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.
39,078 questions
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preimage of an interval overlap under multiplication
$\newcommand\Z{\mathbb Z}$Let $M=pq$ for some distinct prime numbers $p$ and $q$. Consider the homomorphism $\phi: \Z/M\Z \to \Z/M\Z$ given by $\phi(x):= qx \bmod M$. We have
$$|\phi(\Z/M\Z) \cap [-p,...
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Different approaches to counting the number of consecutive iterates of $S(n) = \frac{3n+1}2$ that stay in $n \equiv 3 \pmod{4}$
Let
$$O = \{k \in \mathbb{Z}^+ | k \equiv 3 \pmod{4}\}, \quad S(n) = \frac{3n+1}2, \text{for } n \in O$$
and define
$$C(k) = 1 + \text{number of consecutive iterates of } S \text{ starting at } k \...
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On the Peaks of Sum-Equals-Product Sequence at Prime Indices
The product of $n$ positive integers is equal to their sum, which can be expressed by the following equation：
$\prod_{i=1}^na_i=\sum_{i=1}^na_i$
For example, when $n=3$, we noticed that $(1,2,3)$ is ...
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Eratosthenes again .... patterns in the primes? [closed]
Here is a naive (flawed) proof that prime numbers follow a pattern:
Starting from the blank grid, it is evident that the uncovered numbers after removing multiples of 2 follows a very simple pattern.
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Is the modularity of the Steiner circuit blocking of Collatz-type sequences studied in the literature? [closed]
I couldn't find any previous literature, so I had to do some exploring on my own. I'd like to understand where I might be missing the point, so are there any references to texts on the subject that I ...
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Conjecture: $\quad C_{2n-1}=\frac{1}{2+\frac{3}{4+\frac{5}{6+\frac{7}{\dots(2n-2)+(2n-1)}}}}=\frac{p}{q} \implies \frac{p+q}{2} \mid (2n-1)!!$
For any $n>0$,
Consider the fraction:
$$
C_{2n-1}=\frac{1}{2+\frac{3}{4+\frac{5}{6+\frac{7}{\dots(2n-2)+(2n-1)}}}}
$$
Let $ N_{2n-1}$and $D_{2n-1}$ be the lowest numerator and denominator of the $...
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Prove $B_{2k+1}(1/4) = \frac{-(2k+1) E_{2k}}{4^{2k+1}}$
Question. Is there a simpler way to prove $$B_{2k+1}(1/4) = \frac{-(2k+1) E_{2k}}{4^{2k+1}}$$ where $B_n(x)$ is the $n$-th Bernoulli polynomial and $E_n$ is the $n$-th Euler number?
I have verified ...
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Why is there no Euclidean Algorithm for the least common multiple (lcm)?
The greatest common divisor (gcd) of two integers $a$ and $b$ can be computed with the Euclidean Algorithm.
With the gcd known, one can compute the least common multiple (lcm) via the formula $\mathrm{...
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Find square of format A+n*B [duplicate]
Is there any better way than a brute force scan to find a square (or possible the smallest square) of format $A+n*B$, where $A$ and $B$ are some fixed constant integers? (I know that that is ...
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Can I get some help with IMO 1977 #6? [duplicate]
Let $f : \mathbb{N} \to \mathbb{N}$ be a function such that $f(n+1) > f(f(n))$ for all $n \in \mathbb{N}$. Prove that $f(n) = n$ for all $n$.
Can I get some help with this question? My approach is ...
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Proving that the set of the quotients of Fibonacci numbers is not dense in the positive reals
As in the heading, I'm trying to write up a proof that the quotients of all Fibonacci numbers is not dense in $\mathbb{R_+}$. This is what I have come up with and would like to know if it's correct.
...
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Show that $\lim_{n \to \infty}v_{p_{\lfloor n/k \rfloor}}(p_n)! = k$
Since my last question on here about prime numbers didn't go so well I hope this goes better:
For $k \in \mathbb{N}$, show that $\lim_{n \to \infty}v_{p_{\lfloor n/k \rfloor}}((p_n)!) = k$.
I have ...
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How to prove this inequality $2^n\le 2t+v_2\left( \sum_{i=t}^{2^{n-1}} \binom{2^n}{2i}\binom{i}{t}\right)$ $(1\le t\le 2^{n-1}$)?
Well, when I do the polynomial problem that my teacher gave me. I've tried new way to solve the problem by using p-adic $v_2$ but cannot solve it, here is the question:
For two polynomials with ...
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beginner proof strategy to show $ax \pmod n$ runs through complete system residue modulo $n$ [duplicate]
As I work through the introductory text "Number Theory Step by Step (Singh)", I made an observation that when $x$ runs through the complete residue system $$\{0, 1, \ldots, n-1\}$$ so does ...
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Cyclicity of the multiplicative group of the integers modulo a prime
Edit. It seems that the Lemma 2 needs already the existence of a primitive root modulo $p$. If there's no other way to prove it, then my argument is pointless.
(NB: I'm aware that there's plenty of ...
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