7
votes
2answers
258 views

Is $7^{8}+8^{9}+9^{7}+1$ a prime? (no computer usage allowed)

Prove or disprove that $$7^{8}+8^{9}+9^{7}+1$$ is a prime number, without using a computer. I tried to transform $n^{n+1}+(n+1)^{n+2}+(n+2)^{n}+1$, unsuccessfully, no useful conclusion.
3
votes
0answers
46 views

Solution of a equation in natural number nvolving reciprocal of prime

Let $p$ be a prime and $n$ a natural number . Solve in $\mathbb{N}$ the equation $$\sum_{k=1}^{n}\frac{1}{x^k_k}=\frac{1}{p}$$
8
votes
2answers
284 views

diophantine equation $ |x^2-py^2|=\frac{p-1}{2} $

Prime $p\equiv3\pmod4$, then diophantine equation $$ |x^2-py^2|=\frac{p-1}{2} $$ has a solution in integers en, $x^2-py^2=-1$ has no solution in integers. I'd be grateful for any help you are ...
0
votes
1answer
96 views

Prove that $2^x \cdot 3^y - 5^z \cdot 7^w = 1$ has no solutions

Prove that $2^x \cdot 3^y - 5^z \cdot 7^w = 1$ has no solutions in $\mathbb{Z}^+$, if $y\ge 3$.
3
votes
1answer
77 views

What's the set p,if $37x^2-113y^2=p$ is solvable,with p a prime

if $37x^2-113y^2=p$ is solvable.with p a odd prime. What's the set of all $p$? Does it have a formula?
2
votes
1answer
59 views

$px^2-2y^2=1$,for what p,the pell equation has the result

$p$ is a odd prime ,If $px^2-2y^2=1$ is solvable,we can get Jacobi symbol $(\frac{-2}{p})=1$ ,so $p=8k+1,8k+3$ but when $k=12,p=97$, the pell equation $97x^2-2y^2=1$ is unsolvable.I think it's ...
1
vote
1answer
50 views

Co-primality of coefficients of coprime integers

Given that $a,b$ are co-prime, we have infinitely many solutions for $x,y$ to the equation $$ax+by=c.$$ Furthermore, solutions have the form: $x=ca^{-1}+tb,y=cb^{-1}-ta$. Given that $c$ ...
1
vote
0answers
58 views

How to solve diophantine equation $\frac{x^p-y^p}{x-y}=n$

$$\frac{x^p-y^p}{x-y}=n$$ whit $p$ a prime greater than or equal to $3$,for what value to $n$, it's solvable and how to solve,and whether $\frac{x^p-y^p}{x-y}=q_1$ $\frac{x^p-y^p}{x-y}=q_2$ is ...
1
vote
3answers
96 views

Find all $n\in\mathbb N$, $n>3$ such that $p(n)=2n+16$, where $p(n)$ is the product of all the prime numbers less than $n$.

Find all $n\in\mathbb N$, $n>3$ such that $p(n)=2n+16$, where $p(n)$ is the product of all the prime numbers less than $n$. E.g. $p(7)=2\cdot3\cdot5$ (and $n=7$ is a solution). Let ...
3
votes
1answer
90 views

$\frac{x^5-y^5}{x-y}=p$,give what p ,the diophantine equation is solvable

for$$\frac{x^3-y^3}{x-y}=x^2+xy+y^2=p$$$p=6k+1$give p prime, On what conditions,the diophantine equation $$\frac{x^5-y^5}{x-y}=p$$ is solvable in integers.does it have a linear expression.for ...
3
votes
0answers
161 views

Second longest prime diagonal in the Ulam spiral?

Given the Ulam spiral with center $C = 41$ and the numbers in a clockwise direction, we have, $$\begin{array}{cccccc} \color{red}{61}&62&63&64&\to\\ ...
6
votes
1answer
212 views

Are differences between powers of 2 equal to differences between powers of 3 infinitely often?

Consider the equation $2^a-2^b=3^c-3^d$ where $a>b>0$, $c>d>0$, and $a,b,c,d$ are all integers. A computer search for solutions with $a,c\le20$ only finds 8-2=9-3, 32-8=27-3, and ...
1
vote
1answer
134 views

Pythagorean triples with additional parameters

I want to find solution in $\mathbb{Z}$ to the following quadratic Diophantene equation: $$na^2 + kb^2 = c^2$$ where $n,k,a,b,c \in \mathbb{Z}$, $n,k > 0$ and $(n,k) = 1$ I know that for some ...
0
votes
1answer
51 views

A diophantine equation related to primes.

I have $2$ prime numbers $p_1$ and $p_2$. I have to find the solution of $\large{p_1t_1+p_2t_2=1}$ where $t_1$ and $t_2$ are integers. How do I do this?
6
votes
1answer
294 views

Diophantine equation involving prime numbers : $p^3 - q^5 = (p+q)^2$

Find all pairs of prime nummbers $p,q$ such that $p^3 - q^5 = (p+q)^2$. It's obvious that $p>q$ and $q=2$ doesn't work, then both $p,q$ are odd. Assuming $p = q + 2k$ we conclude, by the equation, ...
1
vote
1answer
153 views

Linear equation with prime coefficient.

Suppose we have a linear equation with two variables say $x$ and $y$ and three integer coefficient $a , b$ and $c$ (constant), where $a$ and $b$ are prime all are greater than zero. $ax+by=c$ how ...
-1
votes
3answers
278 views

What are the connections between number theory and topology ?? [closed]

What are the connections between number theory and topology ? How does topology relate to number theory ? In particular I wonder about primes and diophantine equations. I do not see how the amount of ...
-7
votes
2answers
697 views

Counterexample of Beal Conjecture GCD=1 Prime Sum [closed]

Just looking for opinions...before contacting any news agencies. Beal's conjecture states that if A^x + B^y = C^z, where a,b,c>0 and x,y,z>2, then gcd(a,b,c)≠1. It only takes 1 counterexample to ...
10
votes
0answers
238 views

Primes as quotients

I ask this question based on a comment of David Speyer in another question. What primes are of the form $$ \frac{p^2-1}{q^2-1} $$ where $p$ and $q$ are prime? The first prime not apparently of this ...
6
votes
1answer
144 views

When does $x^2+2y^2 =p$ have a solution in integers?

Show that $x^2+2y^2=p$ has a solution in $\mathbb{Z}\;$ if and only if $\;p \equiv 1 \; \text{or} \; 3 \mod 8$. Can someone help on this. Thnx.
3
votes
3answers
99 views

On the Pell-like $px^2-qy^2 = 1$ for prime $p,q$

Given any prime of form $p_n = u^2+nv^2$ for non-zero integers $u,v$. Consider, \begin{aligned} &p_2x^2-2y^2 = 1\\ &p_3x^2-3y^2 = 1\\ &p_7x^2-7y^2 = 1\\ &p_{11}x^2-11y^2 = 1\\ ...
5
votes
2answers
252 views

Integer solutions of $n^3 = p^2 - p - 1$

Find all integer solutions of the equation, $n^3 = p^2 - p - 1$, where p is prime.
2
votes
1answer
69 views

How many solutions to prime = $2 b^2 c^2 + 2 c^2 a^2 + 2 a^2 b^2 - a^4 - b^4 - c^4$

Let $a,b,c$ be integers, no sign restriction. Let $p$ be a given prime. How to find the number of solutions to $p = 2 b^2 c^2 + 2 c^2 a^2 + 2 a^2 b^2 - a^4 - b^4 - c^4$ ? Note, from Heron's ...
3
votes
3answers
154 views

How many solutions to prime = $a^3+b^3+c^3 - 3abc$

Let $a,b,c$ be integers. Let $p$ be a given prime. How to find the number of solutions to $p = a^3+b^3+c^3 - 3abc$ ? Another question is ; let $w$ be a positive integer. Let $f(w)$ be the number of ...
1
vote
2answers
92 views

How many solutions to prime = $(d^2-2ad+b^2-2ab+2a^2)(d^2-2cd+2c^2-2bc+b^2)$?

Let $a,b,c,d$ be integers $>-1$. Let $p$ be a given prime. How to find the number of solutions to $p = (d^2-2ad+b^2-2ab+2a^2)(d^2-2cd+2c^2-2bc+b^2)$ ? I assumed that this polynomial above does not ...
4
votes
1answer
195 views

Whether a prime number $ p $ can be written in the form $ 3A + 2B $, where $ A,B \in \mathbb{N} $.

I would like to know whether or not a prime number $ p $ can be written in the form $$ p = 3A + 2B, $$ where $ A $ and $ B $ are positive integers.
4
votes
2answers
89 views

Deciding if a univariate quartic has a solution mod p

I have an equation in $x$ and I would like to determine if it has any solutions modulo a large prime $p$. Suppose $p$ is large enough that I can factor numbers up to $p$, but I cannot test all values ...
2
votes
0answers
32 views

Solving $key=(\sum_{K=0}^n\frac{1}{a^K})\mod m$ with High limits

I was solving this equation:- $$key=(\sum_{K=0}^n\frac{1}{a^K})\mod m$$ Given $$ 1,000,000,000 < a, n, m \; < 5,000,000,000 $$ $$ a, m \; are \;coprime $$ I solved it bruteforcely but it ...
9
votes
3answers
737 views

Proving a statement regarding a Diophantine equation

FINAL EDIT : Prove that if $p^z|n^2-1$ $$p^{x-z}(p^{z}-1)=\dfrac{ n^2-1}{p^z}-3$$ doesn't hold for any chosen values of $p,x,n$ and $z$. Here $p>3$ is an odd prime , $x=2y+z, \ ...
4
votes
2answers
269 views

How many ordered triple $ (p,a,b) $ is possible such that $p^a=b^4+4$?

If we have a prime number $p$ and two natural numbers $a$ and $b$ such that $p^a=b^4+4$, then how many such ordered triplets $(p,a,b)$ exist? What should be the strategy to solve this one? The only I ...
8
votes
1answer
450 views

Can $x^{n}-1$ be prime if $x$ is not a power of $2$ and $n$ is odd?

Are there any solutions to $x^{n}-1=p$ with p prime, integers $x,n>1$ and $x$ not a power of $2$? $x$ must be even. $n$ is odd since if $n=2m$ then $p=x^{n}-1=(x^{m}+1)(x^{m}-1)$ hence $p=x^{m}+1$ ...
3
votes
2answers
227 views

All primes $p,q,r$ such that $(p-q)^2+1=r$

How can one find all prime numbers $p,q,$ and $r$ such that $$(p-q)^2+1=r\ ?$$
7
votes
3answers
584 views

When do the multiples of two primes span all large enough natural numbers?

It is well-known that given two primes $p$ and $q$, $pZ + qZ = Z$ where $Z$ stands for all integers. It seems to me that the set of natural number multiples, i.e. $pN + qN$ also span all natural ...