284 votes

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 ...
151 votes

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 $$a+b+c\equiv a^2+b^2+c^2\equiv 2c^2\equiv 0\pmod 2$$
Ennar's user avatar
  • 22.3k
139 votes

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 ...
Doug M's user avatar
  • 57.6k
130 votes

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 $2^k+1 \equiv 0 \pmod{3}$.
Anurag A's user avatar
  • 40.1k
126 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 $...
Farewell's user avatar
  • 4,978
125 votes

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 ...
Misha Lavrov's user avatar
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 ...
Nate's user avatar
  • 11k
105 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 ...
Théophile's user avatar
  • 26.3k
102 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 $n\in\mathbb{N}$ ...
barak manos's user avatar
  • 42.9k
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 ...
Andreas Caranti's user avatar
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 $n-1$
marwalix's user avatar
  • 16.6k
94 votes

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 ...
John Omielan's user avatar
  • 45.8k
93 votes

$r=\pm1$ are the only rationals with $\,r+1/r\in \Bbb Z$ (sum with its reciprocal is 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 $$n^2+1=zn$$ $$n^2-zn+1=0$$ ...
Franklin Pezzuti Dyer's user avatar
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\...
Arnaud Mortier's user avatar
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 ...
WroteAProgram's user avatar
80 votes

Is every integer representable as the sum of four Fibonacci squares?

If $N$ is big, there are only about $\ln N$ Fibonacci squares below $N$. There are then at most $(\ln N)^4$ different sums below $N$. That count eventually grows more slowly than $N$, so gaps will ...
Empy2's user avatar
  • 49.9k
79 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 ...
Joffan's user avatar
  • 39.4k
79 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}...
Noah Schweber's user avatar
78 votes

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 ...
Mastrem's user avatar
  • 8,032
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 ...
Stella Biderman's user avatar
73 votes

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 ...
Caleb Stanford's user avatar
68 votes

Is it possible to describe the Collatz function in one formula?

$$f(n)=\frac74n+\frac12+(-1)^{n+1} \left(\frac54n+\frac12\right)$$
wythagoras's user avatar
  • 24.8k
68 votes

Does associativity imply commutativity?

Well done - you've essentially rediscovered free objects. In particular, what you've observed is that in the semigroup freely generated by one element (call it $1$), addition happens to be commutative....
goblin GONE's user avatar
  • 66.9k
67 votes

Is $\sqrt{n!}$ a natural number?

For any $n \gt 1$ there will be some prime in the range $(n/2,n]$ which will only occur once in the factorization of $n!$ by Bertrand's Postulate. This will ensure that $\sqrt{n!}$ is not an integer.
Ross Millikan's user avatar
67 votes

Show that there are infinitely many powers of two starting with the digit 7

I was intrigued. Not being smart enough to get a theoretic proof I used brute force. Definitely ugly. Only redeeming aspect is it is constructive. Here is how I proceeded. Remark: If two numbers are ...
P Vanchinathan's user avatar
67 votes

Ramanujan's radical and how we define an infinite nested radical

Introduction: The issue is what "..." really "represents." Typically we use it as a sort of shorthand, as if to say "look, I can't write infinitely many things down, just assume that the obvious ...
PrincessEev's user avatar
  • 41.6k
65 votes

Why a $20$ digit number starting with eleven $1$'s cannot be a perfect square?

That's because\begin{align*}3\,333\,333\,333^2&<11\,111\,111\,111\,000\,000\,000\\&<11\,111\,111\,111\,999\,999\,999\\&<3\,333\,333\,334^2.\end{align*}
José Carlos Santos's user avatar
64 votes

Prove that for every three non-zero integers, a,b and c, at least one of the three products ab,ac,bc is positive

The product of three negative numbers is negative. So if $ab$, $ac$, and $bc$ are all negative, then $(ab)(ac)(bc)\lt0$. But $(ab)(ac)(bc)=a^2b^2c^2$ is the product of three squares, which are all ...
Barry Cipra's user avatar
  • 79.3k
63 votes

Do Arithmetic Mean and Geometric Mean of Prime Numbers converge?

We can use the simple version of the prime counting function $$p_n \approx n \log n$$ and plug it into your expressions. For the arithmetic one, this becomes $$\lim_{n \to \infty} \frac {\sum_{i=1}^n ...
Ross Millikan's user avatar
62 votes

What is the sum of all positive even divisors of 1000?

First consider the prime factorization of $1000$. We have: $$1000=2^3\times 5^3$$ Now, how can we list all the factors of $1000$? We see that we can try listing them in a table: $$\begin{array}{c|c|...
user2825632's user avatar
  • 2,861

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