23.13 : For any integers $a,b$ and $c$, if $a^2 | b$ and $b^3 | c$ then $a^6 | c$.

Conjecture 23.14 : The sum of two even numbers is always even.

Conjecture 23.15 : If both $a + b$ and $b + c$ are even, with $a, b, c \in \mathbb{Z}$, then $a + c$ is even too.

Conjecture 23.16 : There exists only one number which is both odd and even.

Conjecture 23.17 : If an integer is not divisible by 3, then its square is one more than a multiple of 3.

Conjecture 23.18 : If $a^2 > b^2$ then $|a| > |b|$.

Conjecture 23.19 : If $3a - b = c$ (with integers $a, b, c$) then at least one of $a, b, c$ is even.

Conjecture 23.20 : If $3a - b = c$ (with integers $a, b, c$) then at most one of $a, b, c$ is even.

Conjecture 23.21 : $|a||b| = |ab|$.

Conjecture 23.22 : If $6a+9b = 22$, then either $a$ or $b$ or both is a non-integer.

Conjecture 23.23 : If $6a+9b=22$, then at most one of $a$ and $b$ is a non-integer.

Conjecture 23.24 : It’s not possible to sum two integers and get a non-integer.

Conjecture 23.25 : $3a^2 + a + 14$ is even for any integer $a$.

Conjecture 23.26 : Every integer greater than $3$ is equal to the sum of the squares of two integers.

Conjecture 23.27 : Every integer greater than $3$ is equal to the sum of the squares of two numbers (these numbers don’t have to be integers).

Conjecture 23.28 : The sum of the first $a$ odd numbers is $a^2$.

Conjecture 23.29 : Ask a friend to give you five integers. You’ll always be able to find three of them which sum to a multiple of $3$.

Conjecture 23.29 : Same as the previous conjecture, but for $4$ instead of $3$.

Conjecture 23.30 : Ask a friend to give you three numbers. The sum of these numbers will always be a multiple of $3$.

Conjecture 23.31 : There are an infinite number of differently-shaped rectangles with an area of $18$.

Conjecture 23.32 : If $a, b, c, d$ are numbers with $a < b$ and $c < d$, then $ab < cd$.

Conjecture 23.33 : If $a,b,c$ is a Pythagorean triple, then at least one of $a,b,c$ is odd.

Conjecture 23.34 : If $a,b,c$ is a Pythagorean triple, with $a,b < c$, then $a$ and $b$ can’t both be odd.

Conjecture 23.35 : $a^7 - a$ is divisible by $7$ for all positive integers $a$.

Conjecture 23.36 : $3a + 5b$ (where $a$ and $b$ are integers) is even only when $a$ is odd and $b$ is odd.

Conjecture 23.37 : Every integer greater than $2$ is equal to the difference of two perfect squares. (c-e)

Conjecture 23.38 : For all positive numbers $a, \frac{1}{a} < a$.

Conjecture 23.39 : The only way for both $ab$ and $a + b$ to be even is if both $a$ and $b$ are even.

Conjecture 23.40 : There are as many even numbers as natural numbers (even though there even numbers are a subset of the natural numbers).

Conjecture 23.41 : When $a > 0 $, $\sqrt{a} < a$.

Conjecture 23.42 : For all numbers $a$, $-5 \leq |a + 2| - |a - 3| \leq 5$.

Conjecture 23.43 : If more than $a$ pigeons fly into $a$ pigeon holes, then at least one pigeon hole will contain at least two pigeons. (The Pigeonhole Principle.)

Conjecture 23.44 : $a^2 = 4b + 3$ has no integer solutions.

A 52-card deck contains 4 suites (hearts, diamonds, clubs and spades) which each contain 13 different values (A, 1, 2,…, 10, J, Q, K).

What’s the smallest number of cards you need to draw to guarantee that you have a pair? Prove your answer.
What’s the smallest number of cards you need to draw to guarantee that you have a three-of-a-kind? Prove your answer.
What’s the smallest number of cards you need to draw to guarantee that you have a flush (for example, five diamonds)? Prove your answer.
What’s the smallest number of cards you need to draw to guarantee that you have three cards that are either all the same suit or all different suits? Prove your answer.

The popular TV show Tame of Gnomes has had 115 episodes. You’ve watched 60 so far.
Conjecture 23.45 : At least two episodes that you’ve watched are exactly four episodes apart.

You’re at a party with 19 friends. However, not everyone knows each other. You decide to play a game: who knows the most people (excluding you!).
Conjecture 23.46 : At least two people know exactly the same number of people at the party. (You can assume that if person $A$ knows person $B$, then person $B$ knows person $A$.)

You live in The Gyzkyr Republic, which issues 5-sym and 8-sym coins, and 10-sym and 50-sym notes.

What’s the largest amount you cannot pay for?
What’s the largest amount you cannot pay for using only coins?

Conjecture 23.47 : Above a certain price, any price can be paid for using an even number of both coins. (Not necessarily the same number of each coin.)

Conjecture 23.48 : To pay for an item which is evenly-priced, you must use an even number of at least one of the coins.

Conjecture 23.49 : To pay for an item priced 72 syms, you must use at least 6 of one of the coins.