All Questions
12
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Form of the divisors of a number (Prime Factorization). Is this algorithm-based proof correct? [duplicate]
I am trying to proof the following result:
For a number $n$ whose prime number decomposition is $p_1^{\alpha _1} ... p_m^{\alpha _m}$. Every divisor of $n$ has the form $p_1^{\beta _1} ... p_m^{\beta ...
- 1,231
1
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0
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63
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Proof for at least $n$ different prime factors in $2^{2^n}+2^{2^{n-1}}+1$ if $n$ is a natural number
I am stuck on this proof and I need assistance please.
Not sure if it's the right start but here is what I've done so far:
I let $x=2^{n-1}$ and $2x=2^x$ that will change the expression to $2^{2x}+2^x+...
- 11
3
votes
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45
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Transforming an existing proof to a slightly different question.
Here is the original question:
Natural numbers $n$, $k$ are given such that for any prime $p$ there exists an integer $a$ satisfying the condition $p\mid a^k-n$. Decide whether $n$ must necessarily ...
- 291
1
vote
2
answers
171
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Application of Unique Factorisation Theorem in Proof
CONTEXT: Proof question made up by uni math lecturer
Suppose you have $x+y=2z$ (where $x$ and $y$ are consecutive odd primes) for some integer $z>1$, and that you need to prove that $x+y$ has at ...
- 359
1
vote
2
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58
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How can I solve this proof? (Prime factorization)
Let $p_1, p_2, ... , p_n$ be $n$ distinct primes. Prove that for all $x ∈ \Bbb{Z}$ we have $ p_i | x $ for all $i ∈ \{1,2, ... , n\}$ if and only if $(p_1*p_2* ...*p_n)|x$.
Hello. I'm very lost in ...
3
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0
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114
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If 13 does not divide m, then prove that $m^4+8$ is not a cube of an integer [closed]
My question is how can we prove that $m^4 + 8$ not a cube of an integer if
$m$ can not be divided by 13.
What I have done so far:
By Fermat’s Little Theorem:
\begin{align}
m^{p-1} &\equiv 1 \...
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0
votes
1
answer
351
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Proof using prime factorization and fundamental theorem of arithmetic
Let $x$ and $y$ be positive integers such that $7x^5 = 11y^{13}$. The minimum value of $x$ can be written in the form $a^cb^d$, where $a, b, c, d$ are positive integers. Compute $a + b + c + d$.
What ...
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1
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0
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53
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Proof by induction without a closed form
I need to do a proof by induction without the use of a closed form (I do know it's possible to find one, but we aren't supposed to do it that way).
Here's the prompt:
Let $p_1, p_2, p_3, \ldots, ...
- 363
2
votes
2
answers
8k
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Prove that if $p$ and $p^2+2$ are prime then $p^3+2$ is prime too [duplicate]
I'm trying to figure out how to prove that if $p$ and $p^2+2$ are prime numbers then $p^3+2$ is a prime number too.
Can someone help me please?
- 501
1
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3
answers
227
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Proving that $p_1p_2\mid n$ iff $p_1\mid n$ and $ p_2\mid n.$
Let $p_1$, $p_2$ be distinct primes. Using the Fundamental Theorem of
Arithmetic prove that a natural number $n$ is divisible by $p_1p_2$ if and
only if $n$ is divisible by $p_1$ and $n$ is divisible ...
- 1,509
0
votes
2
answers
389
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A New, Possible Proof of the Infinitude of the Primes?
$$1=1$$
$$2=2$$
$$3=3$$
$4=2\cdot2$ At $4$, the first prime number, $2$, is there as a factor. So I say that at the square of $2$, $2$ comes into play as a prime factor. At this point, $2$ is the ...
- 849
3
votes
1
answer
470
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Is my conjecture true? : Every primorial is a superior highly regular number, and every superior highly regular number is a primorial.
I have invented two sets of positive integers: highly regular numbers and superior highly regular numbers.
A positive integer $m \leq n$ is a regular of the positive integer $n$ if all prime numbers ...
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