# Tag Info

Accepted

### 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 ...
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 $$a+b+c\equiv a^2+b^2+c^2\equiv 2c^2\equiv 0\pmod 2$$
Accepted

### 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 ...
Accepted

### 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, ...
Accepted

### 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 $2^k+1 \equiv 0 \pmod{3}$.
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. ...