Articles of natural numbers

Collatz conjecture: Largest number in sequence with starting number n

This question is inspired by a CS course, and it only tangentially relates to the actual content of the exercise. Say in a hailstone sequence (Collatz conjecture) you start with a number n. For any positive integer n, if n is even, divide it by 2; otherwise multiply it by 3 and add 1. If […]

Possible not countable extension of the natural numbers?

This question comes from:Is $1234567891011121314151617181920212223……$ an integer? We define $\mathcal{A}$ as the set of infinite strings of digits $$ \bar a_i=a_0 a_1a_2a_3\cdots a_i \cdots \qquad a_i \in \{0,1,2,3,4,5,6,7,8,9\} $$ We can easily show, with Cantor’s diagonal proof, that $\mathcal{A}$ is a not countable set. We define : $$ \bar a_i =\bar b_i \iff a_i =b_i […]

The sum of three consecutive cubes numbers produces 9 multiple

I want to prove that $n^3 + (n+1)^3 + (n+2)^3$ is always a $9$ multiple I used induction by the way. I reach this equation: $(n+1)^3 + (n+2)^3 + (n+3)^3$ But is a lot of time to calculate each three terms, so could you help me to achieve the induction formula Thanks in advance

Proof by induction that if $n \in \mathbb N$ then it can be written as sum of different Fibonacci numbers

Proof that every natural number $n\in \mathbb N$, can be written as the sum of different Fibonacci numbers between $F_2,F_3,\ldots,F_k,\ldots$. For example: $32 = 21 + 8+3 = F_8+F_6+F_4$ Research effort: The base step it’s simple: Let $k=1$ it can be britten as $k=1=F_2$ For the inductive step I considered: Let $k = k F_2 […]

Is the value of $\sin(\frac{\pi}{n})$ expressible by radicals?

We have the followings: $\sin(\frac{\pi}{1})=\frac{\sqrt{0}}{\sqrt{1}}$ $\sin(\frac{\pi}{2})=\frac{\sqrt{1}}{\sqrt{1}}$ $\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{\sqrt{4}}$ $\sin(\frac{\pi}{4})=\frac{\sqrt{2}}{\sqrt{4}}$ $\sin(\frac{\pi}{5})=\frac{\sqrt{5-\sqrt{5}}}{\sqrt{8}}$ Question: Is the value of $\sin({\frac{\pi}{n}})$ expressible by fractions, radicals and natural numbers for each given $n$? If not, for which $n$ can we prove this non-expressibility?

How many combinations of $3$ natural numbers are there that add up to $30$?

How many combinations of $3$ natural numbers are there that add up to $30$? The answer is $75$ but I need the approach. Although I know that we can use $_{(n-1)}C_{(r-1)}$ i.e. $_{29}C_2 = 406$ but this is when $a, b, c$ are distinguishable which is not the case here. Please explain. EDIT three such […]

Consistency of Peano axioms (Hilbert's second problem)?

(Putting aside for the moment that Wikipedia might not be the best source of knowledge.) I just came across this Wikipedia paragraph on the Peano-Axioms: The vast majority of contemporary mathematicians believe that Peano’s axioms are consistent, relying either on intuition or the acceptance of a consistency proof such as Gentzen’s proof. And then went […]

List of powers of Natural Numbers

Greatings,   Some time ago a friend of mine showed me this astonishing algorithm and recently i tried to find some information about it on the internet but couldn’t find anything… Please help. Pseudocode: Consider that 1 is the starting index of a list 1.  input natural number n. 2.  let s = list of […]

Is there a domain “larger” than (i.e., a supserset of) the complex number domain?

I’ve been teaching my 10yo son some (for me, anyway) pretty advanced mathematics recently and he stumped me with a question. The background is this. In the domain of natural numbers, addition and multiplication always generate natural numbers, staying in the same domain. However subtraction of a large number from a smaller one needs to […]

Proving that any common multiplication of two numbers is a multiplication of their least common multiplication

Im trying to prove that if there are to numbers $n,m$ (natural numbers), and their smallest common multipe is $k$, so that $k = n·i$ and $k = m·j$ for some $i,j$ natural numbers, any common multiple $q$ of $n,m$ is a multiple of $k$, so that there exists a $g$ so that $q=k·g$. Logically […]