# Tagged Questions

Prime numbers are natural numbers greater than 1 not divisible by any smaller number other than 1. This tag is intended for questions about, related to, or involving prime numbers.

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### Quadratic equation solutions modulo prime p

the question is: find all primes p that satisfy the equation: x^2-2*x-5 = 0 (mod p) The discriminante is 24, and I know that the equation mod p has a solution ...
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### The limit of consecutive positive integers which are the product of n primes.

The maximum length of a string of consecutive primes is 2: that is, the primes 2, 3. This is easily proven, as no even number other than 2 is prime. In contrast, consider the set of numbers which ...
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### Why can no prime number appear as the length of a hypotenuse in more than one Pythagorean triangle?

Why is it that no prime number can appear as the length of a hypotenuse in more than one Pythagorean triangle? In other words, could any of you give me a algebraic proof for the following? Given ...
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### Discrete log for prime powers

I was fiddling around and found that a function of the form $$L_b (x)=\left(\frac{b^{\phi (p^k)}-1}{p^k}\right)^{-1}\left(\frac{x^{\phi (p^k)}-1}{p^k}\right) \mod p^k$$ seems to behave similarly to a ...
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### Convergence of the sum $\sum\limits_{p}^{}\sum\limits_{k=1}^{\infty}\frac{\log p}{p^{ks}}$

How can I prove the following sum converges, where $s>1$ and the sum is over all primes. $$\displaystyle\sum_{p}^{}\displaystyle\sum_{k=1}^{\infty}\frac{\log p}{p^{ks}}$$ I've tried grouping terms ...
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### find all primes $p$ and $q$ such that $p \cdot q | 2^p + 2^q$

I have to find all prime numbers $p,q$ such that $p\cdot q | 2^p + 2^q$. I don't know from what I have to start.
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### Modified Sum of Products

A given number k is to be expressed as a sum of products of integers keeping in mind that the integers used in above process do not exceed their cumulative sum as 100. For e.g., k = 19 can be ...
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### Prove by contradiction that : There are Infinite Primes. [closed]

Specify $P$, ~$P$,Q and ~$Q$; For this proof I am having difficulty understanding what p and q signify. I was reading Euclid proof http://www.math.utah.edu/~pa/math/q2.html . But I do not quite ...
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### Legendre's Conjecture Theme (Part I)

Main Question Recently I have been thinking about the Legendre's Conjecture. I noticed that a proof of the conjecture can be obtained if we can prove any one of the following, Conjecture 1. For ...
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### proof of the prime number formula(It's real)

Proof that this formula always generates primes.Also it generates all primes(grate prime numbers formula) I have tried this formula it also generates primes orderd except the prime 2. I am really ...
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### Prove that if $p | x^p + y^p$ where $p$ is a prime number greater than $2$, then $p^2 | x^p + y^p$

I was trying to solve the following problem recently: Prove that if $p | x^p + y^p$ where $p$ is a prime number greater than $2$, then $p^2 | x^p + y^p$. Here $x$ and $y$ are both integers. $a|b$ ...
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### Does this set contain an infinite number of prime numbers?

I know that it still is not known whether the sequence $n^2+1$ contain an infinite number of prime numbers. I guess that this is also not known for any sequence of the form $n^k+1$ where $k\geq2$ is ...
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### positive three-digit prime numbers

I can solve this problem by typing "prime number under 200" in google and then examining three-digit prime numbers. My question is whether there is way to solve the problem without remembering all the ...
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### How many ordered triples of primes $(a, b, c)$ exist

Is there any easy way to find the answer to the following problem without trying numbers one by one? How many ordered triples of primes $(a, b, c)$ exist such that $a+b+c=37$?
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### What is the smallest composite number

I got the correct answer for the following problem by trying all numbers. It's very time consuming. Can anyone tell me whether there is a simple and easy way to solve the problem? What is the ...
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### Calcule $\gcd(0!+1!+\ldots+n!, (n+1)!)$

I have to compute $d=\gcd(0!+1!+\ldots+n!, (n+1)!)$, so let $a=0!+1!+\ldots+n!$ and $b=(n+1)!$. Then: $a=0!+1!+\ldots+n!=3!+0!+1!+2!+4!+...+n!=6+4+4!+...+n! \equiv 2 \mod 4$ Thus, $a$ and $b$ are ...
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### A congruence mod p

Let $p$ be a prime number. Show that $$2^2\times 4^2\times \cdots \times (p-1)^2 \equiv (-1)^{\frac{p+1}{2}} \pmod {p} .$$ Any help will be appreciated!
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As Wikipedia explains in number theory, Mills' constant is defined as: "The smallest positive real number $A$ such that the floor function of the double exponential function $\lfloor A^{3^{n}}\... 1answer 56 views ### Divisibility of a summation by$p^2$I try to use the hint of this problem but I could not. Any detailed answer will be appreciated! Let$p$be a prime number which$p>3$, and $$a/b:=1+1/2+1/3+\cdots +1/(p-1).$$ How could we show ... 0answers 55 views ### Primality test for Thabit numbers of the first kind Definition 1 Let$P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$, where$m$and$x$are nonnegative integers . Definition 2 Let$T_n=3 \cdot 2^n-...
So I have this apparently smooth, parametrized function: The function has a single parameter $m$ and approaches infinity at every $x$ that divides $m$. It is then defined for real $x$ apart ...