Questions tagged [elementary-number-theory]
For questions on introductory topics in number theory, such as divisibility, prime numbers, gcd and lcm, congruences, linear Diophantine equations, Fermat's and Wilson's theorems, the Chinese Remainder theorem, primitive roots, quadratic congruences, quadratic number fields, Pell's equations, and related topics.
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Prove that if $d^k = (3^n - 1)/2$, then $d^{k+1}$ cannot be of the form $(2^m + 1)/3$
Suppose that $d^k = \frac{3^n - 1}{2}$ for positive integers $d,n,k$, with $d,k \geq 2$. Prove that it is impossible for $d^{k+1}$ to be of the form $(2^m + 1)/3$.
I originally asked a similar ...
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Frequency of abnormally large prime gaps
Let $\Delta = P - P'$, where $P$ and $P'$ are arbitrary consecutive primes.
The Prime Number Theorem states that $\frac{\Delta}{\ln(P)} \approx 1$ on average for large enough $P$.
Are there bounds for ...
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Affine combinations over the integers
This is not my area of expertise, but I ran into this interesting problem.
Given a subset $S\subseteq\mathbb Z$ define
$$\mathsf{span}\,S = \left\{x + a(y-x)\ |\ a\in\mathbb Z,\ x,y \in S\right\},$$
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Is $\gcd(\varphi(2^m),\varphi(3^m),\varphi(4^m),\varphi(5^m),\dots )=2$? [closed]
While studying the following theorem:
If $m_1$ and $m_2$ are two positive, relatively prime integers, then
$$
\varphi(m_1) \varphi(m_2) = \varphi(m_1 m_2).
$$
I decided to explore the values of $\...
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Does a limit exist for the ratio of numbers based on a characteristic? [closed]
Let us call a natural number $n$ magical if there exists a divisor $k$ of $n$ ($k \neq 1$ and $k \neq n$), such that $k^{k} \equiv k \pmod n$.
Is it true that the ratio of magical to non-magical ...
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Even number as difference between two primes [duplicate]
Is it true that each even number greater than 2 can be expressed as difference of two prime numbers? If it is,what is the visual meaning of the rule(if it is rule).I don't have much experience in ...
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$8n+3$ is sum of three squares [closed]
I know the Legendre's three square theorem, that asserts that any positive integer that is not of the form $4^n(8m+7)$ $(n,m\geq 0)$ can be expressed as the sum of three squares.
As a consequence, any ...
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For which natural numbers $m$ is $31^m-1$ divisible by $2^m-1$?
(This was an exercise on a recent (Sep/Oct 2024) edition of Die Wurzel - a German magazine aimed at secondary school pupils/undergrad maths students. The current edition mentions that no complete ...
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Given any odd number $a,$ do there exist distinct (odd) prime numbers $p_{k_i}$ such that the product $p_{k_1} p_{k_2}\cdots p_{k_n}\equiv 2\pmod a?$ [duplicate]
Denote the $i$-th prime number as $p_i,$ so $p_1=2, p_2=3,\ldots.$
Given any odd number $a\geq 5,$ do there exist distinct (odd) prime
numbers $p_{k_1}, p_{k_2}, \ldots p_{k_n},$ such that the product
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Is $ \sum_{n=0}^\infty \sum_{k=0}^\infty f(k, n + pk)=\sum_{n=0}^\infty\sum_{k=0}^{\left\lfloor \frac{n}{p} \right\rfloor} f(k, n)$Valid for Any $p$?
I tried to find a transformation property for a double summation. I discovered the following identity:
$$
\sum_{n=0}^\infty \sum_{k=0}^\infty f(k, n + 2k) = \sum_{n=0}^\infty \sum_{k=0}^{\left\lfloor \...
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Does there exist a natural number $n$ for all prime number $p$ such that $2^n-1$ is divisible by p [closed]
I tried to proove that there exist such $n$ for all $p$ . I tried to proove this by first proving that $2^{p_a}-1$ gives unique prime factors where $p_a$ denotes a th prime number . But it didn't help ...
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Must there be a prime $p$ such that $a$ and $b$ have the same order mod $p$? [duplicate]
Let $a$ and $b$ be positive integers not equal to $1$. Must there be a prime $p$ not dividing $ab$ such that the orders of $a$ and $b$ are equal modulo $p$?
This seems heuristically true, but I have ...
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Fractal defined by Continued Fractions
TLDR I have a discrete plot and am trying to find a formula for the continuous fractal that is approximated by it.
Table of contents
Question - Define the fractal that is approximated by Fractal ...
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Does this p-adics topic exist?
So I know that according to the rules of the p-adics we don't know what the most significant digit is. (at least I think I know that)
But then I was trying to figure out if we wanted to pretend that ...
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Useful reformulation of Goormaghtigh equation? [closed]
I've been exploring a particular identity related to the Goormaghtigh equation, and the results have turned out to be quite intriguing. I’d greatly appreciate any insights or perspectives from the ...
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