All Questions
Tagged with prime-factorization modular-arithmetic
93
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128
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For $n \ge 4$ find a factorization $n^2 - 3n + 1 = ab$ where $a \lt n$ and $b \lt n$.
Update: We can use Willie Wong's argument to justify the definition of a 'truth cutoff' function,
$\quad \psi: \{3,4,5,6, \dots \} \to \{4,5,6,7, \dots \}$
For convenience we start with a ...
- 11.5k
1
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2
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221
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Find the least odd prime factor of $155^8+1$
Find the least odd prime factor of $155^8+1$. How do I do this without using Wolfram Alpha or something?
- 783
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Sum of members from multiplicative group of prime order $k$ modulo prime $P$? $c$ in: $\sum_{n=1}^{k} (g^n \bmod P) = c \cdot P$ ($g$ prime order $k$)
Let $P$ be a prime ($>2$) and $g$ a value between $2$ and $P-2$.
Let $M$ be the set of numbers which can be generated with $g$:
$$M = \{g^n\bmod P, \text{ with } 0 < n <P \}$$
If $g$ is a ...
- 77
2
votes
1
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106
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Integer Factor Congruence
Given an integer $N$, with unknown prime factors $f_1$, $f_2$ ... $f_n$, and given unique integers $k_1$, $k_2$ ... $k_n$, with $\sqrt{N} \geq k_i>2$ for all $i$ such that
$$f_1 \equiv 1\pmod {k_1}$...
- 365
9
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3
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371
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Can $7$ be the smallest prime factor of a repunit?
Repunits are numbers whose digits are all $1$. In general, finding the full prime factorization of a repunit is nontrivial.
Sequence A067063 in the OEIS gives the smallest prime factor of repunits. ...
- 221
3
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1
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126
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The proof of $(n+1)!(n+2)!$ divides $(2n+2)!$ for any positive integer $n$
Does $(n+1)!(n+2)!$ divide $(2n+2)!$ for any positive integer $n$?
I tried to prove this when I was trying to prove the fact that ${P_n}^4$ divides $P_{2n}$ where $n$ is a positive integer, where $P_{...
- 438
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196
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Calculate all prime numbers $x$, where $x^{18} - 1$ is divisible by $28728$
Question: Calculate all prime numbers $x$, where $x^{18} - 1$ is divisible by $28728$
Apparently, the answer is all prime numbers except $2, 3, 7,$ and $19.$ I did some prime factorisation and found ...
- 315
1
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80
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Given 2 functions $f(x)$ and $g(x)$ find a value where $g(x)$ divides $f(x)$ meaning $f(x) = 0 \mod g(x)$
Problem:
Given 2 functions $f(x) = 2^{p-1} + x*p$ and $g(x) = 2 * x * p + 1$ find the values where $f(x) = 0 \mod{g(x)}$, where $p$ is a prime number and $x$ is a non negative integer in the range $1,...
- 13
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How to find the prime factors when knowing some congruence?
In order to factorize the integer $N = 67591$, choose a factor base $\{2,3,5\}$ and four congruences: $24256^2 \equiv 2^9 \cdot 3^4(mod\ N)$; $59791^2 \equiv 2^2 \cdot 3^4\cdot 5^2(mod\ N)$; $23541^2 \...
- 33
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114
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Proving the divisibility of $4[(n-1)!+1]+n$ by $n(n+2)$ in the condition of $n,n+2 \in P$ where $P$ is the set of prime numbers [duplicate]
Let $n$ and ($n+2$) be two prime numbers. If any real value of $n$ satisfies that condition, then prove that $$\frac{4{[(n-1)!+1]}+n}{n(n+2)} = k$$ where $k$ is a positive integer.
SOURCE: BANGLADESH ...
- 1,188
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Show that any prime divisor of $x^4+x^3+x^2+x+1$, with $x\in\mathbb{N}$, is $5$ or $1$ mod $5$
We can write the "polynomial" as follows:
$$x^4+x^3+x^2+x+1=\frac{x^5-1}{x-1}.$$
For even $x=2y$, we have that $x^5-1=(2y)^5-1=32y^5-1\equiv1$ mod $5$.
For odd $x=2y+1$, we have that $(2y+1)^5-1\...
- 1,674
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Non-Linear Diophantine Equation in Two Variables [duplicate]
How many solutions are there in $\mathbb{N}\times \mathbb{N}$ to the equation $\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{1995}$ ? I could solve till I got to the point where $1995^2$ is equal to the ...
- 344
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Find X in the Equation
I'm not a mathematician and I have forgotten about some basics in mathematics.
I have this equation:
$$x^y \pmod z = w$$
Given $y, z,$ and $w,$ how will I find $x$? How will I get the equation for $...
- 223
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79
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Finding the smallest prime factor of $\sum_{a=1}^N a^{k}$
This question is linked to my previous question, but I wanted a clearer explanation.
Suppose we have a huge number of that type with a huge $k$.
$$\sum_{a=1}^N a^{k} =1^{k}+2^{k}+3^{k}+...+N^{k},$$
...
- 349
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Find the smallest positive prime divisor of ...
Problem:
That's a problem I have found on the web. I didn't understand the solution:
Why??
Given solution:
How all this sequence has been transformed into $$33-{\lfloor {33\over p}\...
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