272 votes
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Are $14$ and $21$ the only "interesting" numbers?

Note that exactly one of $n$ and $n+1$ is even. It follows that for $n$ to be interesting, either $n=3p$ and $n+1=2N(p)$ or $n=2p$ and $n+1=3P(p)$, where $P(p)$ and $N(p)$ are the previous and next ...
147 votes
Accepted

Prove that the sum of pythagorean triples is always even

Note that $x^2\equiv x\pmod 2$ and thus $a^2+b^2=c^2$ implies $$a+b+c\equiv a^2+b^2+c^2\equiv 2c^2\equiv 0\pmod 2$$
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  • 19.4k
139 votes
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What is the smallest positive multiple of 450 whose digits are all zeroes and ones?

Can we agree that it must be an even multiple of 450? Otherwise the last two digits will be 50. What is the smallest positive multiple of 900 such that all the digits are 0s or 1s? A rule of ...
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  • 56.9k
128 votes
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Do we have negative prime numbers?

I don't know why this question has a down vote, because it identifies a subtle point about arithmetic which becomes particularly significant when the notion of "prime" is extended to other contexts, ...
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  • 96.7k
128 votes
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How to prove that all odd powers of two add one are multiples of three

Since $2 \equiv -1 \pmod{3}$, therefore $2^{k} \equiv (-1)^k \pmod{3}$. When $k$ is odd this becomes $2^k \equiv -1 \pmod{3}$. Thus $2^k+1 \equiv 0 \pmod{3}$.
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  • 39.3k
125 votes

What is the smallest unknown natural number?

The smallest infinitely often occurring prime gap, or $$\liminf_{n\to\infty}\; (p_{n+1} - p_n)$$ is unknown as of now. Most likely, it is $2$, but the twin prime conjecture has not yet been settled. ...
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125 votes

Are there an infinite number of prime numbers where removing any number of digits leaves a prime?

It is clear that we cannot have digits $0,4,6,8,9$ in those prime numbers. There can be at most one $2$ because $22$ is composite,, at most one $3$ because $33$ is composite,at most one $5$ because $...
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  • 4,900
119 votes
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A multiplication algorithm found in a book by Paul Erdős: how does it work?

This method is often called "Russian peasant multiplication". It's often justified by thinking about writing the first number in binary. Here's another way to explain it. At each step, we're ...
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108 votes

Prove that there are infinitely many primes with $666$ in their decimal representation without Dirichlet's theorem.

Consider the set $S$ of all numbers without 666 in their base 10 expression. Here's a fun fact: the sum $\sum_{s\in S} \frac{1}{s}$ converges. It's actually pretty easy to prove, so I'll leave it as ...
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  • 10.3k
103 votes

Do Arithmetic Mean and Geometric Mean of Prime Numbers converge?

Your conjecture for GM was proved in 2011 in the short paper On a limit involving the product of prime numbers by József Sándor and Antoine Verroken. Abstract. Let $p_k$ denote the $k$th prime ...
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101 votes

Are there arbitrarily large gaps between consecutive primes?

Can we infer that there exist two numbers separated by a gap of $10000$, such that no number in between them is prime? We can infer this regardless of what you wrote. For every gap $n\in\mathbb{N}$ ...
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  • 42.3k
96 votes

Am I just not smart enough?

Let me give you a personal story. As a young kid, I was always very strong in math but was pretty hampered by one of the worst educational environments in the USA. I ended up entering a magnet school ...
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96 votes

How to prove that all odd powers of two add one are multiples of three

A direct alternative to the answer via congruences is to note that for $k$ odd one has the well-known polynomial identity $$ x^{k} + 1 = (x + 1) (x^{k-1} - x^{k-2} + \dots - x + 1), $$ and then ...
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95 votes
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Do most numbers have exactly $3$ prime factors?

Yes, the line for numbers with $3$ prime factors will be overtaken by another line. As shown & explained in Prime Factors: Plotting the Prime Factor Frequencies, even up to $10$ million, the most ...
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  • 39.4k
94 votes

I'm trying to find the longest consecutive set of composite numbers

You can have a sequence as long as you wish. Consider $n\in\Bbb{N}$ then the set $$S_n=\{n!+2,n!+3,\cdots,n!+n\}$$ is made of composite consecutive numbers and is of length $n-1$
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  • 16.3k
94 votes

Can sum of a rational number and its reciprocal be an integer?

It seems like you are asking for a rational number $n$ with the property that $$n+\frac{1}{n}$$ is an integer. Let $z$ be an integer. Then we have $$n+\frac{1}{n}=z$$ and $$n^2+1=zn$$ $$n^2-zn+1=0$$ ...
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88 votes

Why does adding a suitable multiple of $9$ always lead to the reverse of the number?

Any number is congruent modulo $9$ to the sum of its digits: $$a_na_{n-1}\ldots a_1a_0\equiv a_n+a_{n-1}+\ldots +a_1+a_0\pmod 9$$ If you reverse the digits, the sum is unchanged. Therefore $$a_0a_1\...
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83 votes
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Does the string of prime numbers contain all natural numbers?

It follows from Dirichlet's Theorem. If $d$ is the number we want to find, define $s=10d+1$. By definition, $\gcd(s,10)=1$ and $s$ contains the digits of $d$. Dirichlet's Theorem's implies there's ...
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82 votes

Are there an infinite number of prime numbers where removing any number of digits leaves a prime?

To build on the earlier answers, there are exactly twenty such numbers, and they are: 1, 2, 3, 5, 7, 11, 13, 17, 23, 31, 37, 53, 71, 73, 113, 131, 137, 173, 311, 317 As can be found by the following ...
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81 votes
Accepted

Are Mersenne prime exponents always odd?

Theorem. If $2^n-1$ is prime then $n$ is prime. Proof. Suppose that $2^n-1$ is prime, and write $n=st$ where $s,t$ are positive integers. Since $$x^s-1=(x-1)(x^{s-1}+x^{s-2}+\cdots+1)\ ,$$ we can ...
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  • 77.2k
78 votes

Prove that the equation $x^2-y^2 = 2002$ has no integer solution

One way to solve this is to look at $x^2-y^2=(x+y)(x-y)$. Then for integer $x,y$, since $2002$ is even, one of $(x+y),(x-y)$ must be even, but since $2002/2=1001$, the other must be odd. That would ...
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  • 38.5k
78 votes
Accepted

I'm trying to find the longest consecutive set of composite numbers

marwalix's answer is great, but it is possible to 'optimize' the given sequence even more using a very simple 'trick'. Simply replace $n!$ by $n\#$, the primorial: $$n\#=\prod_{i=1}^{\pi(n)}p_i$$ The ...
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  • 7,732
78 votes

Why write ten as $10$ and not as $00$

The point of decimal notation is not just to assign each number a "code" - it has an attached meaning. When we write a decimal expression $$a_na_{n-1}...a_2a_1a_0,$$ what we mean is $$a_n10^n+a_{n-1}...
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77 votes

How can I write the numbers 5 and 7 as some sequence of operations on three 9s?

$$5=\sqrt{9}! - \frac{9}{9}, \quad 7=\sqrt{9}! + \frac{9}{9}$$
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  • 6,456
77 votes

How to calculate $\,(a-b)\bmod n\,$ and $ {-}b \bmod n$

Other answers have addressed the immediate question, so I'd like to address a philosophical one. I think that the way you're thinking of "mod" is a bit misleading. You seem to be thinking of "mod" as ...
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73 votes

Is there a way to write an infinite set that contains only irrational numbers without integer multiples?

The set of square roots of prime numbers: $$\{\sqrt{2},\sqrt{3},\sqrt{5},\ldots,\sqrt{p},\ldots\}$$ is an example of such a set. Assume $\sqrt{a}=k\sqrt{b}$ for some integer $k$. Then $a=k^2b$ so we ...
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73 votes
Accepted

Is this a new method for finding powers?

It's not something new, but for your discovery I applaud. This procedure is called the method of successive differences, and you can show that for every power the successive difference appears. Let ...
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72 votes

What is the smallest unknown natural number?

The chromatic number $\chi$ of the plane satisfies $4 \le \chi \le 7$, i.e., $\chi \in \{4,5,6,7\}$. The problem is known as the Hadwiger-Nelson problem: What is the minimum number of colors ...
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71 votes
Accepted

Are all "numbers" just one unit value transformed by a function?

the entire set of natural numbers can be represented by a recursive function that only works on a single unary value Bravo -- this is exactly the key insight that is exploited in most standard ...
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  • 44.1k
69 votes

Divisibility by 7 rule, and Congruence Arithmetic Laws

Note $n = c_0\! + c_1 1000 + \cdots\! + c_k 1000^k\! = f(1000)$ is a polynomial in $1000$ with integer coef's $\,c_i\,$ thus $\,{\rm mod}\ 7\!:\ \color{#c00}{1000}\equiv 10^3\equiv 3^3\equiv \color{#...
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