# Tagged Questions

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### Diophantine: $px^2+2=y^2$ where $p\in \mathbb{P}$

Solve the Diophantine Equation: $px^2+2=y^2$, where $p$ is a prime number and $x,y$ integers. I tried this for ages but didn't get anywhere, but I don't know any advanced machinery since I am only in ...
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### 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.
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### 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 $$\sum_{k=1}^{n}\frac{1}{x^k_k}=\frac{1}{p}$$
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### 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 ...
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### 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 $y\ge 3$.
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### 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?
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### $px^2-2y^2=1$,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 ...
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### 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 $c$ ...
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### 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 ...
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### 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 ...
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### $\frac{x^5-y^5}{x-y}=p$,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 ...
Given the Ulam spiral with center $C = 41$ and the numbers in a clockwise direction, we have, $$\begin{array}{cccccc} \color{red}{61}&62&63&64&\to\\ ... 1answer 218 views ### Are differences between powers of 2 equal to differences between powers of 3 infinitely often? Consider the equation 2^a-2^b=3^c-3^d where a>b>0, c>d>0, and a,b,c,d are all integers. A computer search for solutions with a,c\le20 only finds 8-2=9-3, 32-8=27-3, and ... 1answer 142 views ### Pythagorean triples with additional parameters I want to find solution in \mathbb{Z} to the following quadratic Diophantene equation:$$na^2 + kb^2 = c^2$$where n,k,a,b,c \in \mathbb{Z}, n,k > 0 and (n,k) = 1 I know that for some ... 1answer 53 views ### A diophantine equation related to primes. I have 2 prime numbers p_1 and p_2. I have to find the solution of \large{p_1t_1+p_2t_2=1} where t_1 and t_2 are integers. How do I do this? 1answer 304 views ### Diophantine equation involving prime numbers : p^3 - q^5 = (p+q)^2 Find all pairs of prime nummbers p,q such that p^3 - q^5 = (p+q)^2. It's obvious that p>q and q=2 doesn't work, then both p,q are odd. Assuming p = q + 2k we conclude, by the equation, ... 1answer 156 views ### Linear equation with prime coefficient. Suppose we have a linear equation with two variables say x and y and three integer coefficient a , b and c (constant), where a and b are prime all are greater than zero. ax+by=c how ... 3answers 294 views ### What are the connections between number theory and topology ?? [closed] What are the connections between number theory and topology ? How does topology relate to number theory ? In particular I wonder about primes and diophantine equations. I do not see how the amount of ... 2answers 729 views ### Counterexample of Beal Conjecture GCD=1 Prime Sum [closed] Just looking for opinions...before contacting any news agencies. Beal's conjecture states that if A^x + B^y = C^z, where a,b,c>0 and x,y,z>2, then gcd(a,b,c)≠1. It only takes 1 counterexample to ... 0answers 243 views ### Primes as quotients I ask this question based on a comment of David Speyer in another question. What primes are of the form$$ \frac{p^2-1}{q^2-1} where p and q are prime? The first prime not apparently of this ... 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. 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\\ ... 2answers 261 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. 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 ... 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 ... 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 ... 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, $$where  A  and  B  are positive integers. 2answers 90 views ### Deciding if a univariate quartic has a solution mod p I have an equation in x and I would like to determine if it has any solutions modulo a large prime p. Suppose p is large enough that I can factor numbers up to p, but I cannot test all values ... 0answers 32 views ### Solving key=(\sum_{K=0}^n\frac{1}{a^K})\mod m with High limits I was solving this equation:-$$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 $$I solved it bruteforcely but it ... 3answers 738 views ### Proving a statement regarding a Diophantine equation FINAL EDIT : Prove that if p^z|n^2-1$$p^{x-z}(p^{z}-1)=\dfrac{ n^2-1}{p^z}-3$$doesn't hold for any chosen values of p,x,n and z. Here p>3 is an odd prime , x=2y+z, \ ... 2answers 269 views ### How many ordered triple  (p,a,b)  is possible such that p^a=b^4+4? If we have a prime number p and two natural numbers a and b such that p^a=b^4+4, then how many such ordered triplets (p,a,b) exist? What should be the strategy to solve this one? The only I ... 1answer 454 views ### Can x^{n}-1 be prime if x is not a power of 2 and n is odd? Are there any solutions to x^{n}-1=p with p prime, integers x,n>1 and x not a power of 2? x must be even. n is odd since if n=2m then p=x^{n}-1=(x^{m}+1)(x^{m}-1) hence p=x^{m}+1 ... 2answers 228 views ### All primes p,q,r such that (p-q)^2+1=r How can one find all prime numbers p,q, and r such that$$(p-q)^2+1=r\ ?
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 ...