# 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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### A Conjecture of Schinzel and Sierpinski

Melvyn Nathanson, in his book Methods in Number Theory (Chapter 8: Prime Number's) states the following: A conjecture of Schinzel and Sierpinski asserts that every positive rational number $x$ can ...
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1answer
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### Quadratic sieve algorithm

I am stuck with the sieving stage of Quadratic Sieve algorithm. I've read lots of papers to this point but I can't find any guidlines how to choose sieving interval or how sieving is actually done ...
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### Is this olympiad-like question about remainders an open problem?

Suppose that we are given two positive integers $x$ and $y$ such that $$x \mod p \leqslant y \mod p$$ for each prime number $p$. (Here, $x \mod p,\; y \mod p$ stand for the least non-negative ...
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### Proof: 1007 can not be written as the sum of two primes.

The claim is: 1007 can be written as the sum of two primes. We want to prove or disprove it. Edit: My professor provided this definition in his previous assignment: An integer $n \geq 2$ is ...
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### Is an arbitrary number of the form xyzxyz divisible by 7, 11, 13?

So I was given this question Choose any 3-digit number xyz and write it after itself as follows: xyzxyz. Check whether it is divisible by 7,11, 13. Is an arbitrary number of the form xyzxyz ...
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### Why can't prime numbers satisfy the Pythagoras Theorem? That is, why can't a set of 3 prime numbers be a Pythagorean triplet?

Suppose $a$, $b$ and $c$ are three prime numbers. How to prove that $a^2 + b^2 \neq c^2$?
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### A condition for being a prime: $\;\forall m,n\in\mathbb Z^+\!:\,p=m+n\implies \gcd(m,n)=1$

If $\;p=m+n$ where $p\in\mathbb P$, then $m,n$ are coprime, of course. But what about the converse? Conjecture: $p$ is prime if $\;\forall m,n\in\mathbb Z^+\!:\,p=m+n\implies \gcd(m,n)=1$ ...
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### Prove there are no prime numbers in the sequence $a_n=10017,100117,1001117,10011117, \dots$

Define a sequence as $a_n=10017,100117,1001117,10011117$. (The $nth$ term has $n$ ones after the two zeroes.) I conjecture that there are no prime numbers in the sequence. I used wolfram to find the ...
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### Prove that $n^2+n+41$ is prime for $n<40$

Here's a problem that showed up on an exam I took, I'm interested in seeing if there are other ways to approach it. Let $n\in\{0,1,...,39\}$. Prove that $n^2+n+41$ is prime. I shall provide my ...
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### A prime of the form $38111111\ldots$

Let $z(n)$ denote the number given by $38$ followed by $n 1$'s. What is the least number $n$, such that $z(n)$ is prime ? With brute force, I checked up to $7000$ digits and did not find a prime. ...
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### Proving that there are infinitely many primes with remainder of 2 when divided by 3

I need to prove that there are infinitely many primes with remainder of 2 when divided by 3. I started out similarly to Euclid's classic proof of an infinite number of prime numbers: Suppose there is ...
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### Why do repunit primes have only a prime number of consecutive $1$s?

Repunit primes are primes of the form $\frac{10^n - 1}{9} = 1111\dots11 \space (n-1 \space ones)$. Each repunit prime is denoted by $R_i$, where $i$ is the number of consecutive $1$s it has. So far, ...
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### Use this sequence to prove that there are infinitely many prime numbers. [duplicate]

Problem: By considering this sequence of numbers $$2^1 + 1,\:\: 2^2 + 1,\:\: 2^4 + 1,\:\: 2^8 +1,\:\: 2^{16} +1,\:\: 2^{32}+1,\ldots$$ prove that there are infinitely many prime numbers....
1answer
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### Is there a prime number between every prime and its square?

For each prime number $p$, is there always an other prime number between $p$ and $p^2$ ? I tested it for prime numbers $< 500,000,000$, but I wanted to know if there is any mathematical proof of ...
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### Finitely many Supreme Primes?

A challenge on codegolf.stackexchange is to find the highest "supreme" prime: http://codegolf.stackexchange.com/questions/35441/find-the-largest-prime-whose-length-sum-and-product-is-prime A supreme ...
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### Primes of form $a^2 + 24b^2$

For a prime number $p \neq 2$, $3$, is it necessarily the case the prime number can be written in the form $a^2 + 24b^2$ if and only if $p \equiv 1 \text{ mod }24$? I think this has to be true based ...
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### Elementary proof of Zsigmondy's theorem

I've been writing a not-so-short introduction to elementary number theory, supplying proofs for all theorems. When coming across Zsigmondy's theorem, it seemed difficult to find a proof available on ...
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### Is there a better upper bound for the primorial $x\#$ than $4^x$

In the classic proof of Bertrand's postulate by Paul Erdős, he shows that $x\# < 4^x$ where $x\#$ is the primorial for $x$. Is there any tighter upper bound for a given primorial $x\#$? Ideally, ...
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### Need help understanding Erdős' proof about divergence of $\sum\frac1p$

I'm looking at proofs from Proofs from the Book (Martin Aigner, Günter M. Ziegler). The proof I'm having trouble is the sixth proof of the infinitude of the primes they give (on page 5; although I'll ...
1answer
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