Questions tagged [modular-arithmetic]
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11 questions
10
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1
answer
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An identity related to sum of two squares
Motivation: Let $k$ be a positive integer and $m=4k+1$. I want to find the necessary conditions for the following identity to hold：
$$\displaystyle\sum_{i=1}^{k}\lfloor\sqrt{im}\rfloor=\frac{m^2-1}{12}...
- 990
0
votes
0
answers
29
views
Validity of multiplying a congruence and its modulus by a constant factor [migrated]
I am a student currently in a dispute regarding a step in a proof. I have the following congruence involving a fraction:$$\frac{x(x+1)}{2} \equiv \frac{y(y+1)}{2} \pmod{2^n}$$In my proof, I performed ...
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-3
votes
2
answers
701
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Primality test using the Golden Ratio [closed]
Background and Motivation
The golden ratio,
$$
\phi = \frac{1 + \sqrt{5}}{2},
$$
is a well-known irrational constant that appears frequently in geometry, algebra, and in the Fibonacci and Lucas ...
- 559
5
votes
1
answer
578
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Positive integer solutions of $(a^2x + 1)^2 = (yx - 1)(zx - 1)$
I am exploring the Diophantine equation:
$$(a^2x + 1)^2 = (yx - 1)(zx - 1)$$
with the condition that $a,x,y,z$ all are positive integers and $a^2+2 ≡ x,y,z \pmod 4$
Does this equation have any ...
- 327
4
votes
1
answer
625
views
Checking that $2$ is a primitive root modulo $p$ without factoring $p-1$
Let $p$ be a prime number, and consider 2 as an element of the multiplicative group modulo $p$. We know that the standard method to check if $2$ is a primitive root modulo $p$ involves factoring $p-...
0
votes
0
answers
165
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Linear independence in $\mathbb{Z}_q^n$
Consider $\mathbb{Z}_q \equiv \mathbb{Z}/q\mathbb{Z}$, where $q \geqslant 2$. A set of vectors in $\mathbb{Z}_q^n$ is said to be linearly independent if no nontrivial linear combination of them ...
- 503
2
votes
1
answer
630
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Prove ${^{b}a} \equiv {^{b+1}} a \pmod {10^{\lfloor{\log_{10} (^{b}a) }\rfloor + 1}} \Rightarrow a=5$ as $a$ and $b$ are two integers greater than $1$
$\DeclareMathOperator\len{len}$Let $a, b \in \mathbb{N} -\{ 0, 1 \}$ and define ${^{b}a}$ to be $a^a$ if $b = 2$ and $a^{\left(^{b-1}a \right)}$ if $b \geq 3$ (e.g., ${^{3}5} = 5^{\left( 5^5 \right)} =...
- 2,141
13
votes
2
answers
1k
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Given an irreducible polynomial over $\mathbb{Z}$, how often is it irreducible modulo a prime?
Given a monic irreducible polynomial $f\in\mathbb{Z}[x]$, I'd like to know for how many primes p we have that $f \bmod p$ is irreducible.
In the link: How many primes stay inert in a finite (non-...
- 133
6
votes
1
answer
364
views
Congruence obstructions for three consecutive powerful numbers
A powerful number is an integer $m$ such that if $p$ is prime and $p \mid m$ then $p^2 \mid m$.
Powerful numbers can be represented in the form $m=u^2 v^3$.
Erdos conjectured that three consecutive ...
- 25.8k
2
votes
1
answer
306
views
An arithmetic problem involving a system of equations
Fix a positive integer $r$. Describe the solutions to the system of equations given by:
$$\begin{equation}\sum_{1\leq i\leq r}X_i^2\equiv0\pmod{X_k}(1\leq k\leq r)\end{equation}$$
Example: In the case ...
- 2,413
20
votes
3
answers
1k
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Conjecture $ \sum _{i=1}^{n-1}\lfloor \frac{i^2}{n}\rfloor \ge \frac{\left(n-1\right)\left(n-2\right)}3$ where $n\in\mathbb N^*$
This is a question I also asked on MSE . If it is frowned upon to ask the same question on both threads, you can vote to close this thread.
Let $n \in \mathbb {N^*}$ and $$S_n = \sum_{k=1}^{n-1} (k^2 \...
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